 Thank you for inviting me. So again, so I want to just maybe try and go through a few of the things Phil talked about, but maybe like in a little bit more of a concrete way. He sort of like gave you like a huge overview of everything to sort of try and convince you that it was interesting. And I just want to try and describe like something that you could that you could use it to do. Now it's not something that like that has been completely that we know how to do all the details, but I at least like want to try and describe like what the like what the program would be. So what I want to try and do is explain what how one can mathematically describe the physicist's argument that the following two 3d and equals four theories are 3d mirror to one another. And the mathematical statement that I'm going to take is my definition of 3d mirror. So both of these two theories are of the form t star of some vector space mod g. I'll say exactly which vector space is in two seconds. And this other one is a theory of the form also t star v mod g for a different vector space. And remember that the statement that Phil wrote for our conjecture for like a conjecture of like a different way of stating some plectic duality than would have been stated in the past is that one can look at the quasi coherent sheaves on the space of Diron maps into one of these spaces or into the base of one of these spaces. And that that should be the same as the category of d modules on the space of ordinary. Sorry, I think he was writing his disc with an O like that. Maps from the punctured disc into v shriek mod g shriek and that these things should be identified. And then vice versa if I swapped, if I did Diron maps over here then I should get the d modules on the ordinary maps over here. So this is my statement for like what I mean by two theories that are both correspond to cotangent bundles being 3d mirror. Let me actually just emphasize one fact that Phil did not. So this statement is a mirror symmetry of loop spaces. Right, so like one way of like thinking of this statement is that we're trying to say that symplectic duality like as formulated by Brayden, Proudfoot, Lakota and Webster can actually be deduced from a more fundamental statement, which is a mirror symmetry of loop spaces. This is a B model on what I would call a Diron loop space. So basically, and this is a A model on an ordinary, on what I would call a holomorphic loop space. So if you only like remember one thing about this conjecture, it's that symplectic duality, which I'm not telling you what the old conjecture was. There's a better conjecture, which should be about mirror symmetry for loop spaces. And the important fact is that the Diron gets exchanged from being on the, so here these are sheaves with flat connections on ordinary loops. And then these are ordinary sheaves on loops that have to be flat with respect to some connection. Can I ask you an unknown question? In your talk, are you going to distinguish quasi-garrant sheaves from anything? No, yeah, so basically like nothing that I'm going to do is smart enough to distinguish any functional analysis problems, because basically the physics isn't strong enough to, like physical arguments aren't strong enough to answer those sorts of things. And then in like the mathematical model of these physical arguments, there are places where you get to make choices of like different functional analysis and it can't tell you which ones to choose. So you have to do that manually, like by like actually checking cases. So yeah, so basically it's a problem with this. This is like a thing that happens in ordinary geometric colonization, that there isn't any like way to solve your functional analysis problems abstractly from a physical reason. Like you actually have to do a real analysis on function spaces in order to make things work and similarly the physics is not strong enough to do anything like that. Incomplete categories of physics don't talk to each other. Yeah, it's just like it's just not, yeah. So anyway, so nothing I'll say is smart enough to see that. Okay, but I'm actually like not going to go deeply into an example of this statement. I'm going to try and describe a way that you can deduce statements like this from local Langlands. So you would have to prove some things in local Langlands first and then it would let you get. So you're going to go in this direction. So I mean I usually like to go in the opposite direction going from this to local Langlands. Yeah, so maybe that's, so maybe I'm not going to buy myself anything. But well, we'll see. Okay, so both of these two theories are both type A quiver gauge theories. And so like what I mean by that is like when you write like some diagram like this, what you're describe what like a physicist means when they write some space like that is they mean some 3D theory whose fields are representations of some type A and quiver with dimension vectors v and w. And then the v's are the, and you mod out by gl of the v's. So the v's are what they call the gauge nodes and the w's are what they call the framing nodes. And then you take the cotangent bundle. And so any theory that's like of this form where it's like a cotangent bundle of some vector space mod of a group. We call these theories of cotangent type. And for a while I'm going to talk about these but we'll have in order to do this we're going to have to leave this class in a little bit. Yeah, so basically the vector space is often called the matter in the theory in this group GLV is often called the gauge group. Okay, so let me just tell you the physicist's proof. And then we're going to spend the entire rest of the talk just trying to understand what unearth this said. So the physicist's proof would be like take your quiver, turn it into brains. Then you use S duality and it exchanges one kind of brain with another brain. You notice that the crosses are getting exchanged with the lines, with the vertical lines. And then they would say there's something called a Hanani-Witton move that lets you move brains through one another. And if you just do a whole bunch of Hanani-Witton moves eventually you'll end up with this weird configuration at the end. You can turn that thing back into a quiver and thus these two things are 3D mirror to one another. And I remember the first time I ever heard this argument I was Skyping with a tutor we had just barely met and I nearly threw my laptop like off the table and started shouting at him. But now I'm going to say this argument to you. Can I just ask you a question? So this whole thing is supposed to work only on some very strong assumptions. Yes, so this is only going to work for quiver-gauge theories. Yes, I mean usually we're talking about quiver-gauge theories, not a quiver-gauge theory. And so somehow secretly assuming that we have a quiver such that it is, such that it is for example quiver-type. So where exactly, in this picture, where exactly? It's only in this last step. So actually what we're going to do is we're going to learn how to make sense of the mirror to quiver-gauge theories that aren't of cotangent type. That aren't also quiver varieties. Like that's actually like the real point of the talk. So we'll actually make sense of all these intermediate steps. And so that means you have to talk about theories which are not of quiver-type. But in order to do that, there's a certain category that you have to find in the local Englund's category, that once you find it, then like you can make sense of all these pictures. Okay, so let me just describe the first step to you. This part is like purely translation. Like there's no math in this. It's purely like a dictionary. And so let me just tell you the building block brains. So these horizontal lines, if you have a stack of n of them, they call these things D3 brains. And the only important thing that you need to know about them is that they carry a copy of 4Dn equals 4, super Yang-Mills with gauge group GLN. And 4Dn equals 4 is the theory that geometric Englund's is all about. And it's the one that, one of the ones that Phil was talking about this whole time. And then the other two building blocks are called the NS5 brain and the D5 brain. And these are interfaces between GLN and GLM, super Yang-Mills. And by an interface, as Sasha like mentioned the other day, I just mean a boundary condition for the product theory. There's like a folding trick that lets you change one for the other. Okay, so these are my building blocks. And then now if I have a quiver with the dimension vectors v and w, then the way that you put this thing together, oh, I was a little bit, I was a little bit not quite careful enough. So what you want to have is a pair of, yeah, a pair of, oh no, I was good. So you want to have like a pair of NS5 brains, those circles with a cross in them for each gauge node. So each of these nodes with a vi in it. And then stretched between those two NS5s, you want to have vi of the D3 brains. So like in between my first two NS5 brains, I've stretched V1 D3 brains. Between the second two I would stretch V2 D3 brains, so on until at the end I would stretch Vn D3 brains. And then for each of my framing nodes, like so I have a framing node attached to this gauge node V1, then I would put wi of the D5s. So right now I'm just telling you some recipe. I'm just telling you the recipe for like translating between a quiver picture and for these pictures. There's no math here, it's just like what is the, like how do physicists translate this picture? Can I just ask, so I should picture this as like R4 having some like 3D walls? Yeah, that's exactly what's happening. Yeah, so what we're going to see is that these, yeah, so you should think of the direction, you should think that there are two directions coming out of the board. Like so we're looking at everything from the perspective of these D3 brains which are some four dimensional objects. And you can see, oh no sorry, you should think of three dimensions as coming out of the board. So then at each of these NS5 and D5 brains you should think of three dimensions coming out. And so these things are going to be 3D interfaces between a bunch of 4D theories. So you can see that when a physicist writes this thing down, I'm actually doing a lot of my talk a little early. Like what they're like trying to write down is you can build this 3D theory as a composition of interfaces. Like that's what they're telling you. And so if we can identify these building block interfaces then like we would really be cooking with gas. Yeah, so okay. So then as Sasha was just saying, like not every brain configuration like actually comes from a quiver. Like basically if you were to draw this first brain configuration, like you would notice that there's no NS5 brains in it at all. And so like I don't know how to turn that thing back. The other thing is when I was doing all of these, this physics argument in my very first step, I took my quiver that corresponded, sorry, took my set of brains that corresponded to this quiver. And I did s-duality to it. I ended up with this other weird looking quiver that kind of looks like a tie fighter with like a lot of extra bombing pods or something. And this one like does not correspond to any quiver gauge theory because it has d5 brains just sort of hanging free at the ends. And so the point of all these moves, which like are actually like not necessary, but it's just like a good motivation, is to put this is that you can actually like change this for like equivalent configurations. Like you can basically slide interfaces through one another. There's some relations between interfaces. And then if you use those relations, then you can turn this thing back into a- back into a quiver. So like if we reverse this recipe, you'll see that I should have three gauge nodes. Like I have, you know, three spaces between my NS5s. So I should have three gauge nodes with a one or two and a one in it, which is what I wrote. And there's two d5s there. So there's two d5s here. So then there's a framing of two there. Okay, so the point is, is like these things like are not actually quiver gauge theories. And in fact like they're not of cotangent type. Like they're not theories like these. And so at the moment like nobody really had like a very good way of talking about these things. But at least for the type A ones, we'll be able to do it. But one thing that I will say is these configurations have actually been studied a lot under the name of like turkish bow varieties. So like basically like the, so Nakajima I think like always like puts in an extra mirror, but like the Higgs branches of these things are the turkish bow varieties. But well, and also the Coulomb branch will be a different turkish bow variety. But these turkish bow varieties, they're basically like he was literally trying to make sense of these pictures. And that was literally the problem he was trying to solve. And then he wrote down like some varieties that were supposed to be the Higgs branches for these things. And like what these things look like, because I'm not going to talk about them very much more, is they're basically a mix of like a combination of quiver data like you've seen. And then you also add some extra stuff about solutions to nom's equations on like certain intervals. And then you mix those two things together. And like these d5 brains like they end up being about nom's equations. Like that's like the missing thing that we didn't have. Okay, so now like I want to know like can we interpret like the rest of that argument mathematically. And so the answer is like, yeah, you can try to at least after you do a holomorphic topological or a topological twist. So then that puts you in the world that Phil was talking about earlier where you have shifted some plectic geometry. And he talked about like how boundary conditions were just Lagrangians and shifted some plectic stacks. And then, yeah, so in particular like we're trying to look for these Lagrangians in the equation of motion for 4dn equals 4. And then depending on what problems I'm interested in answering, I might want to compactify that topologically twisted theory on either a curve, either like a closed proper curve, maybe the bubble. Like that's something that's like come up today like Phil drew his bubble where you take this thing and then glue them together along the punctured disc. Or possibly just a punctured disc. And so for this thing, all the way that we found this conjecture was by compactifying on a disc. And so like if I want to answer this question, I'm going to want to compactify on a disc. But there's actually like interesting questions in global geometric Langlands that one can attempt to answer by looking on this thing on like a global curve. In particular, BFN have defined some certain sheaves on the affine-grasmanian. And like if you want to globalize those sheaves to a global curve like then you would compactify on a global curve C instead of the bubble. Also there's like the mirror to these sheaves which nobody's described yet that we'll give a description to. Sorry, I mean they know the mirrors on the bubble but I mean like the global case. Okay, so now what I want to try and do is just tell you a little bit about the twist of 4dn equals 4. So as Phil mentioned, there's a holomorphic topological twist and then there's an A twist and a B twist. But one subtlety is that there's actually two versions of the holomorphic topological twist. Basically they use the same supercharge, but it turns out that you need to use different twisting homomorphisms depending on whether you want to deform, sorry. In 4d there's no problem. You can use the Kapustin-Witton twisting homomorphism and go to either A or B. But if you want to do a twisting compatible with a 3D theory on the boundary, then the Kapustin-Witton twisting homomorphism is not compatible and so there are actually like two different twisting homomorphisms that you need to use and one of them will let you deform to the A in 3D and the other one will let you deform to B in 3D. So like there's going to be some slight subtlety. So I've been talking all this time about like twisting homomorphisms but let me just tell you what that means mathematically. The consequence you'll see of two different twisting homomorphisms is that the fields will be globalized differently. Like so you need the twisting homomorphism to know how to move off of flat space to curved space. So basically like things will be like sections of bundles of the same rank but different bundles depending on the two versions. And then the other thing is that the homological gradings will end up different. Like they'll be different twisting datum that are compatible with the supercharges. So does estuality swap the two? Yeah, estuality swaps the two homomorphic versions of the two homomorphic topological twist. So let me describe the first one. And so this first way of writing it is maybe the least nice way but it's the one that is nicest when you try and look at Lagrangians. So what you're gonna do, so first let me just notice, so if you look down here that C star acts on the one shifted cotangent bundle of BG which is just the quotient stack G dual mod G where I mean linear dual there. And it acts with weight two on the linear functions on that stack and in particular it acts with weight two on the symplectic form. So there's an action like that. And now once you have that action then you can, if you have any space with a C star action well then your square root of the canonical bundle that I've chosen like that is a C star bundle like if I remove the zero section so I can twist the space by that C star bundle. And then I can look at, I'm gonna give me some bundle of stacks over C cross C shifted, sorry my curve C crossed with the complex number shifted by minus one. So it's gonna give me some weird bundle. And then you wanna look at the sections of that weird bundle. And there's a result of, in Ginsburg and Rosenblum basically looked at this example except for without the C shifted by minus one bit. Like they were trying to look at some Lagrangians that Davide defined and they didn't quite describe the fields in this twist but they came very close. And for the purposes that they wanted it was the right thing to do. But to describe these fields like you want this. And it turns out that depending on like what kind of a curve you've chosen so if you chose like a punctured disk, it turns out that conjecturally that is a zero, that that is a, when you choose this canonical bundle that will be a zero oriented thing. This isn't strictly true because like there's not like a good theory for things like speck of Laurent series. Like there's not like a good theory of shifted symplectic geometry and AKSZ that works in this case but I'm gonna assume it will. And then in the setting where you have a smooth proper case where everything is rigorous then this C together with its canonical bundle is oriented of degree, square root of its canonical bundle is oriented of degree one, of dimension one. Then that C shifted by minus one is actually oriented of degree minus one. So you end up finding that you can pull that cotangent bundle through the sections. And then that'll give you this T star of two minus DA on the outside. And then if you think about what maps from C shifted by minus one are well that's just the one shifted tangent bundle. So you end up getting like this equality and the sections turned into just ordinary maps into BG because the C star action that I defined is not acting on the BG part it's strictly acting along the fibers of the cotangent bundle. So that's like one way of describing this moduli space but another word like so if I were to complete this thing around the zero section or the double zero section then this thing would be the same as the dolbo stack of bungee like the V inside piece. And so you can actually like you can actually deform that to bungee dirom. And this is the way in which like D modules like actually show up like when you look at this geometric Langlands like in Phil's formalism is that like actually the thing you see is sheaves on bungee dirom. Like it's and like those things actually look a little bit more if you look in a concrete model they look a little bit more like modules over the dirom complex and they do look like D modules. And like so like when so like if you're a physicist like that's kind of the thing you would see if you like expand and everything out into fields. Okay so any questions about this little piece? So basically the first set of the first shifted symplectic variety is the equations of motion for the holomorphic topological A-twist and then the topological A-twist is the second thing that I've written down. Similarly the B-twist fortunately is a little easier. So for the B-twist you can write that as T star 3 minus dB where dB is some number telling you how oriented the C-dolbo is of T star of bungee of your curve. And then you can rewrite that as maps from C-dolbo into T star shifted by 3 of BG. So now let me just like mention like really fast C-dolbo basically looks like the one shifted the formal one shifted tangent bundle of C and so this is a one dimensional vector bundle over C that I have in the domain and that's the exact same thing that I had like in this other case except for you notice the homological degrees are different. Like there it was a minus one here it's a one. And so that's this consequence of the two different twisting the two different twisting datum and then you notice that in this case I had sections twisted by the square root of the conical bundle and here I only have maps. But now this bundle became non-trivial like the bundle this is a bundle over C and it became non-trivial. So basically the forms got moved into the dolbo is like what happened to things. And then this thing admits naturally another deformation where you can deform C-dolbo to C-dol-rom in the source. And so this will give you a second this will give you the B-twist and you can pull the T star three through again and you find that this is just T star three of three minus DB of flat G of C. And so depending on what you've plugged in either if you plugged in the disc then the dolbo stack will be one oriented you plug a smooth proper curve it's two oriented but the thing that you should notice is that in either case three minus DB is always equal to one minus two minus DA. So like if you choose the same curves you actually got stacks of the same dimension or the same, sorry, symplectic shift. For physicists in the audience if you plug in a disc this is basically the equations of motion for 4DN equals 4 compactified on a circle and then looking at the Rosansky-Witton twist. So any questions about that? Okay, so now like what we were looking for is we were looking for these interfaces between different 3D, between different 4DN equals 4 theories with different gauge groups. So just remember like the picture that we were looking at here. So we had these different brains and these things were interfaces between 4DN equals 4 for gauge group GLN and GLM and these are the things we're trying to make sense of. And Phil told us that that interfaces are particular boundary conditions and boundary conditions are just Lagrangians. So now we're on the hunt for some Lagrangians inside of these stacks that I've talked about before. And so there's two theorems that you can use to produce a lot of Lagrangians. They're not the only two theorems and not the only way that I can get Lagrangians but like two good ones are this AKSZ-PTVV theorem that Phil described. So he just told you what happened when I had a D-oriented space and then some N-shifted symplectic stack. He told you that the mapping space from X to M was symplectic. But the other part of the theorem is that if you have a Lagrangian in M then maps from X into that Lagrangian will also be Lagrangian. And so that theorem will let us produce so this flat G is really just maps from C to ROM into BG, or well actually sorry this, sorry actually that's not important. But the equations of motion that I wrote there for the B-twist like they're a mapping stack. And so this thing will let you produce Lagrangians in there from Lagrangians in T star 3 of BG. But then the second, the A-twist was one of these weird section stacks. Like it's not really a mapping stack. And so Ginsburg and Rosenblum proved an extension of the AKSZ-PTVV result where now if you have an additional C star action on your compatible C star action on your Lagrangian and your symplectic manifold such that the weight of the symplectic form is L then if you have a D-dimensional, sorry where the dimension is computed with respect to the structure sheaf I won't say exactly what that means, stack together with an L-th root of its canonical bundle. So now like I don't actually have an orientation now I have this L-th root of the canonical bundle then it turns out that the section stack will still be well if I do sections into M twisted by this L-th root of the canonical bundle well then that will be symplectic and this other thing will be Lagrangian. So basically you can use this other version to produce Lagrangians in the A-twist. And this is like what Ginsburg and Rosenblum actually made the theorem for us. They were trying to describe something related to the A-twist, but we're kind of doing something just a tiny like an infinitesimal amount different. It literally is just like a no-potent thickening of what they're doing. So do they have a vapor on this? Yeah, it's this one like Gallardo Lagrangians. I mean that if you just search for the column Gallardo Lagrangian but the important thing for you is that if you put in the bubble and you do the A-twist then these things should be giving you your ring objects. Like after you quantize the Lagrangians. And then like the important thing is that they tell you like how to globalize the ring objects to like an arbitrary curve. And then like the natural question is is like what's like the other, like what's the other object, of an incoherent sheaf on low excess from a 3D N equals 4 theory that should be mirror to what you guys did. And like it's obvious what to do for the bubble. Like using Sataki you can just kind of guess but it's a little bit trickier like if you actually want to but if you actually want to describe it on a global curve because the obvious thing to do with Sataki doesn't work. Because like with Sataki right you just use the moment amount of the S-dual space. Anyway, we can talk about it later. Okay. So in summary like what this thing tells you is that if I have a Lagrangian in T star 1 of BG which is G dual mod G then I can get and at C star invariant then I can get something in the A-twist. And in the B-twist all I need is a Lagrangian in T star 3 of BG but in order to make the picture more symmetric I want to actually introduce like an extra C star action on this Lagrangian like on this space. So then there'll be a correspondence between like the two different kinds of data and so what you need is you need to put a weight zero action of C star on G-dual mod G one that doesn't act. And basically like loosely speaking there's a transform of Lagrangians with one C star action to Lagrangians and the other C star action that roughly switches the C star actions in the homological grading. And that's basically the like what changing your twisting datum does. Like when you change your twisting datum it just re-grades everything. And like one of your degrees was homological and the other one was C star and when you do the other twisting datum it swaps everything around. And like and I would be actually like a little bit interested in a better mathematical description of like precisely like what this transform is and like what these objects are. Like in practice like most of the objects had to build both of them. But like to actually describe this like transform in like a rigorous way is something that I wouldn't mind talking to people about. Okay so then one big source of these Lagrangians comes from a theorem of Safronov where he says that suppose that I have some N-shifted symplectic stack and it has a Hamiltonian G action. Then like that should be the same thing as being as having M mod G be a Lagrangian inside of T star 1 of N plus 1 BG where T star N plus 1 of BG is just G dual shifted by N mod G. So you're just modding out the moment by G and that'll give you and the fact that that's a Hamiltonian action will be the same as this other thing being Lagrangian. So in particular that tells me that Hamiltonian G actions give me a big source of Lagrangians because I can apply either the AKS Z construction or I can apply the Ginsburg-Rosenblum construction and basically like from this input data and I'll end up getting two things. Now in the case that I have G acting on some space X well that's the same thing as a map from X mod G into BG and then you could make the Lagrang, well G will also act Hamiltonianly on T star shifted by N of X mod G and that's another way of describing the conormal to X mod G in BG. And so basically the theories of cotangent type with a G symmetry end up giving you the conormal Lagrangians inside of T star N plus 1 of BG. So basically you get a nice correspondence for those things and so that's what this cotangent type hypothesis is buying you when you do this boundary condition thing is it means you only need to quantize a conormal and somehow that's an easier thing to quantize than quantizing like an arbitrary Lagrangian in a symplectic manifold. And so like the key thing that like you need to do is like describe like a quantization procedure for like other Lagrangians and maybe if I have enough time I'll try and explain a little bit how to do it but if not we can talk after. Okay so then let me just say a word. So these Lagrangians that arise from Lagrangians inside of either T star 1 or T star 3 of BG these are precisely the boundary conditions corresponding to 3DN equals 4 theories with G symmetry. But there's actually like when C is a punctured disk there's actually like lots of other Lagrangians inside of these equations of motion like one example is there's something corresponding to the trivial basically there's Lagrangians coming from the surface defects. And so like one example of those would be the inclusion of the punctured disk into the disk induces a Lagrangian in the equations of motion and that one does not come from a 3D theory. And like it's also I don't think it has anything to do with like singular support conditions like I would like a way of like trying to like distinguish like which boundary conditions are surface operators and which boundary which things are coming from equals 4 theories. And like basically like the difference is if you have a 4D theory you could take a boundary condition and then compactify everything on a circle and then you get a boundary condition for the compactified theory. Or in this other thing you're compactifying on a circle and then taking a boundary condition but it didn't come from anything in 3D. And like so one of those things is describing surface defects and the other thing is describing these 3D theories. So there's like sort of like a difference in these things but I'd like something like geometric that like lets you tell these apart. I think that it ought to be something about being like kind of close in some kind of neighborhood of the trivial surface defect but I don't really quite have anything precise to say. Okay, so then now we need, so now we have this way of producing Lagrangians inside of T star 3E or 1 of BG. Like the way that you need to do it is you need to find Hamiltonian GLN spaces or GLN times GLM spaces with nice C star actions. And like one and like basically like I'm not going to actually write them down for you but the like basically the answer to this question is the bow varieties. Like basically Cherkis defined the correct guys for you to write down a long time ago. Sir I'm confused about the numeric system. So you can see the sheeps of cotangent but so now you have this sheep by 1, right? Yeah, so let me just show you, so in this thing if you have a T star shifted by N, the co-norm will get shifted by an extra 1. So that might, so basically like if you have an ordinary space that'll give you something in the one shifted cotangent bundle if you have a two shifted thing which is like hard to which would be like if you re-graded using the C star action as your homological grading like right all the symplectic varieties we're usually interested in have this weight two C star action. So if you re-grade those things to make that the homological grading then you'd get then you'd have a T star two of X and then that would be in T star three. But there's something a little bit tricky like sometimes you could pick a C star action that violates that gives like fermions like an odd weight sorry an even weight and then like that like blows things up and so there's some restriction on these actions even beyond like just being conical or whatever that I don't quite understand. And actually you can compensate sometimes and then that like actually affects like the duality and I don't quite understand the way that that happens. Okay but anyway but the point is that you have these bow varieties. Some of the interesting ones are like if you have an NS5 brain between that was supposed to be an N and an M then that's just T star hom from CN to CM. That's a natural piece when you're building like representations of a quiver. Like you see those guys all the time. And then the interesting ones are when you have these D5 brains and then like one of them like Sasha kind of talked about a little bit in his talk although he was doing a different twist where you have N on both sides and a D5 brain and then you get this T star of GLN times C and then the other one Sasha talked about a bit is like if you have a big step so this isn't a single one but like if you do this full configuration where you have like ND3s ending on a D5 then N-1 ending on a D5 stepping down all the way then that's the regular Whitaker reduction of T star GLN on the right and then you have your GLN action on the left. Again, it's just about the structure of the thing so you have a picture on the left you have a picture on the right you have some kind of variety. A variety with the GLN. So these things are supposed to be some Lagrangians in Lagrangians in like maps from D star into beach you know whatever yeah let me let me yeah yeah yeah so basically yeah yeah so like I have these two spaces like where I make CBD star or C like it doesn't actually matter like any curve and so these are symplectic spaces and then I also have these symplectic spaces and what I told you is that if I have Lagrangians in the target then I can then I can get Lagrangians in the total space like in the whole mapping space or in the whole section space through this AKSZ construction or the Ginsburg-Rosemblum construction and so what I'm saying is that if my input space like if my input Lagrangian is the ones corresponding to those spaces then the Lagrangians that you get are the ones that are supposed to represent those brains like those are the ones that the computer would write down. Like they would write down some Lagrangian in these spaces to be the D5 brain or the NS5 brain and I'm telling you that the way that you build that Lagrangian is it's actually just built from something about T star of bungee T star 1 of BG or T star 3 of BG and then like you do this complicated construction and it globalizes it for you or like it so so there's just a procedure that they have yeah that I described that I described a little bit about here that takes a Lagrangian in a target and then makes a Lagrangian into the mapping spaces or a Lagrangian in the section spaces and so I'm just saying that like so I was telling you in this picture I'm just telling you which things to put in the target you need to think of them with you know some GLN types GLM actions Is it GLM-C or GLM-C to the N? Oh, C to the N, I'm sorry yeah yeah I made a mistake but it looks to me like cotangent type everything uh let's see so here's like something really subtle so they are all of cotangent type or it's like co-normal you're looking at the A twist but when you do this re-grading operation that I was referring to that there's like something about like when you swap the homological and the C star gradings something that was of cotangent type might turn into something that's not and so like basically the B side D5 brains are not of cotangent type and like that's like the sort of like the big like the big obstruction to like making sense of this picture and that's why you have to invent like a form of geometric quantization for shifted symplectic stacks that will do Lagrangians that are not of cotangent type and so like this is like the main like identifying like that object is like the main problem that we're interested in um so unfortunately right these stacks so there is like well it's very much in its infancy like there is kind of like a theory of geometric quantization for and shifted symplectic stacks that's in the process of being built but it does not apply to something as infinite as like when you put in a punctured disc and so even so like basically like like there is kind of like a categorical geometric quantization procedure like for finite dimensional things or maybe for like locally almost a finite type things but like if you want to do it for something as big as that punctured disc like like there's nothing off the shelf that's going to work um and even this and even the finite type thing is like hasn't even been described in the paper like just Pavel's told us about it sometimes well there's a paper of Walbridge that does part of it um okay alright so now I want to try and talk a little bit about this categorified geometric quantization I had originally like written a different talk where that's all I talked about for the whole hour and like still like I barely got through it so this will have to be really brief and really it wasn't like it's not exactly like my idea anyway it's like some mix like of like there's some stuff that's like pretty obvious to do and then like there was one trick that Pavel told me um that fixes everything but the roughly the idea of this categorified geometric quantization is it's supposed to be some way of taking an n-shifted symplectic stack and producing a c-linear infinity n category for cotangent bundles Phil told you what it did the hard part of this problem is the second part which is if you have a Lagrangian how do you actually build some object for the Lagrangian that lives inside of the category corresponding to the manifold um okay so well in particular let me just tell you where things would live if you could do it so Phil told you like if you have a two-shifted cotangent bundle then you just get sheaves of categories over your over your base um and then there's some papers of Gates' Gory where he at least produces a functor into the category Sasha was talking about which is either categories with a strong action of um the loop group or categories living over quasi-coherent sheaves um but there are different functors actually in this case so one of them the second functor is taking global sections of your sheave and the first one is actually taking the fiber over the trivial bundle so like these realizations into categories are like in are like different looking realizations so that actually like makes it really confusing sometimes to think and so that's why I prefer to think in this other model where things are uniform um but it's also true that like there's a bunch of like hard functional analysis issues that people sort of know how to fix these bottom two categories and I don't think anybody has really any good idea of how to fix them in the other like in the other realizations but just for um this is also what the physicists would say they would say the sheaves of categories thing um like because that's what Rosanski said in this paper on Rosanski wooden theory um and then yeah I should get a Lagrangian in that category um and in particular if you get a Lagrangian correspondence you want to try and produce a functor like that's the kind of thing you want to do and then let me just make the statement of estuality and then I'll try and come back and tell you something about how geometric quantization works so the statement of like what estuality is is that like if I have um like the d5 brain there are two different objects that I can produce from it like an a side object and a b side object um by through this geometric quantization procedure uh the only one that that is hard is the b side d5 that's the only one that like we don't know already um and then uh these things are functors between sheaves of categories on bun g l m and bun g uh either bun g l m durom and bun g l m durom bun g l m durom or loxess g l m and loxess g l m durom uh sorry g l m and they're supposed to be intertwined by local langlines like the like if you use the local langlines equivalences on both sides you'll get a commutative square intertwining like the functor on one side with the functor on the other so like the big problem is to identify like what on earth is the geometric quantization b side geometric quantization of the d5 brain like that's the only object that people don't know or that like isn't like within reach right now um and then let me just say really briefly like this Hanani Witten move that I was talking about it's not very important what it is it's just some statement about like an equality of compositions of functors like in the local geometric langlines category so it's something that can be checked like I mean like I don't know that it's true so is it some relation inside the group SL2z uh no it's it's just about like moving these brains through one another like it's basically just like if you like write one configure I'll write it down for you in a minute but it's just something about like letting you like I see in that first picture I just took my first brain I just moved it all the way through to the middle but then I lost a then like I lost some d3 brains when I moved it so like uh so like basically like what you should do this picture is I'm supposed to be taking that d5 brain and moving it to the next five and somehow that can't kills some of the d3s and what this is supposed to be telling you is that each of these configurations is just telling you some composition of functors and like the Hinani Witten move is an assertion that those two compositions are the same and like Nakajima has actually already proved something sort of similar like this at the classical level about the bow varieties and if geometric quantization actually commuted with composition of Lagrangian correspondences then the classical thing would give you like the the quantum statement for free like I mean that's like very far off like that's not like going to be a good way to prove it you could probably find like a way of proving it directly like much faster I'm just saying that morally like if one were trying to like build the program that's going to take like five years or whatever like that's what you would do is like you would try you would try and prove this statement okay so now well let me just say really quickly there's a similar statement relating the 3dn equals 4 gauge theories and the 2dn equals 2 2 theories with g symmetry on the boundary and like Telemann has talked a lot about this in his ICM and like Sasha has like some results about like computing the quantum the equivalent quantum cosmology of like the flag variety this way and I have some results about some other in terms of Coulomb branches and like that can all be kind of like interpreted in this picture but it's the same exact type of argument it's just about a field theory in its boundary and this is kind of what I've been working on more with Tudor okay but now let me try and like make a statement about like what geometric quantization of Lagrangians is supposed to be okay so let me just like tell you some recollections on geometric quantization in general in the ordinary setting so you have M being some zero shifted symplectic manifold and I really just want to give this being smooth and let me notate the symplectic form then in order to do geometric quantization so in this case it's supposed to give me a vector space there are two extra pieces of data that you need to choose so the first thing that you need to choose is a polarization actually let me do the pre-quantization first so the first piece is something called a pre-quantization and this is just you're supposed to choose some line bundle with connection such that the curvature of the connection is equal to the symplectic form the second piece of data that you're supposed to choose is you're supposed to choose polarization and this thing is and I'm only going to talk about real polarizations because that's the only kind I would need to use so this thing is just some Lagrangian foliation of the tangent bundle of M but let me give a definition of this that could possibly generalize so what this thing is this Lagrangian foliation is it's just a algebraoid that includes into the tangently algebraoid and then this oh sorry the foliation is on M the distribution is in TM yeah sorry this thing is supposed to be into the tangent bundle of TM no tangent bundle of M sorry that's right and what this thing does is that you can take the Dirom complex of M which is the Chevelly-Eilenberg complex of this the algebraoid and the map Sigma induces a pullback map the Chevelly-Eilenberg complex of F and then your symplectic form is something in here and the condition that this thing be Lagrangian is just asking that this pullback be equal to zero so your map will take this to this okay and then now let me just really quickly tell me tell you how you change this like if it's N shifted so this isn't due to me this is due to well the pre-quantization part is due to wall bridge Lagrangian Foliation stuff is I think an unpublished work of Tony Panta Damien Klok who are your other two collaborators on this that can be done okay so those are the people who have been working on Lagrangian Foliations and the derived setting so they have worked out like what the definition of Lagrangian Foliation is and basically the only change is it's still a Lie algebraoid but in the derived setting it doesn't make any sense to ask for a map to be injective so you just have to say it's just any Lie algebraoid with this property so you can still make sense even if this is a derived stack you can still make sense of the derived or on complex and then you just have to upgrade these things to being from the stacks of closed forms like and then change this to being a homotopy and so everything is just fine in like the n-shifted symplectic setting and then the part that's due to wall bridge is like how do you change this pre-quantization and like roughly like roughly you change this for a gerb with connection with a so the right thing is that if you're n-shifted you want to get an n-gerb with connection I can say a little bit more about like precisely like what that means but let me just sort of not and so there's some sort of higher curvature map for these n-gerbs and you want and I guess like you want it to have a two the numerics are a little off for me maybe you want it to be a one one connection so like basically like there's higher like you can choose how many forms you want in your connection and then you want the curvature of that gerb with connection so there's like an n-curvature and I want that to be your symplectic form so how if you want a compatibility between your pre-quantization and your polarization you just need to lift this path between basically you can make sense of a kind of co-amology which classifies these n-gerbs which is a small modification of this Chebley-Eilenberg complex and you just have to ask for a lifting of this equality or this path to a path in that other complex so that part's not very interesting sorry I mean it is interesting but that's not the not the part that I think is exciting but let me just really quickly tell you what the geometric quantization would be in this case so the geometric quantization would be with all of this data is you can take m and then mod out by your foliation the fact that this thing has a curvature this connection this gerb has a connection that is your symplectic form and that connection is flat along this condition that the curvature is zero along the foliation tells you that your bundle is flat along the foliation so that tells you that your gerb actually descends to this quotient space so let me call my gerb g with connection and what you want to do is there's a canonical sheaf of quasi-coherent n categories on this thing which I'll call n q-co which isn't really very complicated to define and then you want to twist that by the gerb and take the flat sections and this is the definition of the higher quantization of a symplectic manifold with a foliation and a pre-quantization this part is not very interesting if you have a zero case this reproduces the pre-quantization that you would have had before the part that's really exciting is the following so if you have a Lagrangian inside of your manifold m let's call this i then if you pull back your foliation along i to l turns out that this thing is actually a symplectic foliation of one lower degree of one shifted symplectic foliation so these Lagrangians okay I'm about to finish I can't do the whole story and the point is this thing you should think of as just a family of symplectic manifolds of one shift less and then one can polarize that family and pre-quantize that family comparably by quantizing that family like that is what is supposed to give you the the sheaf like the fibers of the sheaf so like the key example is like if this thing is like a one shifted cotangent bundle of x mapping onto x sorry let me do a one shifted cotangent bundle is fine so then you could have a Lagrangian here so the composition of this Lagrangian with this projection map that is actually like a family of symplectic manifolds living over x and if you geometrically quantize each fiber that's supposed to give you a vector space and if you take one vector space for each fiber that's exactly like a quasi-coherent sheaf on x which is what you wanted to get now like actually like going through and like making this like real is very very difficult and you need like conditions on your Lagrangians to be able to do things and there's functional analysis problems and it's really gross but at least like the idea seems somewhat promising and I guess like the biggest thing with Phil that we've been doing is in the global case there's a few Lagrangians that you can actually try this procedure on to see if you get things that the geometric langlines people have seen so in this corner that you said you isn't known there were like more things instead of on the disc so on the disc like there's no chance of any of this working the technology is not there right now if you do it on a global curve like things become like rigorous and still like there's not a knowledge of what that object is or maybe there is but I don't know it this gives you an onsots for guessing what that object is and one could try and like check that with what's known about the global sorry known about the global correspondence and that and really the reason I think it's exciting is because it's a test as to whether categorical geometric quantization is nonsense or not I mean Pavel has a lot more tests so he knows it's not but this is like a hard I mean this is just like something you could try and see whether it actually works in that case the 3D 2D thing is it compatible with the Telemann proposal or it's slightly different very close he mainly actually talks he doesn't compactify on a circle first and he also like doesn't he does so we're doing like this diram thing he does a Betty version so there's like two ways in which our thing ends up being different than Telemann but I mean like he wouldn't be surprised