Stable Envelopes, Bow Varieties, 3d Mirror Symmetry
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39
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TheoryGroup representationSymmetry (physics)Algebraic varietyGeometryPhysicsHill differential equationMusical ensembleComputer animationLecture/Conference
00:51
Social classVarianceInjektivitätGroup actionSymplectic vector spaceFiber bundleNatural numberWeightAlgebraic varietyVariety (linguistics)Object (grammar)Square numberTorusDimensional analysisProduct (business)DivisorTrigonometric functionsSpacetimeTerm (mathematics)Social classMereologyArrow of timeMany-sorted logicSymmetric matrixVertex (graph theory)1 (number)Point (geometry)RankingQueue (abstract data type)PolynomringPartial derivativeBoundary value problemFlagDirection (geometry)Number theoryPolynomialEquivalence relationCondition numberElement (mathematics)Constraint (mathematics)Plane (geometry)Image resolutionMathematical singularityCategory of beingSheaf (mathematics)Multiplication signSymmetry (physics)Different (Kate Ryan album)Greatest elementRight angleGraph (mathematics)Local ringConsistencyConnectivity (graph theory)Ring (mathematics)CohomologyCharakteristische KlasseSummierbarkeitGrassmannianVelocityAssociative propertyFocus (optics)Holomorphic functionCurveUnitäre GruppeEnthalpyComputer animation
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AxiomPhysical lawVariety (linguistics)Algebraic closurePoint (geometry)Dimensional analysisFiber bundleBoundary value problemSocial classEnvelope (mathematics)MereologyAxiomCohomologyUniqueness quantificationDifferentiable manifoldRight angleSpacetimeStability theoryConnectivity (graph theory)Order (biology)HomologieSigma-algebraSlide ruleMoment (mathematics)Graph (mathematics)Limit (category theory)Derivation (linguistics)Line (geometry)Constraint (mathematics)Algebraic varietyParameter (computer programming)FamilyCurveModulformConsistencyThermodynamic equilibriumElement (mathematics)TorusPositional notationNumber theoryCondition numberSubstitute goodLogicVertex (graph theory)Total S.A.SubgroupDirection (geometry)Product (business)Distribution (mathematics)Arithmetic meanEquivalence relationNormal (geometry)Statistical hypothesis testingDivision (mathematics)Degree (graph theory)Table (information)Greatest elementFluxExpected valueMultiplication signGlattheit <Mathematik>Äquivariante AbbildungComputer animation
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Physical lawSlide ruleMaß <Mathematik>19 (number)Lemma (mathematics)Limit (category theory)Functional (mathematics)Parameter (computer programming)Identical particlesSlide ruleLine (geometry)Holomorphic functionAlgebraic structureEllipseStability theoryVariable (mathematics)SineMereologyLine bundleElliptische FunktionElliptic curveLaurent polynomialAlgebraische K-TheorieImage resolutionSocial classEnvelope (mathematics)Variety (linguistics)Equivalence relationRight angleElement (mathematics)Sheaf (mathematics)Product (business)Point (geometry)PolynomialAxiomHomologieObject (grammar)19 (number)SpacetimeTheory of relativityCycle (graph theory)Planck constantDegree (graph theory)CoefficientPower (physics)Charakteristische KlasseNumber theoryDirection (geometry)CalculationLoop (music)Algebraic varietyMechanism designFundamental theorem of algebraDirac delta functionSummierbarkeitConsistencyConstraint (mathematics)Characteristic polynomialAnalytic setWeightBoundary value problemDivision (mathematics)TheoryStudent's t-testRepresentation theoryAngleMusical ensembleMultiplication signGroup representationGeometryConnected spaceCalculusCohomologySymplectic manifoldLaurent seriesThetafunktionDuality (mathematics)LogarithmDerivation (linguistics)Computer animation
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General linear modelTerm (mathematics)Line (geometry)Equivalence relationVariable (mathematics)Right angleDiagramDecision theoryMereologySlide ruleGroup actionAlgebraic varietyVariety (linguistics)Symmetry (physics)Trigonometric functionsFiber bundleArrow of timeParameter (computer programming)Object (grammar)Number theorySpacetimeTheoryMathematicsTheoremModel theorySquare numberDimensional analysisWell-formed formulaVector spaceGreatest elementGame theoryAlgebraische K-TheorieTorusIsomorphieklasse1 (number)Loop (music)Condition numberRankingPoint (geometry)Sheaf (mathematics)Direction (geometry)Cartesian coordinate systemAlgebraic structureOperator (mathematics)SummierbarkeitDifferent (Kate Ryan album)Proof theoryHamiltonian (quantum mechanics)Reduction of orderQuotientDuality (mathematics)Product (business)MetreEuler's formulaHolomorphic functionCovering spaceÄquivariante AbbildungLecture/ConferenceComputer animation
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SummierbarkeitPoint (geometry)BijectionAlgebraic varietyNumber theory1 (number)Binary treeStatisticsTheoremDifferent (Kate Ryan album)Slide ruleRight angleGame theoryCorrespondence (mathematics)Uniformer RaumAutomorphismEquivalence relationSpacetimeDiagramLine (geometry)Multiplication signMultiplicationAxiom of choiceMatrix (mathematics)Point reflectionNatural numberQuantum groupHomologieGeometryWeightAnalogyInvariant (mathematics)Mortality rateVector spaceTensorSummierbarkeitSkewnessGroup representationTotal S.A.Representation theoryContingency tableAlgebraische K-TheorieGroup actionTheoryVariety (linguistics)Social classCalculusOperator (mathematics)Algebraic structureAssociative propertyIntegerPartition (number theory)Symmetry (physics)Fiber (mathematics)Limit (category theory)Duality (mathematics)CombinatoricsLocal ringQuantumÄquivariante AbbildungCohomologyRing (mathematics)10 (number)IsomorphieklasseEllipseTable (information)Greatest elementAffine geometryFinitismusSimilarity (geometry)Computer animation
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SubsetBipartite graphGraph (mathematics)Degree (graph theory)PermutationPartial derivativeFlagAlgebraic varietyOrder (biology)Set theoryVariety (linguistics)CalculusPoint (geometry)Right angleLecture/Conference
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SummierbarkeitLatent class modelPoint (geometry)Variety (linguistics)Symmetry (physics)Envelope (mathematics)CombinatoricsCurveMoment (mathematics)Graph (mathematics)AxiomTorusSpacetimeStability theoryDuality (mathematics)Representation theoryRecurrence relationCharakteristische KlasseQuantum groupVertex (graph theory)Number theoryLinear algebraAlgebraic structureConvolutionWell-formed formulaDiagramSlide ruleContingency tableGeometryGraph (mathematics)Algebraic varietyTerm (mathematics)Higgs mechanismInterpolationGame theoryMultiplication signSheaf (mathematics)Theory of relativitySet theoryTable (information)Binary treeObject (grammar)Right angleDifferent (Kate Ryan album)CohomologyHyperbolischer RaumPresentation of a groupComputer animation
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Slide ruleStability theoryAxiom of choiceGroup actionTotal S.A.Charge carrierVariable (mathematics)Parameter (computer programming)Multiplication signVelocityVertex (graph theory)Theory of relativityGreatest elementMortality rateMathematicsIsomorphieklasseLine bundleVariety (linguistics)Symmetry (physics)Equivalence relationAlgebraic varietyFlow separationAnalytic setDescriptive statisticsAdditionMereologyEnvelope (mathematics)Fiber bundlePartial derivativeTrigonometric functionsMultiplicationCalculationFlagCohomologyCombinatoricsDiagramCalculusElement (mathematics)Clique-widthTorusScalar fieldGeometrySpacetimeSymplectic manifoldLagrange-MethodeAlgebraic structureDeterminantLine (geometry)Äquivariante AbbildungPolynomialLecture/Conference
Transcript: English(auto-generated)
00:14
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
00:20
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
00:42
delicacies I will hide, but I want you to give some, to have some picture of these, to have some understanding of these after the talk. Let me give a, acknowledge my co-authors on different papers that are related to this story. Rosansky, Varchenko, Smirnov, Zhu, Weber, and Xu.
01:06
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.
01:21
So that, the explanation of that double arrow will be the first half of the talk. So I tried to explain what it means, what, what characteristic classes for, for some certain singularities in a space are, what, what they are, and what does it mean that for two different spaces, in this case, in this example
01:40
I chose the Grassmannian 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, the second half of the talk will be explanation of the rest of the, the picture.
02:04
I want to talk about Chirki's 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 homogeneous spaces.
02:21
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, or let me recall this notion that was already defined in
02:42
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.
03:02
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 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
03:22
familiar spaces. They are cotangent, total spaces of cotangent bundles over partial flag varieties. For example, the Grassmannia you get by just one portion of a quiver, 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.
03:43
Here is an example. So, to this quiver variety is, is dimension two, and it is the usual resolution of a Kleinian singularity, c2 over Z mod three. More interesting than the, than the actual definition, let's see some properties of, of quiver varieties.
04:06
In this case, type A, they are smooth, they carry some holomorphic, carry a holomorphic symplectic form, and there is a torus action on it. So let me spend a little time on the torus action, so at least naming what the torus is. At each of the framed vertices, the framed ones are the little squares,
04:24
you imagine a torus 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 fact, one more c star, which comes from the fact that, that in the definition you considered
04:42
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, torus fixed point is finite,
05:01
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.
05:21
So eventually I want to talk about the cohomology ring, equivory, torus equivory and cohomology ring of the, of the Nakajima Curieties, look at the top left part, so that's what I want to explain, how I will talk about an element of the equivory and cohomology ring. So the way we will, we will name elements in the equivory and cohomology ring,
05:42
is we will name their images under the localization map. So consider, consider this map on the top left called loc loc, so that is just restricting a cohomology class to the torus fixed points. It is just the most innocent mapping cohomology, the restriction map.
06:01
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,
06:20
so the restriction to the torus fixed points is, is the sum of the restriction for the cohomology ring of each fixed point, and the equivory and 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's t star gr, grass one, two, c4.
06:42
It has six fixed points, these are the vertices of this graph. So to name an equivory and cohomology class 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's.
07:04
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 is what is explained, or indicated by the edges of this picture.
07:24
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. Says that an element, a coordinate at that fixed point, 2, 3,
07:42
and the 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 kind of constraints must be true for an image in the localization map.
08:02
Here's an example. So you see everything that is in blue is a six-tuple of polynomials, and because they satisfy these consistency conditions, they do represent an element in the accurate cohomology of the Grassmannian. So let's check one constraint, so maybe the bottom right constraint, the edge decorated by 2 and 4.
08:23
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 polynomials, so they are indeed equal. And you can check all the others.
08:40
This particular six-tuple is an element in the accurate cohomology of the Grassmannian of 2, 4. This particular six-tuple is actually something that will be called stable, and it's an example of a stable envelope later. This slide is just a warning, that I said that the components are not independent,
09:03
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 priority, the edges come in moduli. There's a one-parameter family, a pencil of curves, you see in the middle actually at two places as well.
09:21
And in those cases, the constraints on the components are deeper than just a coincidence under some linear form. They also have to happen not only for the polynomials, but for some higher derivatives as well. So I'm not giving the complete statement,
09:40
I'm just saying that the constraints are more involved than just the coincidences. Okay, so now that we have a way of thinking about the equivalent conjuring of the Grassmannian, or a Nakajima priority,
10:00
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 Torus fixed point, so the notation is tab 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
10:22
for defining an element in the accurate and conjuring. So the preparation is the following. I'm going to fix 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.
10:42
If that is fixed, then we can talk about this so-called Bieruniski-Bierula 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,
11:00
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 usually B-B 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.
11:23
So look at only the left, and if you... And so this is, of course, the moment graph. It is the skeleton of one of the Nakajima purieties. And one of the fixed points is called 1, 4. And I don't know how much it's visible, but there is something shaded blue, or it looks like green for me here.
11:43
All the points that flow into the fixed point 1, 4. So that's the leaf of 1, 4. So I'm going back. So if we have these notions of leaves, we also have a partial order, by just taking which fixed point is in the closure of a leaf of another fixed point.
12:05
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. If you take the leaf of your fixed point, but also then you look at the points which are in the closure of that leaf,
12:23
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. Well, like a joke, you're right, it is. So if you are familiar with working with homogeneous spaces,
12:43
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,
13:01
which have the same dimension, and then iteratively 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. So whatever is painted bluish here is the slope of 1,4.
13:24
Okay, so this is the geometric part of a definition that is needed to define the stable envelope. This is the maulich-okunkov axiom's definition of stable envelopes in cohomology.
13:42
So 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. It's mostly supported on the slope of that fixed point. The second axiom is a normalization that the stable envelope of P restricted to P itself,
14:02
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 in the ambient space, and you take the equivariant Euler class.
14:23
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, and it will be divisible by H. So the stable envelope will have a degree half of the dimension of the space,
14:41
so all these restrictions are of the same degree. So divisible by H means that if for some person who doesn't see H, just put flux in H equals one, then this means that the step restricted to Q is smaller degree than expected. So actually we like to call it a degree axiom,
15:00
that the stable envelope restricted to anywhere else but P, it collapses a little bit, so it's a smallness condition, always think of it as the restrictions are small. Actually maybe as a remark now that I mentioned the H equals one substitution, so in special cases for example for G over P, the stable envelopes were known before,
15:21
and they have a name called the Chern-Schwarz-McPherson classes, so in special cases it recovers some old notion with H equals one. Okay, so in this picture I'm going to explain, just further elaborate on the axioms of step 1-4.
15:40
So again, step 1-4 is a cohomology class, so it has components 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,
16:01
it says that the stable envelope of one-four is 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 slope, and the product of the Euler classes of those has to be the restriction there.
16:23
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 one-two, one-three and two-three must be zero, because this class is supported on the slope.
16:43
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 direction which is normal to the slope, which is to the northwest direction at one-four. Since this is normal to the slope there,
17:01
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 u2 plus h. So the support axiom is more than just a bunch of zeros outside, even in the boundary it requires some divisibilities.
17:21
So I hope now most of these should be clear. And of course divisibility by h is an explicit thing, everywhere but at one-four the restriction has to be divisible by one-four. 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 10-tuple of polynomials,
17:43
which satisfy the consistencies, which I haven't really told you the derivative constraints, but anyway, so the consistency, there is just one 10-tuple which satisfies the consistencies together with these axioms. But it's true, and that is the stable envelope.
18:01
Okay, let me check the time. Okay, I'm gonna skip this part, I think. Okay, so let's just regroup. I might come back to geometric representation theory connections, but probably not. So, so far what we have is that we have a way of thinking about accurate co-homological varieties,
18:21
and we defined co-homological stable envelopes of it, associated to every fixed point. The way of thinking of that, I like to think of it as the class of this loop, right, so if there is a big ambient space and you have a sub-variety, then it's tempting to think about that as a cycle,
18:42
and that represents the co-homology and the ambient space, as soon as you have some kind of poincare duality 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.
19:03
So if you take the highest degree, sorry, the coefficient of the highest h-power of this class, it has no h in it anymore, that is like a Schuber class type of object. So this is an h-bar deformation of like 19th century fundamental class type of calculations.
19:25
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
19:41
and in elliptic cohomology. And there is no way I'm going to define those, not even the theories, and even as the actual stable envelopes, but I want to give you some feelings about them. So first of all, how do I think about a k-theory element on a Lagrangian curiety or an elliptic cohomology element
20:01
in the Lagrangian curiety? So the first line says that a stable envelope or anything else, a cohomology 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.
20:22
And in the elliptic, accurate elliptic cohomology 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 variables 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 a Lagrangian curiety
20:41
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 why in elliptic cohomology, when you want to talk about characteristic classes,
21:01
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, to look at the theta function,
21:20
this is a section of a line bundle over the elliptic curve. The way I want you to look at this is that it starts with 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 would be just the k-theory part of it. So the theta function is really just a Q-decoration,
21:43
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 theta function, theta a, b is just, you cook up from the theta functions another two-parameter function, theta a and b. Okay, this is just definitions,
22:01
because then the next slide will be the one which I hope will give an intuition, so you're feeling 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 the stable envelope, so there are many others,
22:21
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 a good class,
22:44
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 elliptic world,
23:01
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 the sum of products of delta functions is zero. It's a good exercise for your graduate students. So theta was defined in the earlier slide,
23:21
and it turns out that about elliptic functions, this is the only identity. Everything else follows from it, although highly non-trivially. Anyway, this is the identity which is behind the fact that in elliptic world, you can define actually characteristic classes. Now, let's do the following, take that top identity and plug in q equals zero,
23:43
so let's go to k-theory. Then what you get is the middle identity, which is, you see, a trigonometric identity. Now you can give it 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.
24:01
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 decorations,
24:23
is that the identities in k-theory and in the rational limit, they are easier, because they tell you more. They say that if the x is added up to zero, then the left-hand side is equal to one. And if the y is added up to zero,
24:41
then the right-hand side is equal to one. So as you see, 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, we were not forced to work with the other set of variables, because these identities,
25:01
the governing identity behind characteristic classes, 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 split to an x part and the y part. You are forced to work with y.
25:21
Okay, this is actually something elliptic cohomology taught us, that we should do cohomology and trigonometry as well, with two sets of variables, with the k-theory parameters, but we need to go out of our ways to introduce them. Okay, so this was the explanation of the bottom right corner of this space,
25:40
that stable envelopes are defined, but they depend on u variables, which I call v's. Any questions? All right, then we come to this fact that, which I call 3D mirror symmetry for characteristic classes,
26:00
it turns out that there are pairs of Nakajima p-varieties, let's call them x, x shriek, that the pairing together comes with some fixed 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,
26:21
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 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.
26:44
So this is the 3D mirror duality for elliptic characteristic classes. Here's an example. So everything above the purple line is about one Nakajima p-variety, there you see the cotangent bundle of p2.
27:02
Sorry, yes? We have a question. Yes. 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 just restrict it to a point, so that you really have an elliptic function. But yeah.
27:21
Oh, what should I? Yeah, maybe I shouldn't. Okay, so let me continue. So over the purple line, it's all about the equivariity of cotangent bundle of p2. So you see that 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.
27:41
And the table on the top is the elliptic stable envelopes, in the following sense that you take the rows of that table. So the first row is the stable envelope of f1, and that means that that statement of restricted to f1 is that product of theta functions, restricted to f2 is zero, restricted to f3 is zero.
28:02
And the middle line is the table envelope of f2, and so on. Now, the 3D mirror dual of that equivariity is this one below the purple line, and that has a totally different looking moment graph.
28:21
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 it. 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.
28:40
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 H. Okay, so this is the baby example of 3D mirror symmetry for stable envelopes. Here are some other random examples.
29:03
With green on the two sides, I'm indicating the dimensions of these varieties. Yeah, so you see 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.
29:22
The bottom line is that the left hand side is of course a Nakajima variety, and I claim that there is no Nakajima variety 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.
29:42
Yes? Do you expect a stable envelope for cohomology? Okay, I guess yes. But having a group versus a formal group law has some advantages, an algebraic group. So these three varieties, the three cohomology theories that I named, cohomology, K-theory and elliptic cohomology,
30:01
these are the cohomology theories that corresponds to one-dimensional algebraic groups, which are parts of the formal group laws. So there are advantages of working with formulas. So indeed maybe there is a formula for the most general cohomology theory,
30:20
but I just want, I'm a formula person, so I certainly want to look at these three. Do we have a question? In what sense are all theta function identities derivable from the trisecant 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,
30:42
and I don't remember the details, I just remember the intuitive statement. So just at the very beginning we fixed this homomorphism from C star to T. So is that important? Yeah, so in the slides which I skipped it is important.
31:03
So if you start playing with that one-parameter subgroup you choose, then you recover representation theory. So after a while I might comment on those. So the stable envelopes of course depend on that? 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.
31:22
And changing them you recover Jungian R 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 to define, so give a feeling about what Bovariates are.
31:40
And actually I want to advertise them, I think we should look at Bovariates instead of queer 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
32:04
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 drew a blue or a red skew line, and so that you don't have to memorize this,
32:22
it's 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, of course, some supersing theory after.
32:44
Okay, so what is the, what is the Chirki's bow variety? Start doing the same thing as you would do for equivariities, 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,
33:01
then you just do the same thing as you would do for equivariities. Take home C and to C M, where, you know, N and M are the numbers, the decorations on the two sides. Take the cotangent bundle and that, of course, has an action of GLM cross GLM. So you do almost the same thing for the other type of five brains,
33:23
but it will be a different space, not just the cotangent bundle of home C and C M. That is, the left-hand side I call the arrow, the arrow edge, okay, arrow brain, and then this is the bow brain. Actually, this whole thing should be called a quiver,
33:40
which you can put both an arrow and a bow into a quiver. So for the other type of brains, I indicated roughly what that space is, but, of course, a lot of things are skipped in this group, so what acts how, but it's also just a Hamiltonian reduction or GIT quotient
34:02
of rather obvious spaces. The key difference is that on this other space, an extra group act 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
34:22
when you want to build up the Nakajima variety, is that you take the product of all these T homes and Bs and Ms, 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.
34:42
So in the space that you get, there will be a C star action for every D5 brain. You see, this is realized that every D5 brain be decorated by an equivalent variable. So if you do this story, then what will this Chirquoise variety C, script C of D, will be the name of the Chirquoise-Beau variety.
35:02
It will be smooth. It will have a holomorphic symplectic structure on it. It comes with tautological bundles coming from the D3 brains. So those numbers are the ranks of the bundles. Later I will show you that it has finitely many fixed points, and it has a total section. And the total section comes from the D3 brains,
35:22
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 mecagem equivorities are sort of true for this one. Everything is very combinatorial, and there is an advantage, which will come very soon, I hope. Can I ask a question?
35:42
Yes. So what is the framing that is usually the place where this is at? The D5 brains, correct. The D5 brains has come, okay. Somehow they collapse together to be the framing. So there is a dimension formula of course you don't have to memorize,
36:00
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-Boe variety. For example, if you see this brain diagram that is on the left bottom of the slide, then you plug in the numbers, and you will get four.
36:21
It's not a surprise, this will be T star P2, the Cherkis-Boe 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 one dot and one square, and now it's somewhat longer. But there will be things that will be at the end.
36:42
Okay, so there's a dimension formula. Oh yeah, and 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 bow? A brain diagram. So the quiver has parts, these K and N parts,
37:01
look at the top left part, and whenever you see this K and N part, 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.
37:22
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, we will just go to the brain diagram on the right, which is, you know, shaded with yellow as well.
37:41
But 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 works only for Type A quiver, Nagashimaquiver variety, or for any... Okay, I only know Type A. So what about loops?
38:00
Yeah, okay, I think the next slide. 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 an observation here that the brain diagrams that we get on the right,
38:22
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 DeFi brain, you will always have the same numbers. Okay, but here's the advantage that we do not have for quiver varieties.
38:40
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 reflected down, reflected by the horizontal axis. Or in other words, change red to blue, and blue to red, and so on.
39:00
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 brain name of T star P2, and we just formally create the 3D mirror dual of it. Okay, I can calculate its dimension, its dimension too.
39:20
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 both varieties, and it's called Hanani-Witten transition. Think of it as like a Rheidermeister move.
39:41
So 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, and it changes the way which is on the right.
40:02
And the theorem is that if you carry out such a change on your combinatorial model, then the associated bovaryotic doesn't change its same. Actually, the total parameterization, the total action re-parameterizes a little bit, but for the purpose of this talk, it's just an isomorphism.
40:21
Okay, then let's continue this example that we saw a few slides up. That the first two lines are just, we found 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. You see it?
40:40
Then I hope you can just carry out this transition. 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 co-balanced. Okay, so I was lucky enough to be able to carry out Hanani-Witten transitions
41:02
to make my brain diagram co-balanced. And if it's co-balanced, then I can recover it as a Nakajima variety. So then we'll cover this thing, this example, the baby example of 3D mirror symmetry between two Nakajima quiver varieties. Any questions? Oh, okay, so I wanted to give you an F-f binary type.
41:22
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 BOO 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.
41:41
The 3D mirror dual is just switching D5 and NS5 brains. And then in this one is such a simple thing that it's accidentally already co-balanced. So I can rewrite it as a quiver variety. This is, of course, a very well-known example of Hilbert schemes and its dual. But you can play the game with more complicated
42:03
type A or F-I 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.
42:21
So the, oh, so your question is that are there quiver varieties which are isomorphic to each other? Different combinatorial codes isomorphic to each other. Yeah, that I doubt. Yeah, I doubt. Oh, yeah, yeah, I'm sorry. I can, of course, I can present the one point space many different ways.
42:42
So maybe I have to be more back on that. I'm not sure. Okay, 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 Grassmann 2C5,
43:01
you just have to write its BOO picture. Then formally take its 3D mirror and then carry out any of its moves if you can. And if you're lucky, in this case you are, and then you will get a quiver variety. Right, what I want to talk about is some other
43:21
very important structure that BOO varieties come with. One of them is brain charge. So it will be a number, an integer associated to every five brain. So an NS five brain, it is just the difference of the two numbers on the two sides of the NS five brain,
43:41
L minus K, plus the number of D five brains on the left of it. So here you see that this is now just type A, not affine type A. In affine type A, there is something local charge, but anyway, so this is let it be just finite type A. And for D five brain, a very similar integer associated. So K minus L plus the number of different type of five brains
44:01
to the right of it. Here's an example. Is everything visible? Not much. Take the left most NS five brain. His brain charge is two minus zero, because these are the numbers on this two side, plus the number of D five brains to the left of it,
44:21
which is nothing. So that is the top red two 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 NS five brains left of this empty matrix,
44:40
the charges of D five 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-Witsen class of the brain diagrams.
45:00
So Hanani-Witsen 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. So if I didn't want to define you Boveri at least, but Boveri it is up to Hanani-Witsen, then I would have just told you
45:20
that they are associated to a pair of vectors. This pair of vectors have this one extra property, the sum of the red numbers is the same as the sum of the blue numbers. So again, so up to Hanani-Witsen transitions,
45:44
Boveri at least 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 equivalent variety. These are the ones which were in the top.
46:01
The top vector is a partition. Those numbers are weakly decreasing. And if you put numbers there fully one, then these are just the topics of Schuber calculus. So why do we care? Because actually, if you're coming from representation theory, then you think that you can...
46:20
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 Boveri are more general than quiver varieties.
46:43
And why do we like that both vectors are arbitrary? Because it comes from one extra operation, transpose, which is essentially 3D mirror duality. So this was not complete for quiver varieties, as you can see. Oh, okay, so since I keep mentioning
47:01
geometric representation theory, I might say the following. So if you see these two vectors, then the height of this matrix is a number, say n, and then you say, take the Yangian of GLn. This is the quantum group. Then those numbers on top, read them as...
47:24
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
47:41
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 Bode variety. So again, this is why these spaces are important in geometric representation theory. On the equivariant cohomology rings of these varieties,
48:02
quantum groups act. So one Bode variety is one weight space of such a quantum group. And this is true in K-theory and elliptic. This is the analog of this Maudic Okonkow Yangian? Yeah, yeah, yeah, this is exactly. So if these numbers on the top were just partitioned, this is the Maudic Okonkow Yangian.
48:28
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 quiver varieties, it was a tuple of partitions and it was somewhat messy.
48:41
It has a different picture here, of course equivalent, and of course more general because for Bode varieties. And I find it fascinating. So the claim is that fixed points are in bijection with tie diagrams. A tie diagram is on the picture. It consists of ties. A tie must connect a blue brain with a red brain.
49:01
And you know, my picture of drawing them in the skew lines, it tells you, yeah, so it's a natural way of connecting them. 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 its multiplicity. So just imagine that you only see the brain diagram
49:21
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 bovarete has 123 fixed points. Here are the fixed points of Grassmann 2C4. And you see there's a natural bijection between 4 choose 2,
49:43
just like in any other names. These tie diagrams beautifully transform under Hananavidan transition, which looks like a Rheidermeister 3 move. And they beautifully transform with 3D mirror symmetry, just take the, you know, the reflection of the image.
50:03
So that means that not just that we have a bijection between Hananavidan equivalent bovariates, which should be because they are isomorphic, but there is a bijection, natural bijection between 3D mirror dual bovariates, the bottom line. That was needed for the 3D mirror symmetry statement.
50:25
And we also mentioned this other combinatorial gadget that this matrix used to be empty, but now 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 green numbers.
50:42
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 theorem that fixed points are in bijection with binary contingency tables.
51:01
Maybe I want to draw a picture, that if you come from Schuber calculus, then you learn to work with, for example, the full-flag variety, and everything about the full-flag varieties is parameterized by permutations. This is a permutation.
51:22
In this language, what I'm saying is that quiver varieties, in the geometric quiver varieties, everything is parameterized by... Oh no, first what I want to say is that a partial-flag variety will be parameterized by an order set of subsets,
51:41
which can be identified with this 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 says, Taurus fixed points, everything is parameterized by these things. And with both varieties, what we are doing is that we are permitting
52:01
higher degrees on the right as well. So bipartite graphs, which are the same thing, essentially, as 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.
52:22
So on the right side, there are the BCT codes of Taurus fixed points on Grassmann 2C4. Okay, so I think at the beginning of this talk, I wanted to convince you that to find characteristic classes from a space, you have to walk through the following path.
52:42
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.
53:03
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 with all the decorations. And then the stable envelopes are defined.
53:23
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,
53:40
so there is some kind of cohomological whole algebra type of structure among stable envelopes. You want to, by convolution, multiply two of them to get a third one. And there are some formulas as well in certain special cases.
54:02
Okay, so then I'm just walking through something that I already showed you, is that if you start with those two diagrams, when 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
54:20
from just the brain diagrams, nothing else. Okay, so this statement that the stable envelopes match for 3D mirror dual-boule varieties, it's proved in certain special cases. For the Grassmannian and its dual,
54:40
it's proved by a paper with Sysmierno, Varchenko, Zhu. For the full-fledged variety in type A being self-dual, it's proved in a different paper by the same authors, and in general type with Andre Weber. The hyperbolic being 3D mirror dual in terms of stable envelopes with its dual is proved by Smeirno and Zhu,
55:02
and we calculated finitely many other cases. Maybe I should emphasize many, so it's quite a well-established conjecture that it should happen. Of course, everybody thinks it should happen. This should, I mean, you might remember the Higgs branch, Coulomb branch interpretation of this in Kannitzer's talk.
55:20
Sorry, they, they are, oh, that's, okay, so it may not, I don't know. Yeah, I don't know, I'm sorry.
55:40
So certainly I'm not sure that that is a special case of what I'm presenting. Certainly the middle, or the general G over L, and when it's logarithm dual, it's not a free variety or a full variety of type A, so that's not a special case of what I'm saying, that these are the cases for which the 3D mirror symmetry for characteristic classes is established or proved.
56:02
So here is a summary, and actually I'm on time. The summary of the, 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 enumerative geometry, they are relations to representation theory of quantum groups.
56:21
Okay, I'm, okay, they are related to some, some very important Q difference equations, and actually two sets of Q difference equations. So they are, they are, yeah, I advise everybody to study them, they are very important notions. We also learned that they come in pairs, such that the stable, I mean the spaces come in pairs,
56:42
and there is a, such that the stable envelopes on the two are related, and that the natural pool of detecting it is both varieties, which is closed for 3D mirror symmetry, and easy combinatorics govern their 3D mirror symmetry. The end.
57:02
Thank you. What's the relation, if any, between the bovariities and the construction of Coulomb branches we saw last week? So that's Nakajima Takayama paper,
57:23
that they proved that these bovariities are the Coulomb branches. Type A, probably more general, but I understand the type A. There was a question in the Q&A, how the stub changes under the, and any width and the move, is this clear?
57:41
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 torus gets re-parameterized, but that just means that in one of the variables, one of the equivariant parameters, instead of u1, you write u1 times h. So it's an isomorphism, so there's no change.
58:03
There is another one. The bovariity description of a given quiver variety includes some additional parameters that were not visible on the quiver variety side. Do you know how this should be dealt with when a completely stable envelope of bovariities?
58:22
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 bovariities? So maybe my first remark is that those v's are also there for the quiver picture, they are at these top vertices.
58:42
So there is here, of course there's not enough, so this is v1 over v2, and this is v2 over v3. So this bottom, the top vertices are decorated by the scalar parameters. Actually, the scalar parameters are, they come from the Picard group, so there's a line bundle, the determinant bundle, here and here and here, and these are the scalar parameters.
59:01
Okay, the question, how to deal with them? Yeah, I mean, in the definition of a little stable envelope, those 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, equivariant parameters
59:20
and scalar parameters just switched beautifully, so as they should. So that's just for me, just another incarnation of the fact that this is the right way, this is a good way of looking at these varieties. So here, the equivariant parameters and scalar parameters, it's not clear how they switch, but in the blue picture, it's rather nice.
59:43
I only did the GLN part, that it is extended to GLN M, with Lev Rozanski 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...
01:00:00
In a sophisticated sense, you apply a Lagrange transform there, and there's a different associated space. And in the cohomology of that, not the Yang-Yens of GLN, but the Yang-Yens of GLNM will act in cohomology. The geometric object, like, can you explain what that really is? No, it's rather sophisticated.
01:00:20
So first of all, we don't do it in the GIT way, because first of all, we have to give up holomorphic symplectic structure. It might be Poisson, but certainly we have to give up the omega. So we rephrase the definition to some Lagrangian intersection definition, and then, this is
01:00:42
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. But then, for where there is a star, first you apply some Lagrange transform, and then you intersect them. So that will be the new variety.
01:01:02
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 Levrozanski, so he certainly deserves his name.
01:01:21
Any further? You did pass on to use the combinatories of these diagrams to calculate the step value? In some sense they define it, but it's just very complicated. So actually there is a student, Tommaso Botta, who has a great way of doing it in cohomology
01:01:41
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 Schuber calculus settings, that if you take some cohomological whole-algebra that Markus will define tomorrow, I'm sorry to take it for granted, and then you take some natural elements one, there are many
01:02:04
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 Schuber calculus settings, when you take cotangent bundles of partial flag varieties. But Markus will convince you that this cohomological whole-algebra multiplication is non-trivial,
01:02:25
it's beautiful, but it's not just multiplication, it's non-computative 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
01:02:43
that's a local calculation. Right, so it's only for polynomials. So it is a shuffle-algebra. But for elliptic, we are trying to set it up here, there are lots of technical difficulties. Any further questions? I must say that Tomaso Botta does it for elliptic as well, but for quiver varieties.
01:03:03
Maybe then you can point me out, please. Oh, hi! OK, let's thank, let's thank Richard again.