 Okay, thank you very much for the introduction and thank you for the invitation to this amazing place. It's a big pleasure and honor to speak here. I will tell you about a large class, some mostly conjectures about a large class of vertex-related algebras that are associated to divisors in Calabi of free faults. And compared to the basically all the talks I've given on the topic during the past two years where I took one perspective out of the three known perspectives that I have at this point. Today I've decided to include all of them in a single lecture, which means that it, I can, I have to skip basically all the details and all the technicalities, but hopefully it still will be understandable. And if not, please feel free to ask me afterwards after the lecture. I should also mention that what I'm going to talk about is mostly, is going to be mostly the work I've done with David Goyalto, Tomáš Prucháska, Jan Svibelman, Nia Pingiang and Gupang Zaho. And let me start right away with the introduction. So my point of view on vertex-related algebras is going to be very algebraic and physics, physicsy. And roughly the vertex-related algebras are extensions of the Virazor algebra generated by the most of the stress energy tensor, ln, with the following commutation relations centrally extended by the following term. And the extension is by some fields, by modes of some other fields where alpha labels the fields and N is an integer labeling the mode. I don't have time to review all the definitions of the vertex-related algebras, but I think this is going to be sufficient for this talk. And for a physicist, a vertex-related algebra is roughly the symmetry algebra of 2D CFTs. Since I'm a physicist, I'm going to start with a physical picture that I have in mind when talking about the vertex-related algebras. And the physical picture to which I'm going to associate the vertex-related algebra is the following configuration. I will consider M-theory, which is an 11-dimensional theory, on, let's say, T star times sigma, where sigma is three months surface, on which the vertex-related algebra is going to live, or the 2D CFT is going to live, times the Calabria free fold. And M-theory contains M5 brains, and I will consider M5 brains, oh, and there's extra, let's say, circle. The M5 brains are going to be supported on sigma inside of T star sigma, and D inside the Calabria free fold, which is a divisor. What do I mean by that? I mean that we have this Riemann surface, and we have N1 M5 brains wrapping one for cycle inside the Calabria free fold, possibly some other M5 brains, and let's say N2 of them wrapping another for cycle, where each of the for cycles correspond to different smooth components of the divisor. So this is the configuration, and in the same way as using the same motivation as the AGT, as used in the paper of Aldaigato and Tachikava, who associated a vertex-related algebra to gauge theory on C2, one can consider compactification of the setup on D. To get some two-dimensional theory living on the Riemann surface, this is subclass of local operators, giving rise to a vertex-related algebra that I label as VOAD, similarly as in the recent paper of Gukov, who studies very similar configurations. But the vertex-related algebras associated to a given divisor have been identified only for a limited class of such divisors, basically only C2, or resolutions of C2 mon ZK, and today. So it's a divisor in the Calabria free fold, so it's a divisor in C2. Oh no, I'm just saying, yeah, right. So the two examples appearing in literature, the following example, where the corresponding vertex-related algebra can be identified with WN algebra, or the following divisor, where we have NM5 brains on C2-mode ZK resolution inside C2-mode ZK times C, where the corresponding vertex-related algebra is some extension of Katsumude GLK hat at level N. When you say compactification, do you mean like dimensional reduction? I guess so, yeah. So today I'm going to tell you, I'm going to tell you what are the vertex-related algebras, and I'm actually going to give you three different characterizations of the vertex-related algebras associated to the divisors in Toric Calabria free folds. So today I will try to answer the question, what is the vertex-related algebra VOAD for Calabria free fold Toric? Why do we even care about this question? Let me give you three reasons. So firstly, it is interesting from the point of view of the vertex-related algebras, it leads to interesting results in the theory of VOAs. Since one can discover new dual constructions of vertex-related algebras, one can understand structure or parameterization of modules of interesting vertex-related algebras, or give a diagrammatic way to understand various extensions of vertex-related algebra, as we will see later, possibly many others. Secondly, the picture that I've drawn above actually points towards an application and a generalization of AGT for respect to instantons of Necrossof. Intuitively, what one can do, one can consider instead of the compactification on the divisor D, one can compactify on the Riemann surface to get a gauge theory with GLN1 and GLN2 gauge group on each of the smooth components of the divisors, mutually interacting along the intersection. And it turns out that there is some correspondence between the vertex-related algebra on the left-hand side and this gauge theory configuration on the right-hand side. And these gauge theory configurations have been recently studied in a series of papers by Necrossof. So what I call by AGT correspondence for respect to instantons is this relation between VOAD and gauge theory on D. This relation has very concrete realizations, both in mathematics and physics, where in the math literature, people can construct actions of the natural geometric actions of the vertex-related algebras on equivariant cohomology of modular space of instantons associated to D, something that we have done for the simplest example of, very the simplest example of C3 with Jan Svabelmania-Pinkangen-Gufang-Zell. And the physics literature, one can find statements that the conformal blocks of the vertex-related algebra equal the partition function, Necrossof partition function on the other side. And the final motivation that is being explored by Davide, but I won't have time to talk about, are some applications in geometric langlines program. Since the configuration that I described for the Toric-Labiau-Phefels have actually dual description in terms of interfaces in n equals for super young males and the appearance of the vertex-operated algebra then gives away how to realize the vertex-operated algebras and employ the language and the techniques of vertex-operated algebras in the gaseous setup of Kapustin and Witten. Okay, so this is the motivation. And let me now start with reviewing some facts about the Toric-Labiau-Phefels and Toric diagrams. So I will think about Toric-Labiau-Phefels as R times T two vibrations over R three. Let me describe how the Toric diagram or let me describe in like concretely the simplest example where the Toric-Labiau-Phefels is simply C three. So we have the natural coordinates, D one, Z two, Z three on C three. And one can consider the following Hamiltonian functions that give the projection to the base and generate the R times T two fiber. In particular, these mu one and mu two generate the following rotation of the coordinates. T two, Z two, E minus I T one minus I T two, I T two, Z three. The Toric diagram is then a diagram that shows the low psi where one of the cycles of the torus degenerate in the mu one, mu two plane. And it's not hard to convince yourself that the Toric diagram for this C three has simply the following form with the first A cycle degenerating along this line, along this arrow. The B cycle is degenerating over this line. And A plus B cycle degenerating along the last arrow. Sorry, what does this picture mean? So you're just drawing the lines or would you have the generations or is it? That's correct. That's right. Just the generation of the two circles. Okay, now let me tell you how to describe in these diagrams, divisors. So a generic divisor that is fixed under the Toric action has only three smooth components and correspond to the three coordinate planes inside C three. So with the multiplicity cities N one, N two, and N three. And we're gonna see that each of these smooth components map to one of the two faces of the Toric diagram. And we can then each of such smooth component associated to one of the faces. So we can specify the divisor by this colored Toric diagram with N one, N two, and N three integers associated to each of the three faces. What can do similar things? And it turns out that any Toric, maybe our three fold can be described by some kind of Toric diagram. Let's specify again the degeneration low side of various cycles above the base in the mu one, mu two plane. And instead of giving you details, let me give you a couple of examples. The first simple example is when we glue together two trivalent junctions, two trivalent junctions. In the diagram, each of the trivalent vertices correspond to one C three. And the diagram describes how they are glued together roughly. So this diagram corresponds to the probably second simply is Toric by our three fold, which is resolved conifold. And one can again associate divisor, one can again label divisor by putting integers inside each of the four faces. The second simplest example containing also only three trivalent vertices has the following form. Let me also specify which are the cycles that generate along each of the arrows. And the divisor, the corresponding divisor that is fixed under Toric action has again four smooth components. And I can continue drawing these Toric diagrams and study the corresponding divisors. Let me come back to the resolved conifold case, which is the following Toric diagram because this is actually going to be the only one that I'm going to speak today about, but you can do the same or similar kind of construction for other Toric Calabria three folds. So just to be concrete, from now on I'm going to concentrate on the simplest trivalent junction and this simplest group diagram corresponding to the resolved conifold. So the resolved conifold has actually the form of the O1, O minus one, the O minus one bundle over P1. And the Toric divisor has four smooth components with one of them being one of the two O1 bundles over P1, the other one being the other bundle, O minus one bundle, then fiber over the north pole of the P1 and finally the fiber over the south pole of the P1. Okay, so that's the Toric geometry that we need and the Toric diagrams. And let me now describe, go to the vertex split algebras and describe the vertex algebra vertex. Okay, and I'm actually going to give you three conjecturally equivalent definitions of the algebra. I'm sure, so what is the connection between this discussion of diagrams and what you're doing now? Well, I'm just, right now I'm going to, okay. I'm going to associate vertex split algebras to each such diagram. Throughout this talk, I'm not going to describe what is the relation between the geometry and the vertex split algebras. I'm going to define them in terms of formulas and I'm going to give you three different definitions. So I will associate three different definitions to each of such diagrams. And conjecturally, these vertex split algebras that I'm going to define are going to be those that appear in this generalized AGT from an action of some algebras on acrylamic homology on the modularized basis. So you're not just going to do it for the result of the quantum fault, but you're going to do it just for the diagram. Well, just an example for it. That's right, that's right, that's right, that's right. Everything should work for arbitrary diagram, but I'm going to write down explicit expressions only for the result quantum fault. And so to get the ordinary AGT to reach, raise your specialization to which case you need to do. So by AGT, I don't mean the duration with the partition function on S4, but by AGT, I mean the curl part, the duration of the C2. And it's relatively easy, right? Because, sorry? C3 inside C2. That's correct, that's correct. So that would be, so the picture leading to original AGT would be N1 M5 brains, N1 would correspond to N1 M5 brains wrapping C2 inside C3. And the corresponding vertex of the algebra is indeed going to be WN algebra. So. And what is the Toric diagram? And the Toric diagram is the following. And epsilon parameters? And epsilon parameters are going to show up. They appear as accurate parameters on the geometric side, and they are going to be, there will be parameters psi, which is K plus N, and it will be equal to the ratio of the actuarian parameters. So. They act equivalently on C2. That's right. Yes, that's correct. Yeah. And all around there is going to be this extra parameter, which is the ratio of the actuarian parameters, and we are working equivalently with respect to the T2 action. Okay. So I'm going to give you three different definitions. We have strong reasons to believe that they're, they actually give the same algebra. It has not been proved. And the first one is the original that we gave with Davide in the paper almost two years ago, based on this dual perspective that I've already mentioned in the context of geometric algorithms. Now I can be actually slightly more concrete. Right, so. I'm going to now tell you what is the vertex of the algebra associated to the vertex or this diagram. So from the dual perspective, now I can be more concrete. This set up is also a description in terms of N equals four super young males, where we have a junction of interfaces between various, between N equals four super young males with gauge group N1, N2, and N3. And by analysis of the boundary condition or the interface conditions along the lines of a Gauss and Witten between the gauge theory, we were allowed to conjecture the following definition for the corresponding vertex operator algebra. So this interface interfaces in N equals four super young males lead quite naturally to an identification of the vertex operator algebra in terms of a quantum Hamiltonian reduction, consisting of a drain filter core reduction with respect to N3 minus N2, with respect to a principal embedding inside N3 minus N2 diagonal block inside GL N1 slash N3. There's the parameter psi, where the parameter psi, as I mentioned, is the ratio of the equivalent parameters on the dual side and on the side, on the N equals four super young males side, is just the gauge coupling or the Kapustin-Witten parameter or whatever, roughly the gauge coupling. And then one needs to take a coset with respect to the remaining diagonal block, which is GL N1 and N2 psi minus one. I don't have time to describe the details, I just wanted to write down the formula for the people who know the Hamiltonian reductions. I just, yeah, actually this one says, we're writing Y of N1 and N2 and N3 and then the square break, what's inside the, what's the symbol inside the square bracket? No, on the left, no, no, on the left. This one? So this is the label of the, this is how we labeled it. But then there's a letter inside the square bracket. This is psi, yeah. And what is psi? So psi is either on this side, on the geometric side, it should be the ratio of the equivalent parameters and on the gauge series side, it is the Kapustin-Witten parameter in any case, first-of-a-angles. And so if we were in the kind of general formula, I'm sorry, is there a parameter? I mean, on what kind of parameters should this vector separator algebra depend? I mean, can we just start with any Calabria free-fold with a device or maybe with the Doric one and then they're saying that you have an algebra, not just, sorry, not an algebra, but an algebra depending, in this case on one parameter. So, right. So I'm not sure how it's going to be if you are not in Toric Calabria free-folds and there I don't even know how to be concrete in terms of formulas. I'm restricting to Toric Calabria free-folds because then I know when I can write down formulas and the algebras explicitly and these are always going to depend on this parameter, one parameter side. I'm pretty sure that this- And if you're using this m-theory perspective somehow, can you predict what kind of parameters should get in picture? Circle? Is it related to the circle or no? Which, no, no, no, no. It looks like it's related to it, how non-compact this guy is. That's right, that's right, yeah. Yeah, I don't have a good answer for that. Where does supersymmetry come from? Supersymmetry? Well- What does it mean, supersymmetry? Well, there's supersymmetry all around. The whole configuration preserves, you can either look at it from the dual perspective or this perspective and you just write down the supersymmetry preserved by the setup. It is going to be 0,4 two-dimensional supersymmetry or something like that. So four supercharges are going to be preserved and they can do topological twists and most of the things I'm talking about should be actually in some kind of twist. Okay, so let me give you the second perspective that we came up with by analyzing a character of this vertex of the algebra in large and limit. So if one computes vacuum character for an i going to infinity, it turns out that one gets the MacMallon function with the following form, which actually turns out to be also a character of an algebra called w1 plus infinity. What do I mean by w1 plus infinity? So you only take n1 goes to infinity again? It doesn't matter. The character doesn't, so even the character of this algebra has very nice box counting interpretation and everything that I don't have time to describe, but it doesn't matter in this case how you take the limit. It will matter in the case of resolved conifol and more complicated diagrams, but in this simplest case, it doesn't matter how you take the limit. And if you take the limit, you'll always get the same character, the MacMallon function. So what do I mean by w1 plus infinity? I mean something slightly different than in most of the literature. So w1 plus infinity, for me, is an extension of the virtual algebra with the stress energy tensor label as w2, consisting of the most ln that I've written before, by fields of conformal weight, one, three, four, and so on up to infinity. What I mean by conformal fields of weight i, I mean that they have the commutation relation with the most of the stress energy tensor of the following form. So I can now ask questions, something that Gabriel Gopakumar asked some time ago. What are the possible vertex operator algebras containing such fields? And turns out that Jacobi identity is fixed all the OPEs, all the basically everything up to two parameters. So the algebra is unique up to the parameters lambda one, lambda two, lambda three, subject to the following constraint. Sir, are you in the quantiful case or in the general case? Right now I'm talking about the simplest C-free. C-free with arbitrary one or two. And the limit is using vacuum character for all m i's go to infinity or just one thing. Doesn't matter. One of them could be the one. Take the limit where only one of them takes goes to infinity and the two of them are zero. You'll get W and algebras and you send n to infinity. So you'll get obviously the MacMahon function. And doesn't matter how you do it. You can, it's going to be always the same. I can actually because it's kind of fun, right? So the MacMahon function is known to count boxes that like 3D partitions, boxes that fit into a three dimensional corner. So we have those boxes. So if we expand this character, we get one plus q plus three q squared, blah, blah, blah. Where the one corresponds to the empty corner. This other one corresponds to a single box put in the corner. And this three corresponds to the three ways how you can put two boxes in the corner. There are two other ways, right? You can stick. The other box here or here from three different sides and you'll get three and you can continue. And it's kind of fun that it turns out again, something for which we don't have a proof but we expect it to be true based on some examples. Anyway, no, okay. It turns out that the character of this y and one and two and three counts also boxes that fit under a shifted corner that is shifted by n one and two and three. So we have this vector shifted by n one and two and three and the character of y and one and two and three accounts the boxes that fit under the shifted corner. Okay, it also turns out that there exists an ideal inside this w one plus infinity. If the parameters lambda are specialized as follows if this combination equals one and one can consider quotient with respect to such ideal and recover the algebra's y and one and two and three as such a quotient. And no, exactly, that's what I'm about to say. We have three parameters subject to this constraint and this extra constraint coming from the, that is required in order to get the ideal. And we have one extra parameter and the one extra parameter is parameter psi which is minus lambda two over lambda one. Let me add a remark. It is known, though I don't think it's proven that this w one plus infinity algebra is isomorphic as an associative algebra with the affine Youngian of GL one hat where the parameters h one, h two, h three are roughly one of the lambda parameters one of the lambda and the h one, h two, h three. Oh, I called an epsilon there of the left-hand side. Of the right-hand side. What is definition of right-hand side? There's no notion. I mean, after all, there's no definition of Youngian of GL one hat. Why not? This is not cosmodial algebra, so. We can write down genitalian relations. No? People have written down genitalian relations to make this work of Tsimba-Lukand. Yeah, but I think the original definition is just this w one plus infinity, but more or less, it's the original definition of this Youngian. I think there is a definition of Youngian for any paroidal algebra by generators, integrations and the papers of Hernandez also relates to that as a procedure of affinization. You can do affinization for any quantum group in particular is a hat. I think for GL one, I have to do it separately, but okay. Ah, for GL one, compared to like simple? Yes. Ah, maybe. Yes. Okay, that's true, okay. That's true for what? Okay, so that was the second definition or characterization of the algebras. And finally, let me give you a free field realization. Again, I won't be very concrete, but I'll just tell you that is why N1, N2, and 3 can be realized as a subalgebra of a tensor product of N1 plus N2 plus N3 Heisenberg algebras, Heisenberg BOAs, generated by JNs with the following commutation relation. And I don't have time to give you the definition like properly, so let me just tell you two examples. The first example being this Y002 algebra that we expect to be the W2 algebra, which is the Virazoro, times an extra Heisenberg algebra. And one can realize it in S subalgebra of two free bosons by setting this W1 generator to the sum of the two Js and the Virazoro to be up to normalization, the following combination where I've introduced J, that is J1 minus J2. Yeah, and wherever I write Hs, I mean epsilon, sorry for that. And sir, do you know somehow some explicit equations by which this is cut out inside Heisenberg algebra like some of these like spring operators? Yes, I do. Worried that I don't have time to discuss, but I do have two characterizations. The first one in terms of screening currents based on the work of Fagin and Litvinov and collaborators. And one based on some kind of generalization of Murat transformation from our paper, with Tomasz Pruchaska. Yeah, I'm happy to tell you later. But yeah, we do have explicit formulas, explicit characterizations of these embeddings that I just don't have time to describe. Because I want to say a few words about the gluing and how to define vertex operator algebras associated to more complicated Toric free faults. And I will illustrate everything, though most of the claims I'm going to tell you can be made much more general. I will illustrate everything on the simplest example of the result calling fault. And again, I will give you three definitions as I did in the simplest case. So based on the dual perspective, one can again give this quantum Hamiltonian reduction definition or the BRST definition, at least in some cases for some Toric, for some divisors. And write down formulas as follows. Now we will have to do two gene-physical reduction, one with respect to N1 minus N2, sorry, N4 minus N3 diagonal block, another with respect to N1 minus N2. I can later on tell you also how to read it from the diagram. Sir, when you write the principle circle off with an index, what do you mean, sir? It means it's a principle circle reduction with respect to what with respect to? With respect to the principle embedding inside N4 minus N3 times N4 minus N3 diagonal block, which means that I'm taking the N generator to be... So I take psi of this super matrix, N1 and N4. I take, let's say, that I want to take, do this difference reduction with respect to N1 minus N2. So I take N1 minus N2 diagonal block and I consider the principle embedding inside this block. And what do you do if N1 minus N2 is greater than? Sure, and that's exactly what I mean by saying that in some cases you can define the principle circle reduction. The cases when I can define the principle circle reduction is when both of the ends on the left and right either decrease or increase. So those are the only cases where I can write down them. And this is going to be true also for more general diagrams, like these slinner tree diagrams. Okay, so that was the Hamiltonian reduction definition. And in the same way as we did the analysis for the large N character, vacuum character of the simple vertex algebra vertices, we can do the large N analysis also here. And conjecture that the vertex-operated algebras can be defined as quotients of some version of W infinity algebra, some infinity algebra that I call W one slash one times infinity rho. That is conjecturally isomorphic to the shifty the Angian GL one slash one hat. This shift being the intersection number of the P one of the resolved conifold with the divisor. In general, there are going to be more shifts and the shifts are going to be parametrized again by intersection numbers of various P ones inside the toricalibial freefold with the divisor. And in this case, one can explicitly write it as N four plus N two minus N one minus N three. So that was the second definition. And finally, I will give tell you also the free federalization. You get this quotients of GL one hat, young N of GL one only for the conifold case or? That's right. So now I'm talking only about the conifold case. And in principle, do you have a definition in a general toric case or? Yes. So to have an N look of this young N of GL one hat. That's right. So if there's the diagrams of the following form where you have one zeros ending on line from the left and right, and there's N of them on the right and N of them on the left, the corresponding young N should be shifted young N of that's right. That's the expectation. And again, the shifts are going to be given by the intersection number of the divisor with the P one. So let me give you the freefield realization. So the freefield realization is inspired by the topological vertex where one takes, one associates some topological vertices to each of the trivalent diagrams and extend by some, and sum over something associated with the internal lines. And the freefield realization of the algebras is given by an extension of the product of the two algebras associated with the trivalent vertices, which is Y and one and two and three times Y and one and three and four by some modules or by modules labeled by lattice, labeled by N one plus N three, which are these two numbers, dimensional lattice and have roughly the following form where J N, J is H three N, where this N is the lattice vector times a difference between the vectors of J associated to to the Heisenberg algebras of the two vertices. There's the Js associated to the first vertex and J tilde associated to the second vertex and one just takes this difference and multiplies it by N. Okay, I don't have time to go through details again, but this kind of construction is a combination of the standard lattice extensions, construction of lattice vertex of the algebras and screening currents techniques. So let me conclude. So I gave you three characterizations of a class of vertex operator algebras or sketched three characterizations based on BRST, W infinity and free field realization of these vertex operator algebras, VoAD. And there are indeed many open questions and things that should be done more precisely and properly. So it would be nice to develop fully the theory of gluing to combine fully the techniques of the lattice vertex of the algebra extensions and screening currents techniques. There are many open questions in this point. It would be nice to explore properties of these infinity algebras, especially for the for the diagrams that are not of this simple form, the versions of the W infinity. And finally, it would be interesting to find actually or prove that the definitions that I gave you are equivalent. Are there some cases where it can actually do it now? Some cases. Well, sure, for example, if you take these Wn algebras, all of those are known to be equivalent, right? Besides that. Yeah, there are many other examples that we have worked out, for example. But what we have done, we have basically worked out only the examples for small numbers of Ns. So for example, one, two, one, one, and so on. And maybe there are even some, like there are many people working on vertex algebras and I can, I expect that there might be some results also for like bigger class of examples. For example, I think it might have been shown at least to some extent for the following configurations as well, since these are simple cosets gLn over gLn minus one that have been studied. So it's very likely that there is some kind of equivalence known for like these classes of vertex algebras, but I don't know. Okay, and finally, there are many possibly interesting applications, for example, prove that these vertex-operated algebras are indeed those that come from the, from the action, geometric actions on the current cohomology of modular spaces. So proof of AGT for necrosis of spiked instantons or possibly use some of the results, something that is being developed by Davide Gaiotto with collaborators in geometric long-lance and possibly many other applications that are hopefully going to show up. So thank you very much. Okay. This is, with the sequential partitions inside to six, it smells like you have the right object and it's not like this only some relation to both of us. Exactly, that's right. So that's what we discuss in the recent paper with Jan Pinkengufang. And if you take exactly this this configuration, the cohomology of our algebra has a quiver description for C3 in terms of the following quiver. And we show that it naturally acts on the equivalent cohomology of a framed quiver coming from like putting in the divisor and studying. That has quiver description in terms of the following quiver with potentials and their ranks and one and two and three. Let me take. So there's a natural action of the cohomology of algebra. There are actually two natural actions, one giving rise to raising operators and lowering operators of the affine-yangian of GL1. And one can then use the, in the standard way of Schiffmann and Wasserholt, for example, one can use the co-product of the affine-yangian and the simple free field realization of the action of the affine-yangian on the simplest ADHM quiver or the equivalent cohomology of the simplest ADHM quiver to find the free field realization of the action of the affine-yangian on the equivalent cohomology of the corresponding modular space. And one can compare this free field realization with the free field realization of the vertex operator algebras that I've sketched definition of. So in this case of C3, in this case of C3, when the tericlabial free-volity C3, we have a proof that, proof of this AGD conjecture and it indeed uses the cohomological algebra in this sense. And it would be nice to generalize it for other tericlabial free-volts and... I have a very basic question. So this is work of Fagan and Bukov, associated with vertex operator algebra, there are four variables. So how are these related? Actually, close my second quiz sheet. Yeah, so firstly, I think in all the examples or most of the examples that they've been studying, they concentrate only on rank one cases. So the cases were the... I mean, double the algebra solving here. And, okay, another complaint about their work is that it doesn't seem to have very many concrete, like very concrete predictions. So, yeah, but the relation is... Can we start with the original perspective? Yeah, the relation is obvious, absolutely. So it gives them by different procedures, right? Like the... Yeah, but like... The motivation is different, but still the dollars seem to be... The physics motivation is similar, I would say, similar, but not the same, but... Well, kind of, because they're also wrapping in five brains on... When these four variables, it's exactly this setting of Bukov and the... No, but... Sorry about that. So here, associated with divisor in Calabria, three-fold, and they're associated with four-manifold. Yeah, they're... Sure, sure, sure. Yeah, so in the case when they're manifold... So there's really an overlap of R-setup and D-setup. If you consider a manifold that can be embedded inside the Ricobel three-fold or inside the Calabria three-fold, then it should fit within our framework. On the other hand... You mean, sorry, can you just repeat that? If your four-manifold is your divisor, is that what you want to... Yes, yes. But then you're saying that it's independent of the ambient Calabria three-fold? I don't know, okay, okay. This divisor was not compact and singular, yeah? Yeah, it's not. But also, an S-setup, you actually should start not with a four-manifold, but with a four-manifold and basically a gauge group, right? Which can be GLM if you want, but somehow it's true. You need some... So it's a four-manifold plus additional data. And here, just after the divisor, so... Here, we also have M's. Well, you could have M's, but the universal could be any divisor, right? I mean, just... You could have a multiplicity of one, for example. Yes, there clearly is an overlap, just... Viola, ask... Do you see a connection to youngians, to find youngians away from GLM for AD series? Or everything was like, GLM? Well, everything here was GLM, but also the reason is that I don't have good understanding how, like, what happens for more complicated PQwebs. Basically, all the PQwebs that I've been considering and examples, or most of them, are of the following form. Once we start considering more complicated PQwebs, it's likely that other youngians might appear. Let me give you one example. This actually went on to this geometric long-lance program. If you consider the following diagram, that is, indeed, not of this type, the corresponding vertex of the algebra turns out to be katsumu di algebra, based on this... I think it's labeled as d21, comma, minus psi, at level one. Exceptional super-li algebra, that's right. So you can get many interesting objects once you start considering more complicated PQwebs, but I don't know how to treat them at the moment, but I don't know how to say something interesting. But possibly there are there.