 Today we're going to be studying the topology of y by which I mean the simplest thing you could imagine that I mean by that is just studying the cosmology of y and that's almost what we're going to be interested in but something slightly different a similar a vector space of the same dimension but a different vector space so I'm going to explain what what I mean by that in a second but the first thing that I want to do is bring back one piece of structure I didn't talk how much about last time which was this Hamiltonian torus action so I recall that I wanted an action of a torus preserving the symplectic form acting on both x and y compatibility and I demanded that the fixed points for this torus action is a finite maybe I didn't emphasize this last time quite enough but if we look downstairs on x this fixed points of the torus action on x is just a single point it's the same fixed points as for the torus action for the c star this conical c star so remember we already had a existing another c star conical c star which scaled everything down and this this Hamiltonian torus commutes with that one and so they'll have the same fixed points upstairs on y the conical torus has maybe not just finding many fixed points as a big fixed point set but this torus action will just have finally many fixed points and then I'm going to find the attracting sets for this torus action so y plus so this is the set of all points in y oh sorry one more piece of data this is what I meant to say inside this Hamiltonian torus I'm going to choose a generic c star choose and fix this c star okay and this choice of c star is actually should be thought of as part of the data we'll see later that under symplectic duality it corresponds to a choice of resolution so with this choice of c star I can consider attracting sets for this c star inside the torus so this c star is chosen generically so it's fixed point set is the same as for the full torus and then the attracting set is just defined to be the set of points in y such that the limit for this torus action for the site for this c star action is exists I mean if the limit exists it must definitely be in this fixed point set y t so that's y plus and and similarly we have x plus for the same definition so just as an example recall our favorite example of symplectic resolution was this cone no point of cone of s l2 being resolved by the cotangent bundle of p1 and in this case um maybe I just draw the okay well inside here maybe I'll draw the zero section of this p1 which is mapped down to the now the um the way that the torus action works here it's kind of in this direction it's attracting you down so this is this c star action I mean in this case this torus the torus that acts effectively on p1 or just hemotonic torus is just c star so in it it's sort of going down in this direction and up in this direction so that the attracting set x plus is just this copy of a1 here and the other hand in y plus the attracting set is the union of this p1 and this pre-image of this a1 I mean it's the pre-image of this a1 but it's the p1 and this part and then the two fixed points are here and here of course zero so the blue stuff is the plus locus the green stuff is the fixed points in general there's a morphism from of course from y plus to x okay um so what are we doing about this y plus well we're going to introduce one um so this uh this maybe it's slightly scary sounding thing but it's actually makes our life much simpler and is is not it's not complicated so it's called hyperbolic stock so what is it it's a functor of call it capital phi and it goes from the drive category of sheaves of constructible sheaves on x to category of vector spaces and it goes like this you take a sheaf and you don't just compute its co-demology but you first take the shriek pullback to x plus and then compute the co-demology along x plus of your i as the inclusion of x plus index okay so um I think this guy hasn't been damaged I don't know internalized in in the literature but I think probably the best treatment of it comes from Nakajima's PCMI lecture notes from five years ago and well you can be defined in a much more general context in our situation the fixed point set is just finite or rather a single point that's why I call it sort of hyperbolic stock if the fixed point set was not just a single point but some locus I might call it hyperbolic um localization but here it's just a single point and the main the main fact is that if um if f f is perverse then this five f is concentrated in a single degree well which degree I mean the degree should be um I guess I should the degree will be um to d my question is here 2d is the dimension complex dimension of y so y is of course even dimensional because it's subtracted and that's dimension 2d so d is the the dimension and maybe I should have said this a second ago but the this y plus and x plus they'll be half dimensional compared to x and y the reason is well it's easier to see maybe in y plus um in for if you consider y plus it's a Lagrangian sub variety because the torus action preserves the symplectic form so the positive directions for the torus action for the c star and the negative directions for the c star are paired under the symplectic form so they each have half dimension so y plus is half dimension principally sorry and also it's Lagrangian but it's half dimension so y plus and x plus are half dimensional x and y and then okay and then also we have this fact here that if you have a porous sheaf and you apply this hyperbolic stock functor then it's concentrated in this maybe I should say sorry maybe I should say not all constructible sheaves constructible with we have a preferred stratification on x namely the stratification we've all mentioned it right now because we we need to discuss in a second so on x I mentioned this last time but when we reiterate it we have finally many symplectic leaves they call them x alpha alpha and i and they give us a stratification and when I say constructible sheaves I mean the tractor with respect to this stratification so why this hyperbolic stock um well you'll see in one second so we're going to consider the decomposition theorem so we have this map from y to x so we can take the push forward of the constant sheaf on y shifted by 2d to make it reverse and it's constructible with respect to this stratification by symplectic leaves so that's the result of cleaning and it decomposes therefore as a direct sum over alpha of the ic sheave of these leaves or their closures tensor the topomology of the fibers here f alpha by inverse of a point x alpha x little out this is little x alpha point and then after this so sometimes you would have taken this decomposition theorem then push forward to a point to up to reach the usual I mean to reach a decomposition of the co-mology of y but we'll do something different which is we'll take this decomposition theorem and then apply the hyperbolic localization hyperbolic stock filter so that's our function so this is an equation in the category of constructible sheaves on x so I'm free to apply my functor which goes from constructible sheaves on x to vector spaces so after I do that this left hand side will turn into the homology of this positive set and in fact the homology of this positive set in a single degree namely the top degree like 2d actually um yeah okay let me write 2d here in a second and then abbreviate something so we get the topomology of y plus and here we get these ic sheaves so the good thing about doing this hyperbolic localization is it sort of gets rid of the ic sheaves and in rather than turning them into the intersection homology of x alpha it turns them into the topomology of this x alpha bar plus the tracking set inside x alpha so here x alpha bar plus is um just x plus in a second x alpha bar and then this guy just is a vector space so he just comes along for the right so we get this version of the deconvention and just because everything in sight just involves topomology it's only going to be concerned with the intersection the irreducible components of these varieties and I'm just going to abbreviate um h of something as h top so then we can just write this equation maybe more simply as h of y plus equals direct sum over alpha h x alpha bar plus tensor h left so let's look at an example so this maybe seems a little strange or abstract um let's take the cotangent bundle of g24 so this is the example it's rich enough that we see a bit of the structure so in maps it's a resolution of the of the um four by four square zero matrices okay so that's my x that's my y um what are my strata well actually what's my x plus first of all x plus is pretty straightforward it's just such a's such that a is upper triangular so because the the way the this this torus is just acting by conjugation so I chose this generic c star and then the tracking sets for the generic c star would just be the upper triangle and then the various strata are as follows there's x zero which is just zero there's x one well which is such matrices with square zero and rank one and x two which is such matrices with um square zero and rank two okay so picture our space x like so then we have a the locus like so where rank a is equal to one and then somewhere here we have this point zero and then what are the um so then we should uh according to this uh equation here we should examine right in this form we should examine fibers and we should examine this positive that that's attracting sets in each in each strata so let's start with the tracking sets and then we do fibers so the tracking sets are pretty straightforward x zero plus well just still zero x one plus so it's a upper triangular um a squared is zero rank a is one well maybe if I put the bar then it's right a is less than one and this variety actually has three irreducible components so there are four by four matrices like so there's these all these free entries in the upper in the upper part of the matrix but I want the rank to be um in most one in the matrix that's square zero and if you put that then there's three possibilities you could have like this like this more like so so there's a three irreducible components there and x two plus um which is the same as x plus it has two irreducible components maybe I'll leave it as an exercise to figure out what those two components are this is two and actually in all three cases the the fibers are irreducible here the fibers g2 4 here the fiber is p1 and here the fiber is just a point so if we were to look at our our equation so it says homology top homology of um y plus but I keep thinking top homology maybe I'm making a big mistake my top homology sorry I think I don't I maybe I'm sorry let me let me uh sorry I think I said something all wrong the right hand side is correct the left hand side just total homology and then I got myself confused no no sorry I think it's okay okay okay this decomposition gives the homology of y plus and I get um so so I get homology of x0 plus top homology of x0 plus tensor top homology of f0 plus top homology of x1 plus tensor top homology of f1 plus top homology of x2 plus tensor top homology of f2 so in this case this is like one one I'm writing the dimensions here three one and two one for interest it works out to six okay so um great so that this is this is the the the main object of study will be this homology of y plus and this and it's decomposition here so one reason well one reason to focus on this homology of y plus is because it's gonna um um it's gonna play a big role in the symphonic quality but another reason is that it admits a categorification so yeah uh that is our question in the queue and the person asks is it top homology or y plus or total homology sorry that's why I've got confused by it for a second maybe I should just stop a second and straighten this out in my head um first of all homology whenever I say homology I mean borough more homology I actually think that the top of borough more homology I actually think that sorry no no I'm back to believing what I thought a second ago so the the the I think this this thing about the this back here that when it's perverse then the I feel like stock is concentrated in a single degree actually proves that the top homology of y plus is the same as the total homology of y plus again borough more homology there's always if somebody thinks I'm wrong they can point it out but now I'm sort of more convinced that it's right but for example for this t star p1 um there's no zero homology this point is borough more homology doesn't contribute anything because it can go off to anything okay no no one says I'm wrong so I'll assume that I'm right okay so this it can do both top homology and total homology because they that would be answer the question okay so what do I mean by emits a categorification so I mentioned last time that we have these um algebras a so this is a quantization of x or or of y and she's algebras quantizing y let's just stick to a single algebra a a quantization of x um actually a depends on a parameter which lives in h2 so maybe I'll fix it a given one let me call it a sub theta doesn't h2 and then the action of the torus on x gives us an action of the torus on a theta and then the action of c the choice of c star inside of here give us a c star inside of here so to get a c star action on this algebra a theta so this means we get a z-grading action of c star and a vector space is the same as z-grading on that vector space to get a z-grading on a theta so in the case of um just a quick example when when y is the cotangent model of the fly variety then this torus action is just the usual torus action on the universal enveloping algebra and this z-grading on the universal enveloping algebras a theta is the universal enveloping algebra sln modulo some central character and the grading well it depends on the choice of c star but if we choose the choice the natural choice which is like rho for c star then the grading is just gives degree of e i's usual shovel a generator is one and degree is of f i's minus one so it's that kind of rating and then inside of a theta we have the positive degree part and we make the following definition um we define category o or a theta is the category of finitely generated a theta modules on which a theta plus acts locally in appointment so um this is supposed to be a generalization of the Bernstein-Gelfon-Gelfon category o for the universal enveloping algebra the semi-simple the algebra and it was the insight of brady and lakata prabford and webster that you should try to generalize this definition to any quantization of a symplectic resolution so we and the definition is pretty straightforward we just look for those modules finally generate modules on which the positive part acts locally in appointment so the hearing of brady and lakata prabford webster is there's a characteristic cycle map the growth in the group of category o right here to this realm or homology of life plus at joy yeah we have our question in the chat great um do you require z-graded modules in category o um um i think um um so the question is whether whether the module should be also graded um usually always that this uh this torus action um there's a there's a there'll be a subspace like a kind of cartons of algebra of a theta where the which is sort of responsible for this torus action and in that way um so that's like a what you might call a quantum moment map for the star section and that way you could recover the z-grading um by looking at the generalized eigen spaces for that um for that c star inside your algebra so you i think you know it's not it's not part of the definition but usually maybe it comes kind of comes for a free that's a short answer to the question okay so let me explain the definition of this characteristic cycle map so it works like this you have m so that's it and the theta module in category o then you sheafify to uh maybe i'll fill it and scripty a theta module so let me just mine what this means so scripty a theta that's a sheaf of algebras on y quantizing the coordinate ring of y and the global sections of this sheaf of algebras is just the original algebra a theta so there's a localization functor and um sometimes this localization functor is in equivalence sometimes not but sometimes in equivalence of categories but whether it's in equivalence of categories or not there's always a functor so you can produce this um sheaf version of it and then you um and i mentioned i mentioned last time that this quantization comes in two flavors the formal quantization where you have an h bar parameter and filtered quantization where you don't so here i'm assuming that we're in the filtered setting so this a theta is a filtered algebra whose associated graded is our coordinate thing and similarly this is a sheaf of filtered algebras whose associated graded is the structure sheaf of y and so at this step we need to choose a a filtration on m on this sheaf m and then we'll take its associated graded and this associated grade of m with respect to this good filtration will now be a sheaf of of um of modules of the a quasi coherent sheaf in fact it'll be coherent because the algebra because m was finally generated so this is a coherent sheaf on y and it will be supported set theoretically supported on y plus it's supported on y plus because i assumed that the a plus acts locally no botanically so a plus axle film only that translates geometrically to the condition that the coherent sheaf is supported on y plus since it's integrated and then i take the support this branch so it's a multi-stage procedure and they prove that under some circumstances this this map is a nice mark so in this way we can categorify this homology and moreover in in their paper which beautiful paper this is from about um well appeared on the archive almost 10 years ago maybe published four or five years ago um they explain uh essentially that that this decomposition i mentioned above of this homology of y plus can be seen um algebraically using the category okay so what kind of assumptions you need to be a management user so for example uh do you still need the specific connectiveness of spata no i don't think that's necessary um the problem i think well i don't there's a few this this i don't know the precise statement of what when you need when you have a nice morphism but i think it's um i think it's almost always it's the conditions on like y and x are basically the ones i've mentioned above which is to be a kind of a simplistic resolution i don't think you need to simply connect this to strata but what's maybe more complicated is for which theta this map is a nice morphism so in general it wouldn't be a nice morphism for all theta but only for a certain theta and maybe a set that you don't know very explicitly but you just know there are some theta or maybe generic theta it's a nice mouse recall that theta is the quantization parameter so um if for example in this classical case of universal enveloping algebra semi-simply algebra theta is the central character okay so now we're ready to move on and talk about some like duality um maybe i should say first of all that this i don't know word seems like duality and there's another word called 3d neuro symmetry and um some people use them differently i'll just use them as exact synonyms for me some like duality and 3d mere symmetry are just synonyms for each other um so what does it mean it means that there's one simplistic resolution and there's another simplistic resolution and these two simplistic resolutions are very different for example there are different dimensions um maybe they're constructed in different ways but they have matching properties so two simplistic resolutions and that's why we call them dual and another aspect of the dual is that if you take the dual twice assuming that you're doing uh uh usually it's come back to the thing you started so so this dual is really a dual in that sense and so this simplified duality or 3d mere symmetry has been observed both in mathematics and in physics and in physics it goes sort of under the name of something called s duality for 3d n equals 4 uh super symmetric field theories so i won't say anything more about the physics motivations well maybe slightly more but basically i won't say anything more about the physics motivations and if you have questions you should definitely ask yehow on thursday so he knows about that um somebody asked uh yann asked when is the localization and an equivalence um in this this localization function i mentioned here and i mean it's a good question and there's some theorems when it's an equivalence but i can't first of all i don't recall the theorems and secondly i don't think there's so um any like really general very explicit results like the exact you know they'll tell the theorem tells you that usually it's an equivalence or outside of uh some collection of some maybe affine hyperplanes it's an equivalence but okay back to simplified duality so two dual simplistic resolutions that have things which match not the same things on each side but so different things which match so and i should also say there's a lot of things which match not just like uh two things and that's it but a long list and i'm going to tell you a lot now and and i know many more that are not one of the ones some that i'll tell you about so many things many things can be matched on on both sides okay and the last thing i should tell you is that um the um this tourist action is going to play a big role so it's not just two simplified resolutions but they should also have Hamiltonian tourist actions with and um so so we assume we have chosen a tourist action and also a c star inside of that source and we'll also assume and this and with with this fixed points it turns out that choosing this tourist action would find any many fixed points this is actually equivalent equivalent in the sense of this seems like duality uh to um choosing the resolution by choosing a resolution i mean like if if if we fix x shriek there's many possible resolutions as i discussed a little bit in the answering session last time there's many possible resolutions and um picking out which resolution we're interested in is equivalent to choosing which c star action inside of this tourist action and if this finiteness of the of the fixed point set that's equivalent to the existence of a resolution so if this tourist action is not there's no tourist action that's finding many fixed points there won't be a resolution on the other side and vice versa and in fact there's um even if you don't have a resolution there's still like uh and you don't have a tourist action to find many fixed points there's still stuff you can do um but i'm just going to stick with the simplest case where we have the tourist action on both sides with finding any fixed points and we have resolutions on both sides okay so what's what are some data which what are some matching properties so the first is that the the algebra of the tourist that acts on y is isomorphic to the second homology of y-shrieck so it looks a little weird but it's just a there's a matching of these two vector spaces the le algebra of the tourist and this here's am i right right t sub c is because i'm just going to emphasize that this t sub c um this le algebra of our tourist comes with a natural integral structure namely the coate lattice of the tourist and this well homology also of course comes with the integral structure and this is not just an isomorphism complex vector spaces but compatible with this integral structure moreover inside each of these uh vector spaces there's more data which is there's a uh chamber structure so um and in in particular we're going to pick out two cones in this so inside here we have the not not inside the integral homology but inside the h2 the original one we have the ample cone and this ample cone is going to match with all those c all those choices of c star inside our tourist which give the same attracting set so let me explain this for a second so backing up one second we have a tourist action on y choice of c star inside of it and this led to uh y plus let me write um what are these let me row the choice of this c star in here and then this depends on row so we get we can say that row one is equivalent to row two if y plus row one equals y plus row so two uh maps from two embeddings of c star into the tourist are equivalent if they produce the exact same attracting sets and this will be true because there's not very many choices of what these possible attracting sets can be and we get a um a fan structure on on the the algebra of the of of of our um tourists well in the real the algebra of the tourists for example in the case of the cotangent bundle of five variety this give reproduces you the vile fan and this part of this data matching that matching of datas is that the ample cone of so ample cone means those line bundles which are which are ample the ample cone of our uh but this is actually by by a work of collated in this case it's just a smoke to the car group so those line bundles which are ample they'll match those c stars which are equivalent to our fixed c star so that's one another example of this matching okay so that's that's a matching from the this vector space data okay second second piece of matching so I mentioned we have um the symphonic leaves on both sides and stratifications of our of our singular varieties x and actually so we want there to be a bijection an order reversing bijection between this so I here denotes the strata of x of x on the street that's the strata of x and uh moreover so we would like that this um exchanges these fibers with this um right now so such that we exchange the topomology of the fiber over some f alpha with the topomology of the positive part sorry the tracking set in x alpha actually and so these are just isomorphism vector space but even better it just comes from bijection between between the reduced performance and and vice versa the the homology of the the x uh alpha the tracking sets is the same as the homology of the fiber so that here we had these fundamental decompositions I mentioned and they will be exchanged for this duality so the direct sum is over the exact same set and so it makes sense to match up these pieces of the decomposition and these um these pieces will match here and these pieces will match here okay so fibers get exchanged with the tracking sets um okay let's see an example of this right now so the simplest example is to take the cotangent bundle of projective space take that as my y and the corresponding x is just the square zero rank one matrices and my dual guy is this resolution of c2 my zoom my in okay so in these cases there's just uh two strata so let's look at how this strata work on the on the left hand side here so we just have x zero is just zero x one is just matrices of rank one and then if we look at this decomposition of the homology of the tracking set in the cotangent bundle of the projective space well we get the homology of x zero which is x zero is the point so it's one-dimensional tensor the homology of the uh fiber over that which is the projective space plus the homology of the x one the tracking part tensor the homology of the fiber over that which is the point and um the the guy that's so this is the dimensions or the number of register components here are one one one and the only interesting guy is this x one plus and it has um n minus one here to spoke components so we already saw we already saw a version of that over here sorry to scroll so much but right here this is the same x one plus it was four by four matrices with square zero and rank and most one and had three irreducible components so there's an obvious generalization by the way I didn't say it before but these these kinds of these things are called orbital varieties you take a no point over closure and intersect with upper triangular matrices and the irreducible components are studied like since um the 70s or 80s I don't know and they're in bijection with the uncouple I guess by Spaltenstein's maybe the name was dissociated so in this case there are three components and here there's n minus one components of the same flavor notice these numbers like you know multiply these numbers and then have these numbers so it adds up to n and on the other hand if we work newly with c2 mod z mod n so then the well there's again two strata the zero and everything else well so everything else is in the second strata and I mentioned before um so this is just a point and here we have the fiber over zero and so I mentioned before that the fiber over zero in this resolution is very pretty it's a chain of n different p ones sorry n minus one p ones so here we have that that fiber so it's those n minus one has to mention n minus one and then here we reach that point and and here the the tracking set of x1 I mean this is just the affine space affine line so one one okay so here we see this exchange this guy is matched with this guy because of this order reversing bijection so maybe the notation not so good because I wrote zeros and ones but the the bijection takes zero to one so here are the the the bijection between the strata must reverse the reverse the order so the bijection is not just the bijection taking zero to zero and one to one it's the bijection switching at zero and one and in this decomposition this also switches the roles of fibers and the tracking sets and there here you see this n minus one here matching this n minus one all the rest of them are one so it's hard to see that it's matching but but it is okay so there's there's like this is a very fun game to play to pick your favorite sounds like your resolutions and try find the guy that looks dual in this sense so there's a maybe um let's let me say the third point maybe so I'm listening well back here I was listening make a list of matching structures on both sides then as I mentioned I could make a very long list but I I'll stop after one more point so I started by saying that the there's this funny thing about the torus matching the h2 then I said this thing about the order of verse and bijection is a strata leading to this isomorphisms of the top homologies and the last thing I'm going to say is a categorification of number two so here's an equivalence of categories between the derived category of category o for the quantization of our original guy and the derived category of category o for I mean I don't know why they say for the quantization of the dual guy okay so this this categorifies two and this equivalence is a little complicated in that it takes the form of a causal duality between graded lifts of these categories so I mean this would be like subject of a whole lot more lectures but I won't talk too much detail about how this is supposed to work or but it just to just to be give you a little bit insight or a little bit of honesty it's supposed to be a causal duality between the graded lifts of these category o's and it should categorify the isomorphism between the homologies that we saw in number two and again this this idea of searching for such things is due to written in the kind of crowdfoot webster and it's inspired by the results of a causal duality for the classical category o due to bearish that balance in Ginsburg's story okay so you're probably probably a good idea to give some examples and so well we just saw this kind of fundamental example okay and so of course this this fundamental example generalized in many ways so one way it generalizes is we can have a hypertoric variety here so a hypertoric variety is given the data defining a hypertoric variety is the embedding of a torus let's say let's say rank k so c star to the k embedded inside of c star to the n and if we have that data we use this to produce a hypertoric variety which will end up having dimension to n minus k so we take remember the we take the cotangent bundle of this c n and we take the Hamiltonian reduction by this c star k three lines and the dual dual to this embedding of c star the k and c star d n there's an embedding of c star to the n minus k inside of c star to the n i mean this c star to the n is like the dual torus of that one and this leads to a well to a hypertoric variety but in the same way does not have the same dimension as you see and like the the fundamental example of this duality is where we have just this c star embedded diagonally in c star to the n and that's dual to this torus with product one so rank n minus one torus embedded c star so that's a that's the example which which reproduces this okay and um so there's some beautiful commentatoric self hyperplane arrangements which are related to this uh the defining these hypertoric varieties related to defining these embeddings of torus so um michael will talk about that in in the question answer session tomorrow so more details on this duality in this thing this is called in terms of hyperplane arrangements this is called gale duality um a next a next example is the cotenderminal of a of a flag variety full flag variety is actually always dual to itself or maybe slightly more precisely to the flag variety and then um last but not least is um if we have here the um perver variety so i explained last time the definition of a nakajima perver variety associated to choice of uh quiver and two dimension vectors i don't know what i used last time because i see but let me call it m lambda mu dual to this choice of of perver variety will be an affine grismani and six so i'd like to explain this but i'm a little short on time for today i don't know whether to try to sneak it in today or or discuss it it will definitely take more than five minutes but i'm a little unsure whether to sneak it in today or discuss it next time start with that next time uh maybe i just pause to see if there are any questions so if there are any questions we can take a few questions now and if we do this next time or if not i'll go on uh i had a question so in the the point three above tita is the same for both sides the tita is the same oh sorry it shouldn't it shouldn't be okay so there are okay so this depends on a choice of tita and tita or strip yeah um but usually um we would just take here a theta and theta street could be generic and integral um yeah another question do you comment on what you said um about being 3d mirror symmetry well well probably not but i can uh um so i don't i don't know i mean you mean from the perspective of of the physics or or just in well maybe from the perspective of um of mirror symmetry in terms of like syz or or otherwise i think it's not yeah that's a good that's a good question but that's not it's not related to that i mean i think from mathematicians viewpoint um it's hard to see what makes this three-dimensional and that two-dimensional for the physics viewpoint um the mirror symmetry has something to do with two-dimensional quantum field theories duality of two-dimensional quantum field theories and this simply duality has to do with the duality of three-dimensional quantum field theories okay and so from the um but from mathematicians viewpoint it's not clear what's like three-dimensional of this one and two-dimensional about the usual mirror symmetry so it's not a great answer um but okay well maybe i'll just say a tiny bit more about the physics since i have not enough time to really talk about the staff members finance license so i'll start with that next time and i'll just say a couple of words just about this physics so in physics people are interested in in this 3d n equals 4 super symmetric field theory and one i don't know too much about these things but one thing i heard is that if you have such a theory it produces you two what could say algebraic varieties two kinds of spaces one space is called the Higgs branch and one space is called the Coulomb branch and one way to say what this symmetric duality is that we observe in math is it's the relationship between the Higgs branch and the Coulomb branch of a single theory so these two are the symphonic dual pairs but there's even a sort of another way to say what it is in physics or closely related is that you can take this three-dimensional super symmetric field theory and then do something called s duality whatever it means and produce another three-dimensional another such three-dimensional n equals four super symmetric field today we call it t and this one t shrink and maybe this guy here is going to be called m h t and this one called m c t and in this field theory it will also have a Higgs branch but it's the Higgs branch of this theory is the Coulomb branch of the original theory and vice versa so another way of thinking about what this um some like duality is about is is that that it's a duality of these field theories and and and what happens so so good so these two guys the Higgs branch of the first theory and the Higgs branch of the second theory will be some likely dual in the mathematical sense and I think I don't know so why is it called 3D mirror symmetry because it has to do with this this duality of these 3D field theories is called 3D mirror symmetry from mathematicians viewpoint there's nothing three-dimensional about it at all I think the subject this history of the subject is like is is is very interesting I mean this duality was served by basically by originally by Braden Lakata-Proudford Webster and one day when Ben Webster was giving a talk at IHS not at IHS at IAS right when he was starting his postdoc there like all the postdocs have to give a introductory talk about their work and he talked about this duality and he had no idea that it had to do with physics at all and um somebody in the audience maybe Gukhav Witten anyway said that oh this is the same as the duality which physicists have observed for like 20 years or something and um so that was a birth I think of a very fruitful interaction um okay so we'll see next time I guess we'll stop now so we'll see next time this I haven't yet to define these affine-grasse-maintenance slices which is a bit surprising because they're my favorite guys so I'll define these guys next time affine-grasse-maintenance slices and we'll see in what way they are dual to clever varieties first thing for next time and then we'll continue um um then we'll discuss this work of Braden of of Ravram and Finkelberg and Akajima about um well constructing these some likely duals in some generality okay I'll stop there any further question or demand don't be shy I have a question about data and data dual for quantizations so they live in different spaces right so one in h2 of y and another in h2 of y dual yeah so um you mean what's what's the relationship supposed to be it's a good question I kind of uh I don't have a very good answer at the moment so I didn't think about it recently I think I think in any case as long as data and as long as they're generic and integral the the category actually doesn't depend on on their choice so I'm pretty sure that this category is independent of it as long as it's generic so just like the usual category O for for um the semi-simple algebra will be independent of the central character so the block so usually we think of the full category O a semi-simple algebra think of blocks so this this thing is like a block of category O so as long as the as long as the block is generic and integral it doesn't depend on the central character so there's not really it's not the choice of them is not important I thought there was some relation that like h2 of y dual is like the choice of one parameter subgroup for x or y and then the one parameter subgroup for the dual is the choice of h2 and so kind of the category O contains both and there's well yeah no no it's true so like the um yeah that that's true too again the category but um in some sense the the choice of this one parameter subgroup is sort of only important up to the this cone that it lies in so like the category O would only be sensitive to the choice of this of where of this cone so I'm assuming that you're already in this sort of ample cone so if you're staying the same cone then the grading will be the same so the category will lose it I think Eugenia satisfied with the answer maybe anyone else have any questions yeah thank you sorry at the end question um yeah but sorry I'm a bit confused uh but in the case of category O um usually it only depends on the choice of central character right so it's actually t quotient by w oh you mean um rather than then t yeah rather than t yeah yeah okay so um technically the yeah so well is it it's it's a there's a small subtlety which is um the out the the the universal space for quant the the parameter space of quantizations of x just as an algebra is just is h2 mod w mod this nemicom of val group in general and then if you want to speak about a quantization of this the variety y so a sheaf of algebras in line then that's parameterized by a choice in h2 so it's almost the same thing so you could think yeah so from what I wrote here it's probably better to think yeah I guess there's something going on um in the localization procedure when you're in unfold column you get an obedient equivalence usually and when you go to the other places you only have the derived equivalence yeah yeah okay thank you so I think yeah I see I see how does the value of the resolution of the e6 singularity so um is a good question I don't usually think about this case and in fact it's an example of okay and I didn't say it because I always mess it up whenever I say it so there's an example of subtracting another like fundamental example of some type of duality which I didn't mention which maybe I should have and it'll come back to answer Alexis question is that you can take the potential bundle of g mod p for any semi-simple p well that's a resolution of some no-pone over closure that's my y I won't bother writing what x is that's dual um to a resolution of a slow to a slice in um the le algebra of g maybe in the language dual the object and um I always mess this up because I don't know the combinatorics of these no-pone over closures very well outside of type a so in type a this is very easy to see outside of type a it's a lot harder to see so Alexis questions of this form because this c2 mod gamma this is the um this c2 mod gamma this is the um sub minimal sub sorry this is a slow to a slice that's a slow to a slice slice in the in the le algebra corresponding to gamma so in this case it would be a slow to a slice in in e6 so that would be dual um to a co-tension bundle of a g mod p and well e6 is simply le so this language doesn't matter very much so it's a co-tension bundle of a partial flag variety e6 um and which partial flag variety I guess it should be a um small I mean yeah the which partial flag right I don't know but some some co-tension one of some partial flag variety in in e6 oh except um now you think about it more carefully okay I don't think um it's like this dictionary doesn't quite very maybe it should have said slowly so this dictionary works very well in type a outside type I think the dictionary which works with we don't always get resolutions so I think in general we have slow to a slices um dual to no potent over closures but not every no potent over closure has a resolution and I think actually this one will be dual to one without a resolution so maybe this answer is not very right so a better answer would be some no point over closure and and I guess we could probably figure exactly which no point over closure it is because this is a slow to a slice to a sub maximal no point over closure so this will be the correspond duly to the sub minimal no point over closure maybe okay so or maybe minimal so maybe it's maybe it's dual to a minimal no point over closure yeah okay now I'm now I'm happy this slow to a slice that you mentioned is dual to the minimal no point over closure in e6 whatever that minimal no point over closure in e6 like and I don't think that one admits this is likely resolution and my reasoning is because I mentioned before that when we don't have a Hamiltonian torus action with isolated fixed points on one side we don't get a symplectic resolution on the other side and if we take c2 my gamma and gamma is not z my end we don't have a Hamiltonian torus action so on the dual side we should expect not uh not to have a resolution so I don't know what this minimal no point over closure in e6 is like but I think it doesn't have I think it's dual and I think it doesn't have a resolution I think we can uh does that answer your question Alexi? Yeah Alexi was asking also for d4 but maybe you can reply to him in your channel or tomorrow Well the answer will be the same it'll be it'll be the answer the same it'll be the minimal no point over closure of that type so the minimal no point over closure in d4 then again I believe it doesn't have a resolution so I think we can thanks uh join again