 Thanks for the invitation, thanks especially for giving both of us a chance to speak on the same topic so we can actually get some air. Also before I actually start my talk following Emily, actually I want to advertise to our two workshops that I've been co-organizing. This workshop, I mean, both happened like this year. This workshop is to understand conformity theory and the relation to vertex argebras and chiral argebras. It's mostly for say graduate students and this one is kind of trying to understand general KFT formalism from a mathematical perspective. This is mostly for researchers, but everyone is welcome to apply for the workshops. Okay, so as I said, everything I'm going to say is joint work with Justin Hilbert. Right, so and we are trying to relate these two different subjects, but these two subjects are pretty big subjects themselves. So my talk is basically going to be introduction to sort of the general ideas and Justin will talk about some kind of more concrete example as well. So it's going to be consisting of several different levels of introduction. Okay, so let me start with a big picture. First of all, we want to relate these two subjects. By the way, maybe I should say that these two subjects in a sense belong to this field called geometric representation theory, but somehow I want to emphasize sort of the relation to physics for this talk. So I'm going to certainly say words from this field of geometric representation theory, but even if you don't follow that part, I hope you can sort of understand some ideas, general ideas of the subject that we are studying. So we want to relate these two subjects. And first idea is to use physics. And the relevant physics are what is called 3D N equal 4 theory and 4D N equal 4 gauge theory. And there have been some work relating these subjects. Let me just mention some of them. So blue things are mostly physics work and reds are mathematical work. So these blue works are kind of finding some relation between these two different looking subjects, one mathematical subject and one physical subject, using physical ideas. And these red ones are giving some kind of mathematical way to understand these physical objects. And these red ones actually have found something that these physicists didn't really realize. So it certainly has some kind of the, it's not just kind of translation, they have more to say. So that's the first thing. The second thing is that to relate this in particular attention to Langmans, the idea now is to relate these physical theories. And Gayot and Witten have related these two physical subjects. And then the natural idea is of course to put like red box somewhere here, so that as a corollary we are going to get some relation between these two. Theory means work theory, guess theory or work theory. What is this theory? Not necessarily. So what is the theory? I mean, this is the first introduction. Essentially my talk consists of like four different levels of introductions. Somewhere along the line I'm going to say what I mean by theory, kind of slightly better than this. So okay, any other questions? Right, so we found sort of the relation between these physical theories. And as a corollary we some got, get some relations between these two metal subjects. And Breberman and Finkeberg also found some relations between these two subjects. So this is I think the end of the first introduction. I think this was clear to everyone. So let's increase the difficulty a bit. It's hard to say, like I don't know. Okay, so I want to say a bit about what we are doing actually. So first of all, let me say that there are several, like vocabulary we need to work with. Quantization just means, I mean, given sort of input data, I think of classical field theory. And for classical field theory I care about, say, solutions to equations of motion. And the second vocabulary I want to explain is this notion of twisting. It means that starting with supersimplified field theory, there is this procedure called twisting, which is supposed to be extracting some parts, a sector of the theory, which is kind of simpler, but more manageable. And the famous example is this 2D n equal to 2, it has two distinct topological twists, called the A model and B model. So given, say, Calabria main for X, it defines sort of two different topological theories. And I can denote that by AX or BX. And third, duality. By duality I mean sort of the identification between two different looking QFTs. So anything you can kind of reasonably extract from QFT should be identified. That's roughly what I mean. So the example here is the mirror symmetry. The statement is that, for given Calabria main for, there is a dual Calabria main for such that this AX is basically the same in this sense to this BX chart. And examples, this grow within the variance, like carb counting variance is same as some kind of periods of original hot structures. And kind of, there is another kind of stronger conjecture due to Konsebic homologation symmetry. It's expecting some equivalence between two categories. Could you clarify what you mean by quantization? Not here. So it's again only the second introduction. Let me get there. Okay, so from here, maybe starting some less dark thing. So I describe this procedure of twisting and say quantization. And what I mean by that is precisely, I guess, fitting in this kind of table. So from here to here is twisting. From here to here is quantization. And our interest is to understand the relation to, say, geometric representation theory. And this subject is mostly algebraic. And supersymmetric theory, physics kind of starts with, say, differential geometric data. And usually, as given, it's not really kind of the, I mean, it is of course relevant, but it's hard to kind of extract this algebraic information right away from this given field theory data. And if you do this twist, for example, topological twist, then it becomes simpler. And sometimes, it can be understood in an algebraic way. And that kind of fits into this subject called shifted symmetric geometry. And quantization, I could mean a few different things, but in this case, I mean some kind of geometric quantization. And as a result, I expect to see some sort of topological quantum field theory. Okay. Then, given this, one can kind of proceed as follows. First of all, we understand classical field theory, namely, try to identify the modular space of solutions to the equations of motion. And claim is that in nice cases, including ours, our cases or our interests, can be understood in terms of shifted symmetric geometry. Second, understand sort of this procedure of quantization. This kind of is going to be categorized version of geometric quantization. And we are not the first to think about this subject. What is EOM? I mean the modular space of solutions to equations of motion. But what do these letters mean? Equation of motion. Equation of motion. Okay. It has been done by several people, Chris Rogers, Pavel Saffronov, Ross-Reiber and James Reberich. And we have learned a lot from Pavel, actually, in particular. Okay. And the third part is once you understand this, now you can try to understand the consequences of the duality that the physicists studied. And if you apply this idea, then you are going to find some new conjectures and one can try to prove them. Okay. So in a certain sense, our sort of the main contribution is more to making a framework for this one and two. And I think in a sense, I think Sasha's last talk kind of is more like kind of the third part. I think the consequence of duality find the conjectures and then prove the conjecture. Okay. So in my talk, I'm going to sort of give basic ideas of these two different subjects and say a bit about like what kind of thing is happening from this general picture. Okay. I think this is the end of the second introduction. And I think this much is also kind of fine. I heard duality is extremely general. Can you restrict the meaning of duality? I mean, it's hard to say somehow. I think for us, we are going to work with a particular sort of the example of duality and I think that's going to be fine. If you have, I don't know, I don't know what kind of answer you want. Is a lecture magnetic as duality in this context? Yes. Yeah. So I don't know. Any other questions? Did you define what is simply duality? Yeah, we'll come later. Yeah, yeah. Maybe we'll switch on the point. Thanks. Okay. Any other? And of course, Justin is going to explain more about. Oh, sorry. Okay. Okay. So let's just cut up the, let me just mention that what's kind of new from our work. We certainly find some new relationship between these two subjects. And along the way, we study sort of the, say, line operators of the 3D and 4D theory and forthcoming work of 2.0.10, Nick Garner and Justin Yerberg. It's actually kind of investigating that much deeper. And also we find some kind of enrichment of simply duality. But I'm not really going to talk about that. And we also find some new conjectures for local regimented longlands. And like kind of one example is description of the corner for local regimented longlands. And I would say like the, in the sense like Sasha described yesterday, it's like 99% due to 99.9, due to Gallaudet-Witton. But my understanding is that actually Sasha and I think Yerberg are trying to prove some version of this, like GR2. There is some, I think progress, I think there. Okay. Okay. I can either confirm or deny. Okay, okay. And we also have some kind of exciting examples using some ideas from student theory. And Justin is going to talk a bit about this. And it's, one thing I want to say is that this is sort of not kind of translation of the result. We are kind of in a sense kind of setting a framework of what physicists have been doing in a sense. And I think Justin kind of really explained more about this. I think along, like to explain this string theory and deep brain manipulation, I think he's going to explain sort of what I mean by this thing. And as I said, somehow setting this framework seems to be kind of useful in the sense that sometimes we are seeing something that like physicists didn't really realize or think about. And in a sense, sort of the most exciting part is this categorified geometry quantization. It seems to be working in a much bigger generality than sort of these examples we are going to consider today. But again, we are not really going to talk too much about this part. Okay, so now it's really sort of the beginning of the solid introduction. So, simplified reality. Again, as I said, I don't really expect everything in these slides and to be really understandable to everyone, but that's fine. So, let me start. So, Braden, the kind of proud food webster made a conjecture called the simplified reality. It says that for certain dual pairs of algebraic simplified varieties, M and M strict, there is a equivalence between two module categories, some sort of the module categories of the deformation quantization. Kind of satisfying a lot of other properties. And examples include this Kotanjenmunder of black variety and the dual groups, and also keyboard varieties and slices of fine grass mania. It's a pretty non-trivial duality. And it kind of relates to several different subjects, several different objects in geometric representation theory in this way. But then, like physicists realized, say, Gukov, I think Gukov was the first one, observed that this mathematical duality seems to have to do with some physical duality. So, in particular, sort of the, there is this three-dimensional theory and four-given theory. There are what is called Higgs branch and Kruller branch, and he observed that these pairs seem to be coinciding with these pairs, known pairs. I think this actually appears in also Gajot and Wittem actually before those words in selective duality. Gajot and Wittem discussed the black duality before some black duality was ever discussed. Oh, is that right? I mean, it depends what you mean. Like they identified like the two dual pairs of varieties, but they had never discussed the module. I think you're right. We're readily kind of proud of that, but that's a relation based on Gajot, was motivated by Gajot and Wittem. I'm sure it's not, because Nick didn't know like about Gajot and Wittem. I mean, I'm just saying, like, so nobody had ever talked about those module categories. No, I'm not talking about Gajot, but about the ideal value. By simplex duality, I mean this thing with kind of a lot of The m's and m's streets had been identified before. Yes, that's what I mean. But the module category equivalence had not been protected before. Any other questions or comments? So there's another point of view of 3D mirror symmetry. It's kind of the duality between two different 3D and 4D theory and in this way. So theory T has dual theory T street and Higgs branch and Coulomb branch are kind of the swap under these theories. Okay, so I want to say a bit about this 3D and 4D theory, but just purely in terms of sort of input data and sort of outputs. So input data, 3D and 4D gauge theory, input data is group G and simplex space with the Hamiltonian G action. So for example, given a group G and linear representation V, you can think of cotangent wonder, and that defines a theory, which I denote by T of G V. Then outputs are affine algebraic simplex varieties, Higgs branch, which is defined in this way. So just this Hamiltonian action, you have moment map, so you can define this way. And in this particular case, it's easy to identify this way. And Coulomb branch, this is kind of the foundation work of for everyone thinking about Nakajima, BFL, identifying this as a spectrum of some commutative ring. But this commutative ring is kind of the non-trivial objects. This is sort of the part that I don't really expect people to follow, but let me kind of read briefly. It is some certain kind of the map to affine grass manian, and you take some equilibrium homology. And in particular, when this representation is zero, then this is just kind of same as affine grass manian. And what I mean by this object is just becoming homology of bun G of the bubble. So G bundles on the bubble, and this bubble is defined as a gluing of two disks along sort of the punctured disk. So it's kind of the usual object in geometric representation theory. And because of the geometry of bubble, it does have usual convolution product. And using that, like you can prove this is commutative, and then you can take spectrum. And then it gives a fine variety. And that's the definition of Coulomb branch. So that's that. And once you have this definition, then you can try to prove this simplicity. And indeed, Ben Webster proved this simplicity in this case. Say for the case when T is TGV, even as, in this way, with this input data of G and representative V, it's implicitly your list of words. Okay, so let me say a bit more about sort of the physics of what's going on. Hex brands are sort of parts of what is called the modular vacua. And modular vacua is kind of the hard object to describe in general. But the point is that, as I kind of hinted, our interest is kind of at the end of the day kind of the algebraic things. So presumably, like we can try to understand this Hex branch and Coulomb branch after some kind of the twist, right? The relevant information should be sort of the visible for after this twist anyway. And twist is going to make things simpler. So why don't you just take twist? And what one means by twist in this TQFT is kind of clear. You can define that modular vacua is roughly a sort of spectrum of some commutative ring, which is given by what you assign to n minus one sphere. But now you shift in your sort of perspective on what it feels like. Now you're using a sort of functorial? Yes, yes. I mean, it's still an introduction, sort of level, but yes. So TQFT is kind of functor. Yes, yes. Z is a functor. So this is going to be a vector space. And it has some algebra structure coming from this configuration of spheres. And I think of that as computer product and then takes back. Okay. Then it is known that this three-dimensional four theory has again two twists, A-twist and B-twist in such a way that if you compute kind of the modular vacua following this description, then A-twist is exactly giving sort of the Coulomb branch and B-twist is giving a Hex branch. And indeed, we can kind of argue following this physics expectation, you can see sort of the Coulomb writing Hex branch. So here comes... You said earlier that somebody needs this reward. So where is the ZA and ZB written down? Who's described? No, it's not really sort of like written down in this way. Yeah, I understand this kind of filtered algebras. This algebras, which you consider it's not, it's kind of limits on finite dimensional spaces. First of all, this is like still not like what we are actually doing like for many different reasons. So we don't necessarily care about like what it is. Because you can see that if you have a finite dimensional space, it's actually gives some kind of filtration and structure of algebraic variety. As a result, this is a non-compact complex variety. You can have different those rate structures. Right, right. There's sort of this configuration that conforms that it has provided. I don't really know. And again, let me say that we are like not really actually using this TKFP formulation. We are doing something kind of... What's the question? Do you want to ask why this algebras find degenerated? Yeah, why it's even... In principle, this is kind of nuclear space, whatever this is, but it's naturally because of conformed, it provides limited filter by finite dimensional spaces and then get filtered out because of a finite dimension system. Algebraic structure. Well, I mean, I think in the cases when I can actually define these things, it's kind of happens to be morals on the nose. But I don't think it is an a priori argument why it should be so. Any other comment or question? Okay, so now we can sort of... This is sort of giving some idea of what we do. So this Cron branch algebra is defined. Functions on that is actually by definition is computed in this way. Homological homology is... You can think of that as a hum space of sheaves, and which means that you can write in this way. So sheaves on this, say, constructible sheaves on this bungee of the bubble, and it's a dualizing sheave, and this is some certain sheave. And I want to explain sort of the... In a sense, this equality can be understood as two different interpretations of a single physical system. So consider a four-dimensional TQFT. If you want it, you can think of that as sort of some version of A twist of 4D and 4D theory, the political theory. And you can try to understand what you assign to S2 times the interval. So, you know, this is kind of three-manifold. It behaves like three-manifold. Once you choose, say, boundary conditions at those two points, and you expect to see some kind of vector space. And this is the claim. This 4D TQFT, what you assign to this S2 times 01, can be read in two different ways. First of all, kind of think of this computation along this direction, and second, maybe this direction. And that's kind of the two different interpretations of this configuration. So let me see what I mean by this. So first of all, the first point of view, you are kind of the compactifying or long interval with two given boundary conditions, and you get a 3D theory. And it is known that by kind of the choice of boundary conditions, you can actually get 3D theory, say A twist TQFT written in this way. You know, if you remember TGV, that can be realized by choosing the two boundary conditions and then do this kind of reduction procedure. Then local operators there, this is local operator for 3D theory, right, what you assign to S2. And that can be identified by the physics I mentioned. This is supposed to be the same as our algebra on the column branch. And the second point of view is once you think in this way what you assign to S2 is either sort of a category of line operators or kind of the compactified boundary conditions. And this B0 and B1 are defining objects of this category. And in this way, this is recovering that. So I'm saying that this equality, the first one is actually really kind of the first interpretation. The second one is from this second interpretation. So this definition. What's B, a boundary B? B is a formal bubble. It's a gluing of two disks along punctured disks. It's right out of the border. Sure. So disk, disk, but it is kind of glued. Ah, but secretly the three-dimensional ball of this minus point in relation is not so simple. Yes, yes, yes, yes. Any other questions? So essentially it's a fine glass mania with the... Equivalent sort of the... Yes, that's right, that's right. Huh? What do you mean? Like this? Yes. Right, so this is really kind of the work of BFN. And this is our interpretation of this BFN, namely 3D local operators, say column branch from these 40, say line operators. But given this, you can try to understand sort of the 3D line operators for these surface operators. So that's only kind of one piece of an idea. Okay? What is line operators? In this language, sort of the... For 4D theory, what do you assign to S2? One can think of that as a collection of line operators. So is the line extending along this interval? I mean sort of the... So when I say 3D line, I already kind of used up this... 4D line. So I'm saying that this B0 and B1 are boundary conditions. But you can think of that as a line operator. I mean it's kind of a general thing and... So what is 3D local operators? Local operators. Like the column branches are really kind of the spectrum of local operators. And I'm saying that this is a local... I'm just going to put the point here. No. Okay. Any other questions? Okay. So this is sort of the end of the solid level introduction. Sort of the... We interpret this work of BFN in this kind of the... By giving two different interpretations. And from that point of view, you can sort of do a bit more. Okay. Now let me start sort of the introduction to geometry langlines. And again, like though you don't really need to understand much here. Just to say kind of what kind of thing we are doing. So again kind of the input data are... They come from around surface and group, like GLN. And you also have a langlines zero group. Then geometry langlines correspondence expects an equivalence between two categories. And well-known version is what is for the best top. It says like this. Let me read. It's a demodulated space of G-bunders on this remodulated surface C. And quasi-currents on the modulated space flat connections on G-check. G-check flat connections on this remodulated surface C. It's an equivalence between these two different categories. Okay. So that's it for geometry langlines. Now like one slide for couples written. So as the relief of this 4D and 4D supersymmetric gauge theories, it gives this version of geometry langlines. That's sort of the main claim of couples in a written. And it's like follows. So gauge theory with gauge group G has like P1 main twist and coupled with some kind of the additional parameter. It's actually parameterized by this parameter C as an element of CP1. So in the same way as like AX defines sort of the A model with the target X to be this A model, G to C sort of specifies a 4D TKFT. And second, S duality is known to identify mainly kind of the under duality, these two different TKFTs. Insert. Compactification along this compacted ground surface C. And if you pick particular C, say 0 and infinity, then it is known that it's an equivalence between these two different 2D theories, duality between them. A model with the target cotangent model of G and B model with the targets of the modular space of flat connections. Then if you apply sort of the homological mirror symmetry, then it is actually, one can argue that these A brains are close enough to these D modules and B brains are like the quasi-current sheeps. So you would expect to see some kind of equivalence. So you get the best of version conjecture. Of course, like mathematically, like a lot of parts are kind of missing. Kind of maybe most important one is the dependence on C is kind of the topological whereas geometry long lines really care about, say, complex structure or algebra structure of this C. So I'm going to say more about that a bit later. But before getting there, I want to say a few words about sort of the framework we are working with. Namely, this how we understand this classical field theory as an object of sort of the shifting simplicity geometry. So field theory is given in this way. So I have spacetime manifold, space of fields, and action function. Then what you care at the level of classical field theory is really sort of the this modular space solutions to occasions motion. So sometimes I use the notation critical S if I want to emphasize this S. And sometimes I write so EOM of M when I want to emphasize sort of the spacetime dependence. That's just notation. And the example is the Chern-Simon theory. If you have three main four, one form with a real algebra value, fields and action function are given in this way. And you can check that critical locus actually kind of cut out sort of the flag connections. Okay, and now I want to say a bit about derived geometry. And just like a scheme is a functional in this sense, a derived stack is a functional, you know, sort of the derived sense. And it's again kind of not too important to know what exactly these things are. But just to give a concrete model for derived rings, you can think of that as sort of the CDGAs non-pository graded. Derived sets, you can think of that as a simple set, or even sort of the topological spaces. What's more important is to understand examples. And user schemes or stack is a derived stack. In particular, this BG classifying stack is a derived stack. And for given derived stack, you can define sort of the shifted cotangent bundle in this way. So if you know sort of the usual definition of cotangent bundle in this way, it's pretty much kind of seems pretty natural. You take symmetric algebra linear of the fiber and that's how you get. And this Tx is a tangent complex, sort of the derived version of the tangent bundle. And one thing that's kind of fun about this subject is that a topological space can be understood as an object of this subject. Namely, to define this function, you just need to understand sort of the assignment of derived sets to derived rings. So given topological space, I can define what is for the baby step. So this S is a derived ring. And what you are giving is just M. You just forget about like ignore like whatever you are eating, this derived ring and then just kind of keep your step back. And that's what I mean by the baby step. So baby stack defines a function and it is a derived stack. Okay and mapping stack, I think this is sort of basically the same as the classical definition. For given derived stacks, you can define this mapping stack in this way. Okay, so now I want to explain sort of the relation between these two subjects. Derived algebra geometry and classical field theory. And the point is once you are in this subject of derived geometry, you cannot really like use equalities in the way you have used, I guess. In this particular case of classical field theory, the equation you cared about is this dS being zero. And that you can read as an intersection, namely fiber product of this graph and zero section sitting inside cotangent bundle. So you'd write the critical locus as a tensor product, the structure shift, there are. But in this derived word, you just kind of think of derived tensor product. And that just amounts to actually keeping track of sort of the homotopy information of how exactly this dS is zero. So just to give an idea, let me give an exercise. Let's say sort of X is kind of like smooth manifold, like affine algebraity, and you can try to compute this resolution, then you actually end up with this thing. Structure shift of minus one shift in cotangent bundle with some differential. And in special case, when S is zero, so action function is H0, then you can find this causal resolution using this natural map. And after taking homology, they kind of cancel out, so you end up with this OX. And using this resolution, if you cancel out this OTX and symbol of this TX, then you would end up with this part, which gives the minus one shifted cotangent bundle. So it's a kind of good exercise to do if you haven't done it. And this feature of having some sort of minus one shifted simple structure is expected from the classical BV formula, isn't it? I think we have seen examples from the OS talk as well. Okay, any questions? So shift is simple structure. So these parts are going to be sort of used crucially in Justin's talk. So let me say a few words about the subject. So fundamental definition was made by Pantheft, when Bakke and Bejouci, PTVB. And simplistic structure on a derived stack is given by a two form, but now of commercial degree N, in the sense that usually you'd expect to see that this induced map is giving identification between tangent bundle and cotangent bundle. But as I said, the tangent complex is a derived version of the tangent bundle. So it's like a co-chain complex as opposed to vector space. So it makes sense to shift things around and still expect to see some kind of identification. And with this N shift, if this is an identification, then this such an omega is a N-shift disimplicated structure. And the shifted cotangent bundle, as expected, has an N-shifted simplified structure. I guess this is non-true result of Damian, actually. I mean, it's expected, I guess, always. But a reductive group clash-fine stack is a two-shifted, and you can kind of easily see that if you understand tangent bundle of BG, which is G-shifted by 1, then to have sort of equivalence between this G1 and its linear dual, only with degree shifting 2, it's identifying. And that 2 is exactly sort of the why this BG is a two-shifted symplectic. So symmetric invariant-violin uniform is giving an identification of these two things, and that's two-shifted symplectic structure. And important parts of this subject is, of course, it's a symplectic geometry. So the notion of Lagrangians is absolutely important. But instead of giving sort of formal definition, let me just kind of observe that the examples are as expected. Sort of there's a general section in Lagrangian, like things like graph of uniform is Lagrangian, and this is a shifted conormally is a Lagrangian. Okay. And some of the main theorems of PT-VVs are about producing these shifted symplectic structures, and they are going to be used sort of the crucial way for us. One theorem is what is called the Lagrangian intersection theorem. It says that given this n-shifted symplectic object and two Lagrangians, by taking intersection, you actually get an n-1-shifted symplectic structure. So in this sense, the derived creative locus, that being minus 1-shifted symplectic, can be explained. Because it was an intersection of graph and zero section, which are Lagrangians of this cotangent bundle. Another important theorem is what is called the AKSC PT-VV. And in this case, it's kind of about giving a shifted symplectic structure on this mapping step. And when x has an m orientation, I mean I'm going to explain a bit about what I mean by this, and why is n-shifted symplectic. This mapping stack is m minus m-shifted symplectic. So let me give an idea. Orientation, if you want, you can just think of that as the information of fundamental class. So let me explain these with examples. So if m is a close orientative in the manifold, then fundamental class is giving the orientation, just because it's giving this map. And from there, I can define this mapping stack, and now I want to use this AKSC PT-VV theorem. So bg was 2-shifted symplectic, and this mb is d-oriented. So if you use the theorem, this was n minus m-shifted, and this is 2, this is d. So you are actually getting 2-shifted symplectic. So in particular, if m was 2-manifold, you are recovering the usual symplectic structure on these modular space local systems. And more examples. When x is a smooth proper scheme of dimension d, then you can define two objects, what's called the Dorbo stack and Drum stack. Like roughly, you can think of Dorbo stack as sort of a thing. Your structure shift looks like algebraic drum forms. Maybe in smooth categories, it looks like a Dorbo complex. And Drum stack is more like you actually have a drum complex with the algebraic drum differential. And you can convince yourself that that gives 2D orientation, each of them. And using that, you can induce shifted symplectic structure on this modular space. So modularity of Higgs bundles can be defined as a mapping stack from x Dorbo to bg. And modularity of algebraic flat connections can be defined in this way. And in the case where they can be compared to sort of the usual definitions, you can check that they kind of coincide. Namely, sort of when, say, d is 1, d is 1. In this case, modularity of Higgs bundles on a curve and modularity of flat connections on a curve, they have the usual symplectic structure. And that's same as the one we are getting here. Okay. And using this sort of the language, you can write down sort of this EOM in this way. So given a Calabia manifold, let's say if I care about this 2D theory b model, like b comma y, it has this aksc description given in this way. So in particular, if you have sort of the two manifold, you are actually getting sort of the minus one shifty symplectic structure as expected, like close to manifold. And 3D transimons, let's say complex transimons, then like mapping stack to bg, or so if you think of sort of the close three manifold, you get this minus one shifty symplectic structure as expected. So now let me say a bit about this 4D and curve 4 gauge theory. You can actually compute this solutions to equations motion as an object of shifty symplectic geometry. So this holomorphic twist, so left hand side kind of has a meaning. It is sort of the solutions to this holomorphic, what is called the holomorphic twist of this 4D and curve 4 theory. And I'm trying to identify what it is. And the theorem, the content is that it can be identified with the right hand side. So as expected, you see t star minus one, which is minus one shifty symplectic, which is expected from this classical BV formalism. And you can write in a few different ways and these two are sort of some ways. But somehow case of interest mostly are this case when your outer surface can be written as a sigma times c. In this case, let me write down sort of the, let me kind of record what's happening for the other twist. This ht stands for holomorphic topological twist. And you can identify sort of the equations motion in this way. So holomorphic topological is called so because dependence on c is kind of more like holomorphic and dependence on sigma is more like an topological. And by bit, you can compute this B twist and A twist. And let me say that this notation is kind of the maybe possibly little unconventional because I mean it's even actually different from what we wrote in the paper. I'm just kind of choosing, have made the choice of this notation because that's kind of the easier to use for our purpose. So this A twist is actually different from sort of the Caput's witness twist. And another thing to note is that we are actually seeing a modulus of flag connections as opposed to modulus based on local systems. And that actually says something. So we are kind of working with topological twist. And topological twist is expected to sort of give topological theory. But my claim is that this actually does see some kind of algebraic or complex structure dependence on c. Does this twist exist only when axis of that product? Some of them, yes. Like this. Of course this is a case and this A twist as well. The B twist makes sense like everywhere. Yes. Now we are so careful so maybe genetic and then some should abundance local system because what they usually use. Yes, this is kind of, that's right. Like what I'm saying is that if you do Caput's witness theory kind of in a careful way, actually looks like this is kind of the one natural model we are kind of naturally seeing. Because actually we are seeing this Higgs motor line as a sort of the first step. And these things are realized further twist, namely deformation of the Higgs motor line. And then it's like just flat G not log G. So we are really seeing sort of the algebraic structures. Yeah, but I think it's impossible to change it. I mean in Germanic language it just did not, I mean local systems means that there are local system by definition. No, no, no, no. It's brought when you use wrong notation but it's kind of confusing. No, I mean it's not my notation, I mean just everybody. Yeah, no, no. It's, listen, it's tradition. Traducence is group of. Just the point that people working in Germanic language, topology and complex numbers in principle does not exist. So it's a power and nothing, nothing non-algebraic in principle. Yeah, yeah, yeah. Ignorant bit in Germanic language, but it's. But bit in Germanic is also kind of pretty fun. I don't know. Yeah, yeah, yeah. Yeah, it's, it's. It seems to be seen to ignore, yeah. Right. But somehow like from our framework it seems kind of the fletch is kind of the material appearing. Okay, if you're a physicist maybe you may ask like we did take topological twist. How can you see topological structures? I mean non-topological structures, like namely algebraic structures. And the claim is that, you know, topological twist if you think about the process is kind of really kind of the trivialization of infinitesimal transformations. And if you know sort of the definition of drum step, it's exactly identifying this, like trivializing this infinitesimal translation. So that, that information is captured by having sort of the drum, drum sort of step. And from this, as I kind of hinted, you can actually like by understanding this procedure and trying to understand this, say, Kapu Smith theory in our own way. Actually we seem to be getting some new structures in geometry-Languan theory, but that's not a topic for today. Okay, so next part is this categorified geometry-Languanization. And it is really, really a thing. And as I said, like these people have done like a lot of kind of interesting, exciting work. But I want to mention this is a science because what I'm going to, what I'm about to say is like doesn't really maybe look like science, but it's just because it's kind of the like fourth level of introduction. Like if I'm, if I have more time, like I've done more, and like Justin actually is going to make this more kind of science looking. Huh? No, I meant I had two talks. Oh, right, okay. But even fifth introduction doesn't, it's not enough to, yeah. Sorry about that. But so it's kind of naive sense. It goes as follows. So you identify this equations motion. You can identify in this way. So equations motion can be defined on sort of the any sub-manifers. Say in this four-digit, you kind of the, any kind of the sub-manifers of this four-manifers. And equations motion on this curve C is expected to be one-shift disimplactic. And one can identify that in this way. T star one of the drums that go bungee and T star one of flat motor line. But if you know sort of the usual geometric quantization, when you have say cotangent bundle, the way you get the silver space is taking functions on the base. That's sort of the naive thing you can do. And this category of geometric quantization, categorified geometric quantization to get something like the TQFT. You take sheeps on the base. So functions are the sections of the line bundles of what they are. Right, but what I'm saying is that if you start with the cotangent bundle. Yeah, it's as easy as the sections. No, no. So functions are. I'm being like really naive here. Like as I said, like this can be done kind of in a more, in a much better way. I'm trying to be nice. Last sections of the bundle one T star and they're flat along the fibers. Right. And from this, we actually get sort of the D-modules on bungee and quasi-corrosions on the flat G check. But the thing is that this logic kind of follows through for the this case of having S1. So it's a one manifold. And by S1 in our algebraic framework, we actually means sort of the puncture disc. And you actually get T star two of a drum signal bungee and T star two of that. And if you take sort of the kind of the two category conversion of these functions, this is kind of what Sasha kind of described. Some sort of sheaves of category. And that's how you get expected sort of the categories to categories of interest for the A side and B side. And so expected equivalence between these two things is a local jointed Langland statement. And this is a sort of global jointed Langlands. And as kind of the hinted by Sasha, this is kind of not like kind of the correct as written. There needs to be modification. And that's also kind of Saturday one can deal with. Okay. So let me say what I hinted at the end of my third introduction. So this A twist, TKFT. Again, I was kind of using this TKFT language, but we realized that actually we cannot because we are actually seeing sort of algebraic dependence. So it's not really TKFT in the usual sense. But you can compute in this kind of equations motion in this naive way, and then also take this categorized geometric quantization in the naive way. Then in some cases, this P of GOV has a mirror theory of known form, namely P g-shape v-shape. In this case, we can write down this categorized geometric quantization through this equations motion like really in this way without much difficulty. But that's already kind of giving sort of new conjectures. And we made this conjecture. And Tudor and again, Justin are actually kind of saying a lot about sort of the properties of this conjecture. Although we were unaware of this work, it seems that sort of the very version of this conjecture or kind of related things, where it seems to be kind of known by kind of the combination of the work of these three. So I'm a little confused about the notation. So you're assuming that there's some kind of dual which looks like also, which also comes from a group in our presentation. I mean, so you have g and v. And then to write this equation, you have to have this v-shriek and g-shriek, which in principle usually would not exist. That's right. I'm saying that sometimes this has a mirror. In that case, Brabham on a gaze screen is proving the Abelian case, I heard. And like I hinted at the end of the introduction, there are these two different approaches. And first approach is kind of really giving this chrome bridge algebra. And in the same way, we just kind of constructed line operators. But you can certainly do construct these things from this, say, 4D point of view. I didn't really use kind of 4D point of view kind of much for the last slide. And like a lot of them, I think Justin is going to explain more about this. And let me just kind of provide some ingredients for that. So Guy O'Dwitton says that 3D and equal 4 theory with flavor symmetry G defines a boundary condition with gauge group G. Boundary condition of this 4D theory with gauge group G. And this duality of boundary conditions are understood, like in examples. So what do we mean by boundary condition? For that, let me explain this. So equations motion on M is, especially on a closed M manifold, has a minus one shifted structure, but that's actually not always the case. If M is not closed, if it has a boundary, then what we have is equations of motion on this boundary manifold, namely codimensional manifold, is simplected, namely general shift is simplected. And we also know that this natural map is Lagrangian. That's sort of the general fact and expectation and sort of the examples that work. So if you had another Lagrangian of this phase space, then by thinking of this fiber product, I'm actually thinking of Lagrangian intersection of these. EOM of M is Lagrangian here, B is Lagrangian here by definition. So by taking fiber product, you actually have minus one shift is simplected object again. So this motivates definition, a classical boundary condition is a Lagrangian of this phase space. And given this, maybe you can read this better now. EOMM, this boundary B, is modular space solutions to equations of motion on M, which satisfies this boundary condition on the boundary. That's how you read this fiber product. And this physics assertion is going to be used for, say, our work and just in stock using this claim, namely this 3D theory, 3D and 4D theory of interest has HT, A, B twist. 4D theory also have this HT, A, B twist. And apparently maybe they are not related for this kind of physics argument, but the claim is that namely HT twist of 3D and 4D theory with flavor symmetry defines a boundary theory of HT twist of 4D and 4D theory and so on. So they are kind of compatible in this way. Okay, thanks. More questions for now? And in this simplected object is an equivalence given by some holonomic demodern or some kind of product of 2. Okay, I don't know. I have quantization of one simplected Algebraic demodern for NASA, yeah? And equivalence, and equivalence is given by model here, in general. Does that matter? I mean, there is a kernel, but only in sort of the sense that Sasha was saying that there was a Langland's kernel, where if you know how some fact duality works, you can do. No, no, no, but if you have actual finite-dimensional simplected algorithm. No, but see, the problem is it's not in equivalence between all categories. Once it's only, also it's not in equivalence. It's casual duality. So, as long as you can see there's some kind of, I mean, it really kind of works on a restrictive setting. So, you have to define some kind of category of modules over the quantization and then the same for the dual and then these categories are not equivalent, but they're casual dual. So, in other words, so, one thing not to say that they're casual dual, they have to define additional grading on that category, which is somehow a pre-warrior. It's not clear where it's coming, where it could be coming from. And so, secretly, you have to define the right category of modules, then define additional grading and then somehow do a little sequence which sort of changes the homological grading with this additional grading. So, it's not really cool ones. No, no, but the rule can, for me, for example, the potential bundle for whatever it is. No, even in case of potential bundle. So, potential bundle, for the black variety, of P1. P1. Even there, yeah, even there, that's it. Because the functor exchanges symbols and projectives. And basically, projectives have no x's. So, this functor does not preserve homological grading. So, it changes homological grading for something else. What's the other grading? Is there like some physical interpretation? I mean, it is a north symmetry, but I think people also think it's supposed to come from some kind of hodge theory. I mean, in all the examples, the way that you see the grading comes from looking at like mixed hodge modules. Like basically, in the case that your theory is a cotangent bundle, so the quantization is demoduled, then you also have a theory of mixed hodge modules. And then, like the hodge grading is the, and it ends up being pure, or the grading on all the homspaces ends up being pure. The hodge structure on all the hodge is just being pure. And then, it's the hodge I think physically it's a hodge-like tree.