 Thanks, and thanks for the invitation. So I want to talk about ongoing joint work with Brian Williams on current signal models and their interpretation in terms of presentations of Carl Lee's Algebraids. So I'm going to begin with listing some of the main players in this talk. I will not say much about them. At first, this is just going to be an introduction and then I will say a few more precise things. Okay, so the first main player is the Chern-Symons theory. So you start with the Lee Algebra, let's say it's a simple Lee Algebra, and you have a level, which is an integer, and then a Chern-Symons theory associated to this data is a certain 3D topological field theory, and the category it attaches to the circle is the category representations of the loop group. So it's a ribbon category. So the category representations of the loop group at level K. This is the modular integer category? Yes, so this is not just ribbon, this is even modular, but it's not going to be important in this talk. You can also present this as modules over the corner group at the root of unity, if you like. Okay, so the next player is going to be the Karel-ZW model. So it's going to be a Karel theory on the boundary of Chern-Symons theory. So try to formalize this notion of a Karel theory living on a boundary of a topological field theory later, but that's roughly how you're supposed to think about this. And let me just say a few things about why you can think about this as a Karel theory living on the boundary. So for instance, you can look at Chern-Symons theory on some topological surface, sigma. So sigma is some topological surface, let's say close topological surface. And this is some Hermitian vector space. And what Karel-ZW gives you is a functional on this space. So Karel-ZW gives you a functional and this functional, so it depends on the surface, it also depends on the complex structure on the surface, while the space doesn't depend on complex structure. And using this functional, you can just compute its norm and this gives you the full WZW model. So the partition function of the full WZW model, not just Karel on sigma, is the norm squared of this functional. So this is partition function of the full WZW model and this relation is known as holomorphic factorization of the full WZW model. All right, so here are a few more players that's gonna be important in the talk. I'll also talk about certain two-dimensional CFT, which is slightly different from Karel-ZW. There's a curved beta gamma system with target X. So it's some complex manifold. If you don't know what curved beta gamma system is, you can think about this as a holomorphic twist, zero comma two sigma model. 2D zero comma two sigma model with target X. So it's a certain two-dimensional CFT and its vertex algebra is known as the vertex algebra of Karel differential operators on X. X is any dimension. So I said that there's a vertex algebra which has been by Karel differential operators. This is not quite a precise statement. Here you need some extra data and let me explain what this extra data is. So this theory exhibits an anomaly at one loop and to actually define the corresponding quantum theory, you need a trivialization of this anomaly. The anomaly is given by the second churn character, which is some characteristic class living in the second homology of X with coefficients in close two forms. And given the trivialization of the second churn character, you get a certain vertex algebra associated to X. Trivialization, you mean in one form? Yes, so think about this in, so for instance in the del bo model of differential forms. So given a primitive for this del bo model, you get a certain vertex algebra and this vertex algebra will depend on which primitive you've chosen. Okay, and the final player will be what's known as Karel Diron complex. So holomorphic twist of 2D22 sigma model with target X gives another 2D CFT. In this case, there are no anomalies and the vertex algebra is known as the Karel Diron complex. So the vertex algebra is the algebra of observables in this holomorphic theory. Okay, so what I've written is three 2D CFTs, Karel Lava ZW, the theory of CDOs and the theory of CDR. Two of them, this current beta gamma system and this homomorphic twist of 2D22 sigma model, they're what's known as holomorphic CFTs. So you can think about them as being on the boundary of some invertible 3DTFT while Karel Lava ZW is on the boundary of some non-invertible transimist 3DTFT and Karel Lava ZW lives on the boundary of the non-invertible 3DTFT, which is the transimist theory. Okay, and the goal of this talk will be to explain some generalization which is called the Quran sigma model. So it's gonna be a 3DTFT and what I will explain is that assuming certain anomaly cancellation, this Quran sigma model has two classes of Karel boundary conditions. In one case, when this Quran sigma model becomes invertible, those two Karel boundary conditions correspond to the Karel Durand complex and Karel differential operators. And in the case when the Quran sigma model is the transimist theory, one of those boundary conditions will be Karel Lava ZW theory. All right, so this is just an introduction. Let me spend some time explaining the framework, how one can think about Karel CFTs living on a boundary over 3DTFT. So, by the strategic drive construction, one to three CFTs, so three-dimensional topological field theories which extend down to one manifold, so to the circle are determined by their values on the circle which is going to be a ribbon category satisfying some point in its conditions. So given the ribbon category satisfying some point in its conditions, I can construct a one to three TFT. That determination is shall represent. Yeah, so actually, so the full statement is due to, let's see if I can remember, Bartlett, Douglas, Schumer-Priess. But in this talk, I'm going to concentrate on Karel CFTs. So I want to replace a ribbon categories by their complex analogs, by what's known as Karel categories or factorization categories. So I'll say in a moment what this is and another name for this is factorization category. If you're so accurate about that one, it's a chain-saming source, so it's not topological series. It has some legality of framing, so to what I mentioned, the virtual series which you don't mention at all about how I mentioned other anomalies. Yes, so that kind of anomalies, it's important but it gives an extra later structure that I don't want to discuss. Okay. All right. So let me explain what these Karel categories are or factorization categories but they should think of them as being holomorphic versions or algebraic versions of ribbon categories. Okay, so let's say C is some complex curve, then a Karel category on C and I will just give an explanation what Karel category is. I will not give a precise definition because it's long. Let's call it factorization category. So it's a collection of categories. It's going to be C, parameterized by points X1 through Xn on this curve equipped with some extra structure and I will just indicate the kind of structure you have without writing the full definition. Not as a morphos with the coolances of categories. I'll just say such as meaning that you have some kind of a coolances and this is a representative example. So if you have a category associated to a pair of points and the points are the same, then you want to identify this with a category for a single point and then if you have a category that's associated to two points and the two points are distinct, then you want to identify it with the tensor product of the corresponding categories. So we've seen this definition in Emily's talk when she talked about factorization algebra. It was the same kind of collection but not of categories but of vector spaces. So you should think about this as roughly being analog of the tensor product and then there's an extra structure which is the unit and again, I'll just write a representative example of this unit function. So the categories, this collection of categories were a plug in just the empty collection of points. You require this to be just a category of vector spaces and you have a functor into C of X for any point X and correspondingly, you can just increase the number of points and you have functors from C of any collection into C of any collection distance union, anything else and this is analog of the unit. Okay, yeah, so let me write two examples. One of them is silly. One of them is really important for the talk. So the first example is the trivial factorization category and this is just the category which to every point associates vector. So this is the category where all this data is just equal ones, tautological isomorphisms everywhere. What about power one, when the two points get closer and collide, you're gonna use the actual tensor product in vector. Yeah, but when I, I'm a little bit lax in which category I'm writing the tensor product. I'm thinking about tensor product of presentable K linear categories where vector is a unit. So this tensor product is just again, vector. More generally, you can look at any symmetric monotl category and this will give rise to a factorization category. So in this case, C X is vector. So you're gonna take any symmetric monotl category, let's call it A, it gives rise to a factorization category with C X being equal to A. And for precise construction, I'll just refer to Sam Raskin's paper. But another important example is representations with a loop group. Is this the same as what people call a chiral category? Yeah, so chiral factorization is the same. The only thing is maybe I want to add unit all. So sometimes people consider this data without the unit, sometimes with the unit. So this really references a whole graphic structure on the curve? Yes. Yes, so I wasn't per, I didn't put this in. This collection of categories should form, it should have a flat connection along the curve. So when you vary the points, yeah. Yeah, up to some kind of Riemann Hilbert, it doesn't matter. Okay, and the second example is the following. So if I take a point on a curve, I can look at a punctured disk around this point, around X. And then you can look at the factorization category, which to a point X associates representations of the loop group at X. Is it really necessary that you fix level K here? You can take K to be zero, for instance. Or what? Or can you just not have a level restriction at all? I mean, take the entire category of representations of the loop group now. Well, so if you don't put the level, that means at level zero. But the level means you put the central extension and then you fix the level. Okay, so I'll give one more example when I talk about CDO since CDR. But these two are informed examples. Okay, so another thing that I'll need is a notion of a lax factorization function of factorization algebra. So you can imagine what this notion is. So given two factorization categories on the same curve. So I'm gonna talk about lax factorization functions from CDO. Well, as you can imagine, this just means that for any collection of points, you have functions which will lax commute with the data. So there's a natural transformation between the corresponding diagrams for C and for D. Okay, and a factorization algebra in C by definition. So if C is a factorization category, in C, this is the same as a lax factorization functor from back into C. So if you don't like factorization categories of factorization algebras, think about E2 categories or bridge monodal categories. Then a braided commutative algebra in a braided monodal category is the same as a lax braided monodal functor from back into that category. So for the next five minutes, you can just think about braided monodal categories when I say factorization categories. Okay, and the example that's gonna be important is the unit. So if C is a factorization category, the unit determines the unit factorization algebra. So this is all I want to say about factorization categories. This is a very quick overview of what they're like, but again, think about either monodal categories or braided monodal categories. Do you have one question, Pavel? Yeah. So your example about cement monodal categories, suppose I'm not sure if this, can I take the curve just C, for example? What is the least you need to assume about the category in order to construct one of these kind of trivial ones? So I believe for C it's enough to have braided monodal category and for an arbitrary curve, it's not to have a ribbon category, but this construction has been written down. This has been written down only for cement monodal categories. Yeah, that's what I meant by ribbon. Oh yeah, yeah, you can, yeah, instead of ribbon, you can just stay balanced. Yeah, of course. Okay, so now, given this definition of factorization, category of factorization algebra, I can define a notion of a 3D TFT having a chiral boundary condition. Okay, so I'm going to encode into a factorization category, C. So again, if you think about factorization category as being analogous to a ribbon category, a 3D TFT, at least which extends down to circle is determined by ribbon category. And then one encodes chiral boundary conditions and I'll give some examples to this in a second. One encodes them in terms of lax factorization filters from C into Vect. So for me, the data of a 3D TFT will be a factorization category and the data of chiral boundary condition will be a lax factorization function from C into Vect. So out of, given such a pair, while in C you have a canonical unit factorization algebra and since this is lax factorization, you can take its image in Vect, then you get a factorization algebra in Vect. And so this is just plain factorization algebra. This is the factorization algebra describing observables in this chiral theory, any curve. Yeah, this is a precision category on human curve but now we say any curve. Yes, so it's- The long-term structure plays no role, yeah. So it plays no role. Yes, so let me just finish this sentence and then I'll say a word about whole work structure. So your functions are all write ad-write adjoints, is that right? You're asking if these functions are write adjoints to something, not to usually know. So probably for interpretation, should I think about this as some sort of categorified version of the map you gave before between insurance and- Yeah, exactly. So this is the data you get on a circle. On the circle, you get a category and a last factorization function. If you go to a remand surface, there you have a vector space and a functional. So this is the same kind of data on the circle. In, for the full picture, you're supposed to couple the two. You have something on the circle, you have something for punctured remand surfaces but I will not define this in a talk. Okay, so it's a word about homework structures. So one usually adds extra data, which is the fact that it carries an action of automorphisms of the punctured disk. So just all definitions are invariant and other changes of coordinates. But what happens in the topological field theory is that this extra data is actually better than just an action of automorphisms. It has an action of the homotopy type of the automorphism group. So this enhancement from the action of automorphisms to the action of the homotopy type is given by the Syngovarra construction in the case of representation of the loop group. And what happens for this picturization function is that it's equivalent to respect to changes of coordinates, but it's not equivalent to respect to the full homotopy type. So the Syngovarra construction acts non-trivially on the representation. But when you change a coordinates, of course the representation doesn't change. When you change coordinates of Lg. Do I understand correctly? If I say that's why you're not asked them to be compatible with the flood connection? Yeah, so this is the flood connection. Again, maybe it's more familiar what happens on the surface. Let me maybe go back to the case of the surface. So there you have a vector space and this vector space has a functional. This vector space has a flood connection over the modular curves. So in some sense it's uniquely determined by the topological type. Or as Maxim mentioned that there is a carol normally, it's usually projected with the flood, which I'm ignoring. But this functional is not a flood functional. So this functional is not preserved by the connection. It's still a holomorphic map. It's compatible with the homomorphic structure, but it's not compatible with the flood structure. So on the circle, this exactly means that it's compatible with the actual automorphism of the function disk, but it's not compatible with the single water data. Okay, so now it sounds a little bit abstract. Let me concentrate on transimans and just give two examples in the case of transimans. In particular, the spectrization algebra that you construct in fact is not a topological. Yes. Because of this failure. Yes. It's just a CFT. Yes. You can also try to, in a similar language, you can try to define topological. You can ask which carol boundary conditions are topological. This is those where this map is flat, or this lux spectrization function is coherent with respect to this data. But again, I'm ignoring this part of the data for now. So sorry, even when you start with something like the modular category to a quantum group, you still in this way construct something that's conformal. Exactly. So again, let me give an example. Okay, so as I said, transimans is determined by urban category representations of a low group. So you have the facturization category of representations of a low group. Maybe let me just say, let me not say that. Okay, I'll say it and it says something else. Okay, so this is my 3DTFT. You can think about this as a ribbon category or be a fusion or an easier algebraic way. Just think about this as a facturization category. Okay, now let me describe some carol boundary conditions to transimans. I'm going to begin with the carol double ZW and then I'll say one more boundary condition. So I want you to find a lux spectrization function on representations of a low group. This function is just a forgetful function. You have representation of a low group. You have a vector space on which it acts. You forget it down to vector spaces. No, it's lux. Let me just say what happens for the unit. So let's look at this category. So the unit in representation of a low group. So the unit is given by the vacuum module at level K. And so its image is going to be the vector space underlying the vacuum module. And if you're familiar with the carol double ZW, this is exactly the facturization algebra for carol double ZW. It's just the characterization algebra structure on the vacuum module. So notice that the same category can be described in terms of the corner group. It's a purely topological description, but the facturization function from representations of a low group to a vector doesn't have any reasonable description in terms of the corner group. Just because down the line representation is totally different. You can write any finite fusion category as modules for a quasi-hop top. Yeah, but I don't think this function will have any finiteness. Just because in the vector space you get are even dimensional. Okay, the next example, I'm going to start again with terms simons theory and I'm going to give a different carol boundary condition. So this carol boundary condition is called the carol total theory. So this is also to DCFT, living on a boundary of terms simons. And this corresponds to a laxation function from the same category. This function is what's known as Jujuflso-Kolovka reduction. I'll just denote it by DS. And roughly speaking, you have representation of the low group. You're restricted to representation of the loops of the new potent. So N is the new potent sub-algebra. And then you take comology, because this is topologically algebraic yet, it takes a mean fin-comology. Okay, so the other line frustration, the algebra is going to be semi-infinite comology of loops of new potent applied to the vacuum module. This is known as the doubly algebra. Is K inter here or not necessary? K? Yes, so you can talk about a version of this where you replace the integrable representation of the loop group, which is a modular terms category by something which is not modular. And this is the so-called cardinalistic category, in which case K can be arbitrary. And then indeed you can talk about arbitrary level. Hey Pavel, can you imagine computing the space of black factorization punctures or the homed space between rep, k-logy, and fact as factorization categories? Not really. So on the classical level, you have some maybe one shift of some black space and want to describe the space of all possible Lagrangians. There's some class of Lagrangians. Can I just give you some examples? So because these two factorization categories were kind of like simply built, is it true that really this function is fixed by the function in the fiber and it automatically inherits the factorization structure just from these categories being somewhat like trivial? Or is there extra data in equipping the forgetful function just between the categories? Is there extra data to enhancing that to factorization? Yeah, you're asking how easy it is to enhance this? So I think enhancing this to a lags factorization puncture should be more or less automatic. Again, once you write down the definition of this as a factorization category, you should be able to also write the lags factorization puncture. And again, for the claim about the drifus occular production, it's more or less in some papers I'm asking. Sorry, wish claim about that. That you have a lags factorization puncture. Oh yeah, this is also automatic actually. Yes. Okay. I mean basically anytime you can do something which doesn't require a choice of coordinate on a curve or a disk, that will be the transition. Okay, so now I want to say a few words about what chiral differential operators are and what the chiral Durham complex is. And then in the last five minutes, I'll say some about the Quran Sigma model which is the generalization of the two. Okay, so let's fix some complex manifold. You can talk about its algebraic loop space. So you should think about X power series T or maps from the disk into X. You have the full, so this is the arc space. You have the full algebraic loop space, maps from the puncture disk. And then you have the formal algebraic loop space. So this is the formal completion of the algebraic loop space along the arc space. So for me, X will be some smooth variety. So it's finding dimensional. So X is finding dimensional, but this formal loop space is something infinite dimensional. So to talk about the objects, I'm going to say you have to do a lot of work, but it's possible to do. So Capron of Bacero defined the notion of a D module on a space like this on a certain class of infinite dimensional varieties. So they've defined a fluctuation category. And again, when I talk about fluctuation category, the, what it's fluctuation on, it's gonna be irrelevant. So there is some curve in the background. It has nothing to do with X. So they defined fluctuation category of D modules on this formal loop space. So just think about vector bonds with a flat connection. Okay. And then the Carl Drom complex has the following description. So you have the inclusion of the arc space into the formal loop space. So the Carl Drom complex of X is the Drom homology over this formal loop space of distributions supported on the arc space. And just because everything here is about punctured disks and it can be made fluctuation on a curve, this is gonna be a fluctuation algebra. Okay. So next let me say a word about CDOs, a similar description of CDOs. I'll just wait. At least if you show me text as a scheme and not a stack, then why can't we just write it as a, why is it enough to work with L-Mod? Why, what do you mean by that? There is some renormalization happens when you do this push forward. And I don't think it is just any kind of Drom homology on L-Mod text. But this presentation will be convenient when I think of them as Carl Bonner conditions. So I want to give a uniform presentation of both Carl Drom complex and CDOs. Okay. So now let me fix the trivialization of the second churn class. Then the algebra of chiral differential operators, it has a very similar description. So that's what David asked. You think about, for instance, the law complex competing this homology and you fix that previously for that. So in this case, you just, instead of taking the wrong homology, it should take coherent homology on the algebraic loop space of the same object. And again, this is gonna be a fluctuation algebra. So the choice of the trivialization of the second churn character enters in interpreting D-Modules as some version of quasi-coherent sheaves on the algebraic loop space so that you can actually take coherent homology. In general, you cannot take coherent homology. All right, so the upshot of this is that you have fluctuation category. You have one lags fluctuation function, which is the Drom homology. And you have another lags fluctuation function, which is coherent homology. This fluctuation category has a unit object and the unit object is the push forward, so it's distributions on the arc space. So this exactly fits into the form, as I mentioned before, you have a 3DTFT. This is a somewhat silly 3DTFT and then it has two carol boundary conditions, which produces carol drum complex or carol differential operators. So for this boundary condition, as I said, you need to choose trivialization of the second churn character. Yes, it will have a name soon. Is it okay if I take five more minutes? Okay, so the name will appear in the next three minutes. Pardon me? You really mean Vect and non-DGVect, the homology? Yes, so you can do non-DGVect, at least in the second example. Yeah, I mean, I have to think about this. Okay, so let me now briefly state the generalization. Okay, so I'm not going to say what the current signal model is. Let me briefly say what the data is for the current signal model. It's what's known as a current algebraoid or a twisted current algebraoid. Again, I'm not going to say the axiom so for a twisted current algebraoid, but you have a pairing, you have a bracket, you have an anchor map and you have a floor form and they satisfy some axioms. Okay, and this data is equivalent to the data of a DGV algebraoid which is concentrating two degrees with the map being the dual of the anchor map. So this is in degree zero, this is in degree minus one. Yeah, so this is in a joint with work with him and equipped with a two-shifted symplactic structure. I'm not going to say what this means, but if like this two-shifted symplactic structure reflects the pairing. Okay, so the upshot, so the conclusion, the current signal model is described by a factorization category which is very similar to turn assignments. The definition of L, should that be E or E dual? Because you have a pairing on E, you turn E into E dual, E dual into E. So you have a factorization category which is representations with a certain central extension of the lee-algebraoid L, but not on X, but on its formal algebraic loop space. So if you think about L just being over a point, so you have a lee-algebra, this is going to be your representations of loop algebra with some central extension. So if, let's say fixing a trivialization of K plus the second turn character, you have a certain boundary condition and having another, so this is kind of an anomaly, a consolation condition. And then there's another CDR-type boundary condition assuming some other anomaly consolation. And let me just finish by saying that this current signal model reduces to turn assignments when X is a point, then you just look at the lee-algebraoid with a pairing. The CDO boundary condition corresponds to the Karel-Dawels-W model. And then for any X, there is a notion of a standard current algebraoid. In this case, the CDO-type boundary condition for this current signal model with respect to the standard current algebraoid is going to be CDO. And then the other Karel-bound recondition corresponds to the CDR. There are questions. In Phil's talk, he explains to us this story about boundary conditions in terms of Lagrangian correspondence. In the sort of appropriately like Karel case of the right dimension and possibly if you have some anomaly consolation condition, can you take his data and produce your sort of data? Yeah, you can also describe both boundary conditions on the classical level. And when you do that, you get sort of Lagrangian. Yes, so Lagrangian-type data corresponds to a slightly different anomaly consolation because there is actually an H bar in here. So in the classical level, you just need to trivialize K, which corresponds to a notion of current algebraoid rather than to the current algebra. Can you talk about dynamic linear and Baxter equation using this formalism? Like you've told me before that there's also some lags with all the transformation. I'm not sure how this fits into this story. Great, let's thank the speaker again.