 Hey, thank you. Thank you everyone for coming to the last talk of this lecture series Was it there a bit of echo? Yeah, I'm trying to take care of it. Okay And then I'm trying to go guy I Still see some people getting into the room. So maybe I'll wait in another In a minute, okay, so Last time we did we just we discussed the cologne the definition of the Coulomb branch due to a problem in Finkelberg Nakajima So let me just recall the notations. We had G a reductive group and in a representation of G And then we produced from this first a space called RGN and Then we studied the homology of the space which was an algebra a GN so the Geovo like a variant homology of the space RGN We also had a quantized version Where we Which was defined by adding C star covariance to the homology So that this H bar quantization parameter is the C star equivalent parameter And then we defined the space the Coulomb branch MC GN to be spec Of this algebra We also considered a few variants of it. Um, maybe I'm not Need them today, but we had some variants of it where we were able to make for example a Candidate for a resolution of the space using project of a graded ring Okay, so today We're going to be interested in in quiver what's called a quiver gauge theories so what that means Is we fix a gamma directed graph cover And at the moment I make no assumptions on gamma can have multiple edges can have loops so We just find And then we have two dimension vectors v and w so they've been Natural numbers the i i is the vertices of the graph And the reason why we have two is because we have this frames picture in mind So we have a circle for every vertex of the graph here. We have some edge of the graph And then for each we have a frame Here and I put wi in the frame and from this data of the graph the dimension vectors We construct a group g which is simply the product of the gls of the vertices and a representation and which is simply the Direct sum of first hams inside the graph And then hams to the framing I don't know why there's so much drugs in this Can't at least get rid of one of them. So I remind you Let me discuss this before but there's two spaces. We're going to consider one is the Higgs branch Um, I'm sorry, maybe one piece of the notation before I get there associated to gamma. We have G gamma that cat's moody the algebra symmetric cat's moody the algebra So it it doesn't care about the orientation of the edges in the graph Um, mostly I'll be concerned with the case when there's no loops when there are loops Maybe this exactly what the algebra we associate. Maybe it comes a little bit more complicated. I'm not Too much expert. So mostly we can say the case without loops Um associated and then um associated this v and w we build a dominant weight lambda And mu just a weight And they are related to v and w by these familiar equations lambda Is the sum of fundamental weights with coefficients given by the w i's And lambda minus mu is the sum of simple roots with coefficients given by the odds So Yeah, so as I said, we have two spaces now associated to this data. We have the Higgs branch All right, m h lambda mu and this is just a Nakajima core for variety by definition I take the cotangent bundle of n and I take the Hamiltonian reduction by At the moment map level zero and at the g it parameter chi and I use my By by the group g And then I have the Higgs Sorry, then I have a Coulomb branch just the Coulomb branch for this gauge theory to find above I just write lambda mu for the instead of g and n Okay, so the first question of course is like what can you say what is this space? What is this? um Coulomb branch associated to the quiver gauge So this is some well That's that's going to be based on the topic of today's lecture. So, um Let's start with the case when g gamma is a finite type so finite type means it's a The dink and it's the dink and diagram of a ad e the algebra in this case, um, I'm going to write g gamma for the corresponding group Or maybe slightly more precisely the langland dual group. But again, this langland duality is going to be irrelevant because we're in this simply this type So g gamma is going to be the corresponding group And then we're going to study the affine-gross mining of g of gamma Okay, so let let me just be clear that there is a Um two affine-gross mining's in the story one affine-gross mining which was used to define the Coulomb branch And that's the affine-gross mining of g the gauge group and now i'm talking about the affine-gross mining of the group whose Dink and diagram is given by gamma When I first heard about the bfm construction it was at a conference in paris. It was in January of 2015 and and sasha was giving a talk It was organized by vastero, I think And and at some point in the talk I interrupted sasha and I said you mean to tell me there's two affine-gross mining in this story And they're not related as sasha said yes okay, so this affine-gross mining is affine-gross mining of this ade group g gamma And I write uh g gamma so it's g gamma k my g gamma o And I just recall notation. We have gr lambda. So that's the g gamma of o orbit through the point t lambda So it's going to be turned to be the same lambda that i'm using above so this lambda Was obtained using just this dimension vector and it encoded a dominant weight for for g gamma and And then I also will have This slice w mu I mentioned him before but let me mention him again. So how's he defined I take? G gamma of t inverse and then I take this this first congruent subgroup condition And take its orbit through t mu. So it's the kernel of the evaluation. So give me these transverse orbits. They go around this And then I define this affine-gross mining slice w lambda mu to be girl lambda bar intersect w mu So it's a it's a subs It's a finite dimensional sub scheme of of w lambda. So it's an affine scheme So that's the definition Of the below. So here's a the first the first theorem about this these coulomb branches Is that they are these affine-gross mining slice? So Well, sorry in this circumstance. So again, we have the assumption that g gamma is a finite type And also the assumption is that mu is dominant Okay, so recall the construction of this um Of this thing didn't require mu to be dominant. We just took mu and used it to I mean mu is obtained using relate to this v's but nothing to do with Being dominant or otherwise. So here only if it's dominant, we have this now. So if mu is dominant then The coulomb branch is isomorphic to that affine-gross mining slice Okay, and and If you've been following my talks then you should find this theorem not surprising at all because I already said many times That the coulomb branch is supposed to be some psychic dual to the Higgs branch And I've also said a few times that the affine-gross mining slice is some psychic dual to the necrogen macro variety so This theorem just a confirmation of those two facts So what happens when mu is not dominant but still in the finite type situation Then we can define something called a generalized affine-gross mining slice The definition looks a little bizarre First of all rather than working in the affine-gross mining actually worked inside the group g of k and I formed the following thing So I considered all um major all elements of g of k which emitted Gauss decomposition like this So inside of any so maybe I should fix some notation before I write this So here, um u and u minus in g gamma to note opposite unipotent subgroups And so inside any Inside the group of matrices or any reductive group you can consider those elements which emit Gauss decomposition like upper triangular diagonal lower triangular decomposition So upper triangular diagonal lower triangular So I'm looking at those matrices which admit or those elements of g of k which admit such a decomposition such that um, the lower triangle and upper triangular piece are one my t inverse and the diagonal piece is like t the mu times this diagonal thing which is one much So at first time seems like a slightly strange thing to do Here's some reasons why um, it's a good idea. Oh the meaning of the subscript So if g is any group and then I have g if t inverse and it emits an Morphism to g which takes t inverse to one and the kernel of this map is g1 t inverse Sometimes called the first congruent subgroup So you can the matrices so for example this u1 t inverse would look something like this one one one down the diagonal here And here up up in the upper part would be like t inverse plus t of minus two So So Where was it? Yes, so this is the definition of This w mu and it looks a little bizarre But here's a few reasons why it's not so strange. So so first of all if mu is dominant It's not very obvious But it's not hard to see that that in map Well, of course this w maps to the afringer's mine in just because g of k maps the afringer's mine in So you map if you if mu is dominant in your map w mu into the afringer's mine in Then um, I realize my g sub gamma looks a little like my g r Hopefully no one gets confused besides me. Okay. If mu is dominant then w mu maps into that Maps isomorphically on to mu w the thing I previously called w mu in the afringer's mine in So there's not two different things called w mu. So that's good to Um, this space w mu has a modular interpretation. I won't recall it um It's not really Um, it's in terms of well, let me vaguely recall it. It's in terms of principal g bundles on p1 With a reduction to a burrell into an opposite burrell Um, I don't find it any more enlightening than this matrix definition In fact, I find the matrix definition more enlightening, but anyway, this is how it's originally defined by by one chemical burden epigema And three which is closely related to and sort of also closely related to this matrix definition Um, and this I think the first incarnation of this w mu. It's some kind of modulate space of singular monopoles and and I say this in This is appearing in the physics literature. So I don't not sure if there's Yeah, a precise mathematical statement here, but this is in the physics literature a little more Demofted Okay, okay, so that's some reasons why why this w is not as like random as it appears Anyway, once you have this w mu then we define the generalized affine-garse-mining slice w and the mu to simply be this W mu well, normally we would take w intersect or lambda bar But now we're not in affine-garse-mining. We're inside this g of k So we take w mu intersect the double coset space and put the closure If I don't if I write w lambda mu without a bar on top, it just means the same thing without the closure So this thing's called So one, um One funny feature about this space is it's often not It's I started the talk that my series of talks by talking about conical symplemic singularities Well, I haven't even talked about or conical some type of resolution I haven't even talked about a resolution space, but actually this space is not actually Usually conical. So it's not conical Or at least doesn't actually admit an obvious cease to action making it conical I don't know exactly if it's possible to prove that it's literally not conical But at least doesn't admit an obvious cease to action making it conical not conical Unless you might think it's unless mu is dominant, but it's actually unless mu is almost dominant Unless mu plus row usual row is dominant Um, I mentioned last time the physicists and and Nakajima sort of introduced this notation in the mathematical literature have a division of these theories Um, it's a good bad and ugly. So good in this case good means mu is dominant Bad is mu plus row is not dominant and ugly is the intermediate case when mu plus row is dominant from mu is not dominant Okay, let's see some examples Oh, I mean, maybe I should say to But okay this theorem With this definition of w lambda mu I won't rather writing it again With this definition of w lambda mu this generalized slice I mean, obviously I wouldn't be telling you this but this if it wasn't true this theorem extends So now we can erase No, I'm very sorry, but we I miss uh question in the chat That was a question of six minutes ago But I already answered it I really okay I saw it and I I answered it Okay, it's fine. There's a for me No problem um, so this theorem Of bfn extends to the case of mu is not dominant when we use this definition of the generalized that fungus minus Again in the finite case If it's not finite type, it's not totally clear. I guess what the meaning of anything I wrote here is Okay, um So let's see some examples So the first example Was was the with this g gamma being sl2 And so my quiver just says one vertex and I'm going to take Let's consider the general case. So I have Framing in in the m here. I discussed this last time a few times when I was talking about Example when I was talking about the bfn construction So in this case lambda is n times the first fundamental weight Mu is n times the first I mean the only fundamental way minus m times the only Simple alpha like if you like you can just think of this as n and this is n minus 2 n is perfectly good way to give it um The corresponding Nakajima quick variety we discussed before is just the cotangent bundle of the grismanian so the mh And then Okay, and what's our coolant branch or our generalized ethnograph money and slice Well, it's actually not very hard to read off like this thing looks like a big mess Guesty composition business, but it's actually not very hard to to analyze it Especially in this as all two case and when you do it if after maybe possibly taking transpose in your matrix or something You reach the following nice description. So it's the set of all matrices a b c d And these are all matrices over polynomial ring and t so Got rid of all powers of t and rest just by scaling through And a is monic Of degree m exactly degree m b and c are of degree Less than m d. I impose no condition at all and ad well except for that ad minus bc must be equal to t to then So it's a it's a nice space You can have fun playing around with what it looks like for different values of m and n for example, you might try to figure out what How the degree of d is sort of affected by the degrees there is in this space At least when When this In the dominant case in the m minus 2m. It's greater than zero Then this space is isomorphic to a m n minus m slow to assist Okay, so that's one nice one nice example and this Generalizes in a few ways. Well one way we could say that this generalizes is that um There's a similar kind of matrix description of it for any glm but also For any glm or any finite or any even affine type a so g gamma's Or affine They pay this this space or or precisely I mean in the app in the affine in the affine type. I don't usually write Wm I just write mc lambda mu This space is a bow variety Joe, I think we have a Might have a question in the q and a please great, um Oh, okay, it's nice of you. I always David. Okay. Let me I'll come to that in one second So let me just finish this right if g gamma's is finite or affine type a then this is a bow variety This is proven by Nakajima and takayama and we'll hear more About bow varieties in richard romani's talks talk next week So it's some explicit description of it, which is um Sort of different than this matrix description Okay, there was a question about does symphlectic duality statement still hold if it's not conical um Well, it depends exactly which part of the symphlectic duality statement um But but the short answer is yes like the um If we focus on like the third part the part about the category Oh Or even the second part about the part about the Homology Those parts still those parts do hold as I'll say in a few minutes Um, even even when mu is not dominant Okay, um So that's I guess it's a sort of second example or a meta example a third example um is um Interestingly enough how to get the the gress mining a finite potential bundle of gress mining as a as a Coulomb branch And and it's like this if you take lambda to be If you take g gamma to be sln and you take lambda to be omega m plus omega n minus m Then you take mu to be zero if you convert that to quiver data what it means is you have a one here two framings and an increasing Sequence like so And then a constant sequence like so so that's the quiver data coming from this lambda nu And this affine gress mining slice in fact this case it's not even a generalized affine gress mining slice, but mu is zero It's dominant this affine gress mining slice is just a cotangent one And you can see this just from the geometry of affine gress mining um, oh and another example I was inspired to add this example because of Eugene's talks, uh, he's like Said I've been neglecting the Hilbert scheme indeed. I have been neglecting the Hilbert scheme So here he is in this for this quiver data so or for this, uh generalized or generalized affine gress mining slice if you like of type a one one or Yeah, um Anyway, for this for this quiver data the the corresponding Coulomb branch is the Um, is the Hilbert scheme as is the I mean, technically the Coulomb branch is singular. So it's really not really the Hilbert scheme. It's really it's c2 I mean see Just symmetric and Just sim and Is another question. No, no, it's the same question okay, so Yeah, that's that's it for this examples. Oh, maybe one more one more example another another classic example We take this quiver data, which is the same as produces the cotangent bundle of the fly variety as the As the as the Higgs branch It also produces the cotangent bundle of the fly variety as the Coulomb branch This is Same as W and oh my god N minus one zero Usually Okay, great. So What are some properties? What are some results we know about these? These generalized affine gress mining slices and more generally these these Coulomb branches associated to cover data All right, sorry Joe, is it true that cotangent bundle to fly varieties are self dual? for any group And yeah, simply quality. Yes. Yeah but only only in type air they realized this as Higgs and Coulomb branches of of gauge theories Yeah Oh, sorry outside in the general type. Maybe it's better to say that the cotangent bundle of gmod b is the Is the langans dual group? Okay, so here's some general properties. So So for the first one, we go back to the case of the finite type And I mentioned of course that there's this more there's this map from W mu to the affine gress mining and if we restricted the to W lambda mu will land inside girl lambda bar Well, this map is of course far from injected Maybe maybe I should have mentioned this from the very beginning But one fact is that the dimension of this W lambda mu is twice rho lambda minus mu Whereas the dimension of girl lambda bar is just twice row of lambda So if mu is not dominant like this negative mu will actually be contributing to the dimension So the dimension will become bigger than girl lambda bar, right? Like if you imagine the extreme case and mu is anti-dominant then this is going to be making a positive number Okay, so So this map is not injective, but if we restrict to the attracting locus And I should say this attracting locus only exists If mu is actually a this this w other mu exists even if mu is not a weight of The lambda representation But the tracking locus only exists if mu is a way of lambda presentation You can prove in this finite type case pretty easily that That this torus has a fixed point if and only if mu is a weight So if we was the next and when when it has a fixed point and then we take this attracting locus Then the map is injective this Gives a nice morphism between the attracting locus and And I say s mu which I never defined before but this s mu Probably I mean that may be s mu minus to be technically correct. I think People have maybe seen it before it's the u minus k orbit And this this this so this result the fact that this thing gives this restriction gives a nice morphism as a theorem of pre-life And as I mentioned before this is true for even if mu is not a weight both sides will be empty if mu is not a weight of the lambda So in particular if we take the top homology of this Attracting locus, I mentioned like a long time ago that I really like these top homology of attracting locus because that's what enters this decomposition of the homology If we take this top homology of attracting locus, it will be the same As a top homology here Which will be isomorphic to The view of lambda mu and this is by Mirko Richelon So one way to think about this generalized afringer's mining slice Is it something which contains this Mirko Richelon and locus as the attracting locus It's just something cooked up to be of twice the dimension of this This girl in the bar intersection Um, and this leads to the following ejecture Which I think is very important still open trajectory of well formulated by the fn They call it the geometric's attack a conductor. So it's the statement that for any Gamma not necessarily a finite type So in finite type case, it's true There is some kind of vector space isomorphism between the homology of the attracting set in this Coulomb branch associated to lambda mu and the V100 And then I should of course point out that this we already know Um by Nakajima's work is the homology of this central fiber F-land and the central fiber in the nakajima variety So that we get this isomorphism from left to right is what's Predicted by some flexibility. This is like my second Homological some Homological some flexibility prediction But this is not known The only case as far as I know outside of finite type is where it is known in affine typing And to show you that it's how unknown it is. Um, we don't even we would expect that this torus um, so There's always a torus c star to the i which I was calling t I hope it wasn't too confusing because it's the torus of this g gamma not the torus of the gauge group g so this torus So one like part of this conjecture would be that this torus and has a fixed point in this Coulomb branch if and only if The mu is actually a weight Um, and that's not even that even that's not known And I should say that this this conjecture I would say is just a conjecture about an isomorphism vector spaces um We wouldn't expect sort of a geometrically defined action of our Lie algebra on the direct sum of these homologies Because it's sort of not what we see in geometric sateke but There is a sort of partial I don't know result or understanding of what should be going on. Um, so In the usual geometric sateke you can say something about restricting to a levy subalgebra and even in in this story You can also say something about what you expect about restricting to a levy So you would expect this isomorphism to be compatible with levy restriction. And that's the statement you would like So sir, uh, this is defined in terms of slices in some, uh, affine-grasmaya or some cat smoothie group No, no, this this is defined in as Coulomb as a Coulomb branch So not not as a Coulomb So, so yeah, maybe I was a little okay. Let me just back up to the beginning then so I said two things at the beginning one one right here So we define this um mc lambda mu as a Coulomb branch so SPAC of this, uh Homology like this official Coulomb branch definition over here So SPAC of the homology of this Vfn space That's the definition of mc lambda mu When gamma is a finite type When gamma is a finite type then And mu is not necessarily dominant then this Coulomb branch is a generalized affine-grasmaya and slice um, I don't think anyone knows Or has tried to work it out whether this theorem really holds Outside the finite type Whether whether we can consider these general study these generalize affine-grasmaya and slices for cat smoothie groups um At least I don't think there's any published results in that direction So for now this this thingy is just defined as a Coulomb branch. Okay, any other question? Okay, so this is all I don't know property one So property one was this in the finite type of business and it led to this connection. Okay property two um So property two, um Well, it's just a special case of this but it's sort of very interesting special case So I'm going to highlight it. So what happens when we take lambda to be zero Let's let's go back to the finite type situation So whenever another way of knowing that I'm working in finite type is usually I'm the finite type I'll write w lambda view whereas in the general type I'll write mc lambda Okay, so in the finite type situation in the finite type with lambda zero I mean, there's no representation to think about there's no gray lambda bar really to think about just a point But this w zero new is actually still very interesting. It's actually nothing but the space of base maps Uh from p1 so I mentioned here because this is uh Uh To to g my to g to g my b g gamma So I mentioned here since we're in uh It's supposed to be about numeric geometry and this is obviously an objective study in numeric geometry So base maps of degree of a minus mu from p1 to g gamma mod b um And this space has no no attracting set. I mean no fixed point. No, there's no no attractions But one feature it does have Is it has well as every coolant branch has which I didn't emphasize too too much. It has this map this integral system map to T mod w so careful this t may be better to write just like let's see uh Okay, so this is the this every coolant branch has a map to the spec of the Um Geo covranc homology of a point And and which I identify in this case with this c to the mu c to the minus Um, so every coolant branch has that map. It's always an integrable system And in this case this map is well studied for these Oh sometimes these spaces are called open zestatum spaces based on base maps and one one interesting feature Is that if we take we're maybe not interested in the We can't look at the homology of the attracting set because there is no attracting set But we can look at the homology of this fiber of this integrable system map And this is actually something you can do with any coolant branch But w any w or any coolant branch study this Zero locus of the integrable system, which will contain the attracting set always when it exists So this is something of general interest But in particular this case is quite interesting and it turns out this is actually esomorphic to the weight space of a verma module So here this m is a verma module And m mu denotes the weight space of the verma module not not the Not the not the mu a mere module. Hi, so I knew it's the notation looks like a little bit So m is a verma module of highest weight zero So interesting enough. We got these weight space of verma modules coming up this way and actually there's a generalization of this Um That the homology of the zero locus in w lambda mu will be a weight space of a verma tensor of u so another Property or sorry generalization of this story Is of course we started the story by with we might be interested in g gamma not of simply this type Maybe finite type and not simply laced Um Sorry type b bc and so on And um for that there's a modification Of this There's a modification of the Coulomb branch story um, so that's due to naka jima in weeks This is called Coulomb branches with symmetrizes And with this modified version of Coulomb branches you can recover Uh these w w lambda mu say that these generalize aphangous mining slices for any finite type G gamma Um, and then another another result I'd like to mention Is okay, so bad. So still of this The g gamma of finite type simply laced So ad situation um But now but new not necessarily dominant then the symplectic leaves So in general it's a very interesting question to study symplectic leaves of Coulomb branches for any g and n and um, there's not any general results in that direction But in this case, um, there is a result. So the symplectic leaves of this w Lambda mu bar r What you might expect there w new mu for New between lambda and mu in new dominant Okay, and that's a theorem of with dia in weeks okay any questions about uh these results and these uh Coulomb branches and these generalized slices So what is the Higgs branch that is taken in the case of non-sympleased? Oh, that's a very good question. I mean there's no, um There's no exact definition of a Nakajima quiver right in the nonsense of this case I mean, it's probably no there was a recent work of geist Leclerc and shore on this topic and There's a relationship between the definition they use for the Coulomb branch and this geist Leclerc and shore but There's not exactly a Higgs branch. I think and there's not exactly a statement of symplectic duality So when you say that the homology of this middle fiber is a vermin module weight space Yeah How much of the do you see like the action on the vermin module or the just the dimensions or what is how much of the vermin module do you see? um just just the dimensions um although from the Quantization which I'll talk about soon um, so so this um Maybe there's a good time to segue into the next topic. So let and slightly answer your questions. So let me let me get back to it um, so suppose I have Um Sympathetic resolution although for my story now the the resolution parts can be sort of irrelevant So let's just stick with just the x the the conical symplectic singularity I forget about the resolution for a second And um, as I mentioned before we have this maybe attracting a look to say it but say it has a with a torus action And actually I suppose I'm gonna have to erase the word conical That example certainly wasn't conical. Okay, so I'm collecting singularity with a torus action and I have this attracting locus x plus And then maybe I also have um Uh Some integrable system map So I don't know if this is exactly a general feature of some glyphic singularities, but these Um cool um branches come with come with this integral system as I mentioned. So we had this integral system map Brandy cool um branch to expect Jacob and cool Back to another question, uh The relation between sympathy duality and t duality and Are some familiar examples of integral systems the list like kitchen system, etc. Um So first of all the Um I don't think there's a relationship between the t duality for these integral systems In simplect duality um Probably I mean there is some things you can say about like some kind of t duality for these integral systems But let me not try to say anything because I'll probably say something that's not true. Um, are there familiar examples of integral systems? I think so. Um um Hitching system, I don't Know of any exact way of the hitching system appearing this way um But for for example One example of this picture, maybe it's a good example. So put it right here um, is if we take the nilpot and cone of sln and It has an integral system called some I guess usually called the gelfand setland integral system and it goes to Uh c to the N choose two And how is it defined you take your matrix a to the characteristic polynomial Of the upper i by i minor of a A ai So for all i i equals one Up to n minus one ai is the upper nine by i minor And notice if you take the zero level of this Um It will contain the upper triangular matrices, which is the attracting set for the Hamiltonian torus action Um, okay. I'm just going back to my general picture then um, I had would have This example is a very good example to keep in mind. So I have x and it has this integral system Well, if it's a cruel average it would go to this this spec of the this guy So in general it goes to I don't know c to some m and um then This attracting set will be contained inside of the zero level of the integral system And if we now quantize x So here a That's we can use this a a here is it going to be a quantization of x and I mentioned the last Beginning basically that we consider a category o for a And that would be thought of as a categorification of the Topomology of this attracting set well to be a bit more precise I said before that I should consider this category as a categorification of the topomology in the tracking set in the resolution so it's really sort of the um Top quotient of this category which catifies this but roughly speaking categorizes There is a map from the growth in the group to there and on the other hand we can consider um Because of this integrable system We get it quantizes to a gulf and sentlin subalgebra to a to a to a maximal commutative subalgebra a um Which maybe I don't find you to choose some special notation for it. Maybe I'll just write as a polynomial So this is a maximum commutative subalgebra and then we can consider the um Recall this the gulf and settlin subalgebra and then we study gulf and settlin modules for a So modules for a which are uh locally finite for the action of this maximal commutative subalgebra And we think of this as categorifies This zero level So now um, so for for example in this case of nilpot and cone um, this was just Then this would just be the gulf and settlin modules or uh, just universal developing ultravesolin one of those central character um So to answer michael's question now finally, uh, michael asked category of finds michael asked In this uh situation I wrote above with the base maps um Can we say something about the the zero level? It could be something about the action on that when I think of it as a vermin module And the answer is well, not not directly on the homology, but once I categorify it. Yes so When I categorify it and think of these modules for this gulf and settlin modules for the quantization then Then it says yes Okay, so this takes me into the last topic which I have a few minutes to explain 10 minutes so, um So the last topic is about the quantizations Of these uh coolum branches for quiver gauge series these Which is generalized african-gryspanian slices. Um, so Um Well, I'm going to explain basically two results in five ten minutes. So I guess five minutes each so the first result was that um And now we're going to go for the finite type case for the first result So then there's this algebra called uh, why new which is a called a shifted union So it's an algebra with this explicit generators and relations That's a so related to the union. I mean when mu is dominant, it's a subalgebra the union and when mu is not dominant it's not a subalgebra, but it's closely related to the union and um It it acts um by difference operators on a big polynomial ring Okay, so this this actually this thing only depended on on Mu, but this polynomial ring depends on lambda. So the R ranges here From one up to Vi or vi's are calculated using lambda in the usual way or lambda in new usual So it's a finitely many variables whereas this is why mu has like infinitely many generators And acts by difference operators in this polynomial ring and finally many variables In the image Of of of it inside this ring of difference operators And this called by lambda and we call it truncated shifted union So here's a theorem which was proved in an appendix and by the following Authors I'll just put their initials because there's too many of them But I guess they've all appeared already in this lectures except for one of the k's not me. It's codera Like one of the k's is me, but the other one is codera. So I write isn't it? so otherwise, you can guess who everyone is and The the theorem is that this quantized Bfn algebra Associated to the same where gnn are associated to the same quiver data from lambda mu is isomorphic to lambda And I just want to very briefly explain the idea of the proof so that we can embed Using localization We can embed in localization in homology echo variant homology We can embed this Coulomb branch algebra inside The Coulomb branch algebra for the torus and of the same time We can also sort of forget about the representation just sort of convenient to do so but it's not that important and But the cost of inverting some stuff and and I guess I should mention that it doesn't really act on this polynomial ring, but really on this polynomial ring More precisely on the ring of rational functions So anyway, you have to invert some stuff. I won't bother writing. We have to invert and this guy so this is just the Coulomb branch algebra, of course, planning through a torus and the Coulomb branch of a torus What with trivial matters they call it no representation is just the cotangent bundle of of its Lie algebra so And so this can can be quantized to a ring of difference operators isn't that so um Yeah, we can go into a lot of detail here. There's like a lot of crazy formulas you can write down, but let me not do that now um, but that I just want to explain that the basic link between this shifted union and Quantized Coulomb branch is that both of them act on a ring of difference operators. This is a very important idea Leads to this isomorphism And maybe just as a quick example of this if we take this quiver associated to the cotangent bundle of the five variety Remember both as Coulomb branch and sx branch then The corresponding truncated shifted union is pretty easy to see. It's a you know some developing other sln module of the central character There's also a whole story with some parameters and so on and so on So you can get any central character this way very good parameters So that's what you get as the quantized Coulomb branch algebra for this theory And then the last results That I want to mention So Um, so this is my work And again a one new initial. This is peter tingley This is a story. Um, so we we studied these We studied these these Truncated shifted unions are more generally these quantized Coulomb branch algebras for any for any gigana So we proved the fun first thing that there is a categorical problem gigana action on The direct sound of these category os and this categorifies Well, I categorize or The representation it categorifies depends on these parameters that I suppressed but it usually categorifies the tensor product representation Thank you for the generic parameters of course tensor product. You can also categorify other representations um, and so this works for any g gamma and the second result we proved is that this category o Is uh, it seems like a dual To the category o for the quantized Coulomb right and the while the key to proving both results um We We related this category o to modules For uh Kovanov lauda ruke webster algebra Okay, so to some category of modules over this diagram some diagrammatic algebra And this diagrammatic algebra was already linked to categorification and linked to these quantized covariates and um Okay, I guess I stopped there. Sorry for rushing through the last results Any question or comment I had a question about this shifted the young and I saw you say that the g gamma was a finite type So is there makes sense to ask similar question when it's a fine type? Like a fine type a something like this um, yes um There there's um, yes, um I'm trying to remember what I know or don't know I think that I mentioned that the way that this isomorphism works is that both algebras act both this um Coulomb branch algebra and and the shifted hanging algebra they act on by different operators here And they have the same And that's how we identify them And I think if I remember correctly that in general it's this truncated this shifted yang will have a sort of smaller image than this bfn algebra at least um, this is some Thing we discussed a little bit with alivia At some point um that maybe one of these bigger we should use some bigger yang And and alex weeks thought about this a bit, but I think there was no Like definite conclusion here So I think there may be some generalization this is all but I haven't thought about it recently and I don't know if anyone has so um, I think it's not completely clear which Kind of yang and you should use to generalize this result outside of finance it Okay, thanks. Any other question? Yeah, I'm in the q&a you have a You have two questions in the q&a Okay Are these shifted the anions related to the usual union some cases like those can be defined by rgt relations. I mean Yes, I mean in this finite type case um, maybe I should have said this before that y0 is just the usual union so, um It's in the finite type case. We just have a the usual union and if mu is dominant Then why mu is a subalgebra of y0 Of what which is just the usual union, but if mu is non-dominant then it's Something it's not a subalgebra um So continuing to sorry I said another and and eugenius is a related to molecule kukov yang yin I mean in the finite type case again, this is the only one I think there's only one And so it's the the we can use what I mean these y0 is just the usual union to find however you like um If if mu is not dominant, I mean in the finite type case When mu is not dominant these shifted the anions are not the usual union They're not some other buddy's union, but they're nothing that complicated. They have a pretty explicit description by generous relations is generalizing the Drinfeld new presentation of the union In outside of finite type again, I don't know You know, uh, alex thought about it when we talked about it with maybe other people Um, I don't know how to generalize this whether these molecules could be useful in general outside of finite type There is another question If you change the cool What is the quantization? Uh, I certainly never thought about that. I can answer the question if you change the cosmology theory to k theory So if you've changed it, um everything here, there's a there's a almost everything goes through if you've changed things to k theory There's a k theory ready cool own branches Which end up being um well So so this this version of what happens when you go to k theory was studied a lot by um finkelberg and finkelberg uh zimbalook In fact, I remember the last time I was at ihs I was in the hallway just outside the room where you guys are and talking to sasha about this um So sasha zimbalook that is so if you work, um, if you do everything in k theory We'll just back up to this step Let's go to this step. So this part Can be done. Um, you can change things to k theory and uh The coolant branch makes sense coolant branch algebra makes sense And the corresponding coolant branch can be identified in the in the finite type k's can be identified In finite type and dominant case can be identified with a slice in the affine flag fountain And the corresponding coolant branch algebra can be identified with instead of a truncated shifted yangon with a truncated shifted um Quantum affine algebra, so instead of truncated shifted yangons here So this step here, so this, everything here goes through in the K-theoretic setting where the shifted Yangons are placed by quantum affine algebras. And again, this is like some long papers of Finkelberger and Symboluk. What happens in elliptic case? I have no idea. So in the K-theoretic case, is it non weather if there is a microcalor structure on the places? I think in some special cases, yes, but in general, I don't think no. I mean, even in the non-K-theoretic cases, also, I mean, in general, it's conjecture that these Coulomb branches would have, in general, would have a microcalor structure. In the conological case, there are also moduli of manifolds, so maybe... Oh, okay, sorry, sorry. You're right. Yeah, okay, sorry. So, yeah, okay, so if mu is dominant, then I think it's known that they have a microcalor. And I think also in the K-theoretic case, I think that there is a, okay, I'm not completely sure. I think it's known that, yeah. If mu is dominant, then it's fine, it's fine, it's fine. Any further question? Let's see. No? Okay, let's thank Joy again.