 First speaker this afternoon is Thomas Kreuzig, and he will talk on categories of line operators and VOAs. Yeah, thank you very much. And thank you very much for inviting me. It's really nice to be invited by you. I'm mostly working on VOAs and right at Kenza categories. And I think these appear a lot in the physical theories and higher dimensional gauge theories. And it is in general quite a hard question to understand what kind of categories appear there. And so my talk will be about the relation about what type of theories that you will hear. And this will be an overview of maybe it will be. So in general, if you have a physical system, we have our symmetries that might be manageable or some group. And VOAs are an important example of this because VOAs are the symmetries that arise from two dimensions. And then usually your symmetries are not that big. The two-dimensional conformity theory at least have an infinite dimension of the edge of us and the symmetry of the whole edge of us. And that makes the two-dimensional conformity theories very interesting. And the respect that in principle, they are exactly the same. And usually the symmetry is more than just the whole edge of us and the bigger structure. We just might hear this. And now VOAs arose a long time ago. And more modernly, we're not the only ones with the same kind of modules. We're kind of talking about a peer about what kind of positions which I would have, for example, whether these modules stand up in the general structure. And in general, we just want to combine all the subject categories of the patient and the other modules. And we want to study those. And in particular, these categories, it's not only this edge. It was had a few of the hierarchical systems that the people, the category and final structure, was catering like a braiding or other kinds of work here. And so the problem is, of course, the more final structure you want to refine, the more hard your problem becomes. And in particular, the type of categories that appear in current systems are not, I mean, it's extremely hard to understand what they are. On the other hand, if you come from physics, a physical theory is often given, for example, by some lung organ. And then you can split your lung organs into some simpler part and some more complicated part. If you can think about it as the simpler part between some type of field term of your story and a similar type of story, or for the small VOA setting, that means you kind of always have a nice structure sitting on the top, and you hope that it governs a lot of your structure that you are eventually interested in. And so what my talk is about is that the VOA's that appear in modern context, usually always allow for feedback variations. And I want to explain to you in what sense the good algebra, together with some additional structure, somehow completely above us, the complicated category of your story. So the idea is somehow we could find the simpler algebras inside the regular structure. And if you would take a course about VOA's, then in the syllabus, probably it would be written that the requirement is a good understanding of the theory because you think about that VOA is some kind of natural contribution of say FFIN, the algebras. And so I thought I would just tell you what my picture boiled down to SS2 would be. SS2 we all know is three dimension the algebras and the phases. We call this phases and then PFH, H for the diagonal part, the sub-algebras. And in particular, the current sub-algebras, nothing but a one dimension, a billion algebras. So it is as simple as it can be, right? And of course, what is SL2 itself? It's a three-dimensional representation of its own one-dimensional algebra. So in particular, we can do a SL2 as a three-dimensional object inside this representation type. And we also have a smaller structure that we can see the algebras generated by a unit. We can pass to the university. I don't think that just polynomials in a single variable, like each polynomial takes a series of degrees under the data of the sub-algebras. And I mean, of course, you don't really use this in order to do representation of SL2, the point I want to make is that this is the passive picture. I want to, and this type of structure turns out for the much more complicated, I also wanted to say there is a kind of a converse that point of view as a group cannot only be viewed as an extension of a very simple structure. And it's part of the sub-algeba. And you can also know it's a knee algebra. Knee algebra arises naturally as a variant of a knee group. That means it has differential operators. That's what happens in particular in the knee algebra where you can use a sub-algeba of an algebra differential operator. Again, the edge of a differential operator is a kind of very simple structure. And so what you want to do is you want to be used as similar concepts, but for the work itself it was for some kind of a complicated reason. Okay, so I don't know who she knows what the work expected of her is. This is not even the complete definition. What it wants to do is it wants to formalize the motion of what the tidal energy of the fluid dimension of the form of field theory should be. So it should be a space of state, the vector space v, together with the state field of our components, that means to every vector and vector space in the associated field, in the case of the formal power series with the professional operators on the vector space. And then we want to have some additional structure here for the UAA structure. This identity is the most important point. It's called locality. And you can read it, right? It looks like a field for mood. And you should think about it as long as they are separated space-time, then the field of mood for the component. And it turns out that this locality implies that the vector algebra has some operator product algebra for family of algebra functions. Anyway, I just wanted you to see the definition of what we haven't seen yet. And of course, if you have an algebra, you're not particularly interested in the algebra itself, but in the case of the station theory and what should be a representation should be a very similar structure. I can make those things together with fields on it, but now that they're back on this vector space. And then you would even want to do that to have more because you also want to have fields that intertwine between different representations. They are called then intertwining fields or intertwining operators. And these are the things that the physicists called the fields of a conformal field theory, because these are the things to use to build non-sugar correlation functions. And for the mathematicians, these correlation things are also the few things these are certain fields and the morphology of the intertwining value of the formal power series. And then if you do some carrying of those, you get the functions on this common variable that maybe actually become meromorphic functions. And then depending on which order we define correlation, take your correlation functions in these meromorphic functions. Not usually not the same, but you can somehow meromorph because they're converging them by regions. And it turns out that this meromorphic combination is the key thing of the deeper structure of the documentation theory of what it says to us because it is what defines a monoidal structure and a little combination of all the ideas very much from physics. In particular, you would say the axioms of physics imply that the representation for the new A must be a greater kind of that it must be even a limited category. Unfortunately, that's something that's very useful for the board, but it should hold on to all of that. Because the standard product is defined in terms of analytical combination of correlation functions, it's nothing. It seems like a social activity or communicativity that usually never holds on the nulls but on the top two isomorphisms. And that makes it quite interesting category for us. So as a physicist, you would say a category of monoidal structure or that should be a ribbon category. I want to give an overview of many things that one would sketch for a ribbon category. It's a category that has a standard product, has a standard life type that satisfies the usual coherence properties. This is another product that's sort of that you have to make sure of isomorphisms. And then, but because you also want taking different orders, if you are isomorphic, so it's kind of a product you thought that would use it after isomorphism for this part of the writing. And then you want other products that you want to have to do it because you want to be able to take traces and then there's a balancing product which this is another product, but just actually totally on the negative. Okay, now, even if you haven't heard about the robotics edge of us, that much you, they found their place in history. They arose with string theory just because a string propagating a space time that it gets to a worksheet, to them in the worksheet. And so it turns out that one of the key of the worksheet must always be a conformative theory, so two dimensional conformative theories and to make we contact connected to a string theory. And it has this nice advantage that it allows for a real mathematical formulation and particularly that means that you have certain injections that some will arise and this is a tool that you might be able to apply. And so a while ago, once this moonshine was, for example, an important problem, that's because it connected this largest or they simply move to automatic forms and string theory totally exactly and it's important to improve it and for us to be able to do it. But that was probably the first highlight of the conformative theory or new A's. All the other highlights are really related to the general structure. So then quite early, you and Xybox were able to axiomize what a C is future means and it was not that everything. And they really defined more than 10 by 10 by 10. This means two R and A's took the moonshine by formulizing it more than 10 by 10. And one of the 10 by 10s that really found a role because it's the thing that we're used in order to improve the values. But not score means the impact on the variance of the two new moons. So roughly speaking that you have programmed and not too many associated red and black for red and black color modules of your category and that spares over the morphosing. That's where I mean. And probably much less known but still very important highlight was that a little later, Kassel and Lucic were able to connect the most basic new A's that are the one related to F and V algebra's point of views. That's kind of a highlight because you can imagine that every module is connected by the vector space and the V algebra action is given by the fields and the product is defined by the correlation function. And it's very, very cumbersome to understand exactly what the pattern is. A quantum group is nothing but a quantum definition of the V algebra. That means it is in most inspector but the quantum group of the V algebra behaves the same way as the V algebra. So this means if you have such a close point is you should move it as an extreme insight on the F and V elements. And that's in fact also my main motivation for what I'm presenting just turns out what I'm presenting is what I do for a long time. Okay, so these highlights there that were from the early days of things really. And then not much happened for a while until we met Rasteli and others wrote a differential paper. But actually that's not a true that they were. And in general, they are quite a few of the most common two-dimensional series that we have four-dimensional series. And then one can translate them to all the interesting mathematics. For example, the AGP, AGP for one is an example of a four-dimensional four-dimensional correspondence and that it translates to the third most important lesson of the extension was X1. Another one that provides a style of four-dimensional H theory is quantum dramatic variance cross-pointing. And you can formulate this in two ways. It's either dealing with equivalences of spaces of four-dimensional times of the U of A because we will then do the dependent spaces of the two models on the remand surfaces. Or you can actually, that's a phrase of in terms of categorical relations basically they are effectively the same thing. Other things that got quite a lot of attention are that they actually has invariance of three and four-dimensional. Instantly modernized space problems with special case of this four-dimensional problem. And for example, taking the loop of the idea that you think about that a four-dimensional has an attached invariant, which is the U of A itself and that has a three-dimensional boundary and to each boundary component and attach the relation to the U of A. And then gluing many parts is consistent with gluing the U of A categories in kind of a higher dimension. And then one song will tell us a bit about topological recursion, which is automation. Representation categories for my perspective, most of them are related to writing of line operators. Line operators can engage here. So yesterday we had a talk about three-dimensional gauge theories, but there were any of the two that were symmetric. So let's say that the connection to U of A is much less here, but if you have any, but more gauge, super symmetry, in that case, it turns out that you can do a twist, but that's topological. This has the property that at the boundary, it still has the following of the two A's. And so that means you have topologically just a gauge here that's both in U of A's. And this is quite nice because as a physicist, you of course want to understand your structure. So the point is you want to understand how the line operators and other categories which are topological means that we take derived categories, derived by the template categories. And now you want to have an honest model for it. We can come up with a perfect method. And models of derived categories of the sheets on the product menu for it. But you can also simply come up and say this category should be the category of U of A modules because each line operate the end on the put them in the subspace, the end point should be derived by fields. Now, so as a mathematician, you would like to have an honest model for both the categories of line operators and you would have two choices for this. Do that by the geometrical or how you take a little bit of a solution for it. Take I do the letter. And now why is this challenging thing? Basically always these categories are either finite, this is basically them in the next sentence. They are not the same in sentence, it's not each module derived in a simple module. And then take the fact that some point in yours, this extension can be done in a different way. So they are kind of complicated. And in fact, there are some notions that we have to start this way. And generally, so I mean, it just means you can get up, understand it. But maybe we don't want to keep it in the span. So here's a picture, you just think about the line operators and you can choose them or you can go for the cross-interface. And the cross-interface is the item of the kind of product that should be non-trivialized model. And that should be exactly the kind of activity as a model in your thing. And so, yeah, what I stated two minutes ago. So the question anyone wants to ask is, so what are these categories of line operators? And there aren't geometric proposals, which I didn't study. Do I say much about this? The drawback of the geometric models, I would say, geometry is good for many things, but it's not good to actually understand. It's a good idea to go, whether you are able, but in principle, we call it the cross-interface. So in principle, we call it the cross-interface. This question that you want to have immediately, here's a picture of what you think about it, what you think about it. And then the line operator, here we have a good image of subspace. So the line operator's end on that and the end points, it's there to identify modules in the line operator. Now, they propose that for the n equals 4, 3D gauge series, there's this polymorphic twist, that the boundary, when you have the boundary, you still have polymorphic objects in your A's. And so one really wants to understand. And in the very first place, this means you need some kind of physical information. What should be the view A's that appear there? And then you have to study the patient category of this view A's and then you can then study whether these representation categories you pay it as you would expect. And for example, I'm fortunate that I'm talking to a good or the most planning a lot. He had a good information about this view A's. And so what we did was for what is a quite a billion gauge series. There's a concrete proposal about the view A's. And the construction of the view A's is extremely different. They are very different. But the interesting thing is that the category of view A model of the A-trust is the equivalent of the category of the view-trust of the mirror of the view A. So in some sense, we can use the category of line operators to test the mirror of some kind. And in fact, we can use the line operators to test the mirror of some kind. And in fact, when I got here and I thought I would present this result, the problem is it's a very technical result. What comes is simply that these categories are equivalent. And so I would, I would not, probably not, I mean, I thought I may be better if I explain a little bit more generally where categories come from because I think this is a very good insights. And one point of physics is that whatever the view A's that come up, they somehow always allow for a nice picture representation. And this should be true. So what, now a little bit more precise, what I mean by a view A's is that I have a given view A's and you have the most simplest type of view A's are called impermeence and equivalence. So by a view A's, you have the embedding of your view A's kind of product maybe many components of the audience or maybe something close to the label that's there. And then what you want to understand is you want to understand the category of new modules called a view and what do we have is because the view is a sub-argument of the field algebra which means every module of the field engine that's at least an object. But of course they are by far not all of them. So we have to go on. Anyway, we have here C, the category of modules of the field algebra and this is typically a super easy category to edit that space. To edit that space. I'm not believing that this is the way to do it. I mean it's the simplest type of kind of category that you can possibly imagine. And so for the question one has to ask is how is the category of modules the category of the complicated view A what are you what sense is it completely completely determined by C and A by how does the view A lie inside the field algebra and what does the field A do. And I can get some kind of possibility at the end. And so we need to so what do you have to do is you have those view A setting the formula Now you have expected to categorize the question question and what's the extraction is you have your small view A that has a ten dot category and that is a sub view A of the bigger view A Now it turns out it's a bigger view A we can view as an algebra and there's an agent category very much as likely as at the beginning I view the S2 itself as an algebra in the category of C So what is the categorical algebra it's an object in case of modification satisfying the obvious it's extreme so here I put down the S of the facility that we have modifications and similarly we call an algebra Now whenever you have an algebra in a category or what do you want you call it an algebra you also want to have a notion of algebra and so you can also go on with that and introduce A modules in the category of view A and it turns out this category always has a ten dot structure but it's much more greater However it always has a smaller category of so-called global models and in our setup this is exactly what we call algebra and with the super nice category C this is exactly what we want So really the first category the question we want to ask is can we somehow recover you from this category view A and then the next question you want to ask is how can I how can I know that view A is by simply looking at that you can understand both things Okay and since this is somehow a little bit technical I thought I would know simply illustrate everything in one example then I give you a little bit Now there are there has been a lot of attention to this but surprisingly even though the subject is 30 years old there are only two the rhythmic view A that are much bigger this one and so forth saying that there are exactly those that appear and so I don't have much choice so I don't have much choice and what I can do and that's what I do here very early when people were talking about or not any invariance come from from the consignment theory and it was also more or less the first example of the relative C of B from then but you see it was from anyway it's an example that I can give you a nice one sees many things so so it's a super algebra just 10 for the real two part algebra two part and then it has even an auto-imagine tool the diagonal one even one but patient relations in particular and I only wrote down the non-zero one from particular central which you could have guessed from and then and now you can write down some patients what you can do is you take a high-slate vector on your car and then you have these two odd ones so the high-slate means it's really one of these odd elements and the other one really but since each of these drama of the two odd elements does get to the central element it's here that the central element x by 0 then with two dimension what you're going to use to control and that means you immediately get a nice example of two-dimensional models that have one level of top model and one level of top model and you're almost already done with understanding the final representation of the finite dimension we have the only question is now are there any extension of this level of top model and you need more that you don't see here and the answer is well there is an obvious one because what we're going to do is we're going to start some kind of generating vectors and let the central element x by 0 of these two elements they can actually so that if the top dimension module and it will be a top dimension module that has simple sub module and the portion value that has a sum of two so you can see the central model so one one likes that will be such a wide definition. And now what one uses for that's a usual dictionary with the study of the presentation period, the final time in the algebra, then the study of the presentation period of the FND algebra. And then for any FND algebra, it's also the vertex algebra and the other side of which representation I encourage you to go to my station. In this case, it comes out of all the presentation of the FND algebra, and if you pass from the new algebra to the FND ones, what you just questions to do is to take the new algebra and send it to the standard for having one element for FND five plus five minus five. And the assets that are there are related to the new algebra just can be such a wide definition difference. The interesting thing here is that the reference of the representation period is FND algebra is enormously similar. So that means the generic rate labels you have in view of a model that's similar and projected. Nothing interesting happens. It's just that these four that have this projective model is the four competition factors. Before you only had one such model for each family but now you have one such type of problem by each. And each type of that family can have a certain bigger family. It's kind of nice that the presentation period that you're going on is completely as it is for the final dimension of that family. And then you will pass to the vertex algebra and the, yeah, I'm just writing down the whole complete form of the FND, I think use with your summing, some flavor of what's happening now. Okay, now, the question you want to ask is not only what is the representation category, this FND algebra, this FND algebra is the best opinion category. Answer what you want to understand what it is as a family. And this goes as follows. So now you need to find the fixed realisation. In this case of transport for each family, you need to figure out for the real ones, you need to figure out what ones are the fermions but the real ones are the fixed ones. The real ones, then the second one are all the fermions and the first one are all the fermions. And then one can quickly compute that what happened with that, and you get the right answer. And in fact, you get this formula relatively quickly which you compute the realisation of finitely the long one, finitely the very long one. Now, next, you want to understand whether this sub-agent is somehow nice to get the right answer. And that you kind of get from physics to the physics component, that's the written theory. And this, let's say, here is a lower part of the order of the combination, and that part of the order of the combination is really the function of these two fields together with the detection. And the recognition process is actually some kind of spinning charge. And what it is, it's just the zero-mode detection of the final one. And then you want to check that the detection embedding is precisely the final one. Okay, so and the point I want to make is that the carrier of P11 is completely determined by the P22 edge of us to get out of the knowledge of the speech. So this is kind of what replaces the P11 as the PVR. So first of all, the P22 edge of us is extremely simple. For example, if you go along, what I said in the module, this is the first thing that you learn, you know, it's a completely not UAA code. And the end product is just added and addition in the variables, and particularly in other parts of the categories. And I think the category of greater than just basis, but with a non-period grading, which is the whole dimension of what I'm talking about. This is really as easy as a 10-byte 10-byte 10-byte 10. So that would probably be one of the first ones. I mean, we would see that actually it could be a 10-byte 10-byte 10. Okay, so, but we want to get a complicated category. So we cannot, we need to somehow try to do something a little more complicated. And this is called the Nicher's Edge of us. So and what the Nicher's Edge of us, what is it? One has a screening chart. What is it about emphasis? The character is the calendar. And the screening chart itself has a square for real and it is an intratina associated module. So this means you identify this screening chart with generating an algebra, then just the the algebra generated by x or the relation x1 with 0. Very simple and simple algebra. But now as an algebra in your category, which just means maybe think about it as the algebra generated by one of these modules. And in particular, you can now think about what kind of modules can you generate. You can have a kind of cross-platform type of module, a kind of deferment, and act with your nicox algebra generated. And it gives you another such module. It will make better changes because you know your nicox algebra generated can is the rate. And but because it's square for real, you're extending has to terminate the problems. Not the most. And with this, you're already completely understand the opinion category of modules, nicox algebra inside the category of the head of the transporters. And it turns out now it was category of nicox algebra representation. It is a canvas that we would like to describe. Okay, now the point, what is the point? You have a kind of a category, and kind of a category of people know very well if you want to pass from a kind of a category to a greater kind of a category. It's simple double it. You go to that point in the center. And then if you have a sub-category in this center, you can also get the centralizer. Another, which is then just a relative center. This is a very, very non purely categorical production. And so that the point here is simply, you have a few realizations. Inside the speech, you have an extremely simple category. You have a nicox algebra and a nicox algebra. It tends to attend that category, mainly the relative center. Relative aspect to the category of modules. Now, why is this so interesting? Because relative center are realized by the project. And that's a very quick computation. That's why this nicox algebra, realizing quantum group is 100211. And this page means in particular, we have a theorem that the representation category of f and v are a of 211, the same thing as the one on point. It's kind of a very good theorem in the sense of that it's something very hard to do. But quantum 011 is really, really good. And the main reason why our theorem works is because we are able to get modules. And, of course, we didn't want to take the point as a general statement. And this is the whole point. So if an, unfortunately, very big amount of assumptions hold, then, and we have a VUA, very much like the presenter point, that embeds into a nicer VUA, for example, then the category of the modules is equivalent for this given by the nicox algebra. Now, in practice, the big caveat is that provided the series of technical assumptions can be verified. Those are many, many that I effectively do, trying to develop a theory that kind of guarantees these assumptions. But you can also take the standpoint that we'll see. What would the physicists say? Well, it would kind of be the axioms of your performance theory, that all these natural assumptions are identified. And then our statement would simply say that all our statements say, even if you can't verify these assumptions, we simply, our statement defines that. And just by having those realizations, where are all of these spinning charges that needed to get you a nicox algebra, then you needed the path and associated location, which looks funny because for you, this is just a bunch of thin webs. But what the point is, these categories of different modules are as explicit as possible. It's a very, in particular, you can relatively easily write down and you're realizing algebra, and in particular, for me, it means I immediately have a projector, which should be, and it's a extremely safe projector. And in particular, since all these physics categories, for example, we use these VRAs associated with speech theories, and essentially always allow for the realizations. It means we know what the category should be and we know what should be working with, and that the kind of nice moves. Okay, one more remark. I presented this because I effectively have thought about special for the last three years. I want to put Tasha Murshchikov on them so we can understand them. But I also kind of like physics a lot, and I'm talking a lot with Google and we had this excellent research with real academic and real, and so we have a program. And so for recently, we showed that mirror symmetry for this, which really is this kind of processing, the length of the line operators, like we have been in case, but as I mentioned before, these categories of line operators also have some geometric models, and you want to connect with us. And we very much believe that, I mean, it's very hard to connect directly. But what turns out this we have geometric models that translate into a method characterization. There you have some polar duality, and it turns out that you can use that to expect of the realizing. And in a toy model, you already see that it's the same. That's the one that we get as the blue one for the delay. And so I think our program at the moment is a very big, slow one. Ever on the end, yeah, we know what we're going to write down and realize what we hope to do, and now we don't make expectations. We study what kind of edge of our business, and I find that's where I find this line. Okay. Yeah. Thank you very much. All right. Thank you very much. Time for questions. Oh, they're never unitary. They can't be unitary. So the associated varieties, they are also one of the big branches of the Gage series, that's why you asked the question, and they are sort of protected varieties. And so I didn't look at this, but they are pretty good. Yeah, that would know. I didn't look at this, but I would know how to do it. And in fact, for people for the week, the Bonaville A is just from F and C, they're very unyielding one. In that case, you had this very basic of this basic example that you said, this is well understood, and you have this exact sequence, and you can go to the slide. So did I understand you correctly that this representation, I think it was, yeah, there, there. Did I understand you correctly that this Pn then is a representation that I can think of a logarithmic C of t? So this Pn, does that correspond to a logarithmic C of t? Yeah. And how would I now see that I get logarithmic from, can I see from the sequence that the correlation function have these logarithmic branch cuts? Yeah. You can already expect that. So the logarithmic correlation function have the exponents in the, so usually the exponents in the correlation function are completely specified like, like the formal way that means by, you know, so little more. And if you get a logarithmic singular, it's going to have other, you know, more, that's not excellent. So the conformal way of the exponentiation of the conformal way that's exactly the variable that's in the category. And so this means that the logarithmic similarity comes from non-sense, logarithmic stress. But what is the logarithmic stress? It's more for you on the model. What are the endomorphisms on the end? Because the identity, the logarithmic stress will be a larger pillar of the identity. But then you also have a new function, a small person next to the top, with a sofa, and everything else to do. So that means if you knew what the logarithmic stress was, for example, for the point that you know, then you needed to compute that it does not have the same sense in here. And then you would know that it's very nice. Okay, thank you. Other questions? Do you see any? Well, let us thank our people on the channel. And we take a minute to move on.