 Okay, thank you very much. So today I'll talk about some relation with the Iain Algebras and vertex models form statistical mechanics. And I want to start with stating two conjectures, which are due to Maxime. One of them is about quantum field theory and it is kind of imprecise just because we don't have a definition of quantum field theory. But the other one is a discrete version of this and which is very precise and we actually, I also suggested proof of it. So the first conjecture has to do with quantum field theory and so I assume we're given a definition of what a quantum field theory is. So let's start with a QFT on some flat space Rd with translation and dilation symmetry. Is it all right with the L? Yeah, yeah, okay. I'll stick with D for the moment, but sure. With translation and dilatation symmetries. And let's say that in this definition of a quantum field theory, there would be some vector bundle of fields on our space. So with a vector bundle of fields, let's say V. Then the conjecture says that translation invariant forms with values in V gets an action of the d-dimensional little disk upright. Yeah, I think you can actually drop this. So it is true, yeah. So let me make a bunch of remarks. Okay, so first of all, like if we were working with complexes, which is the case here. So if we're in the category of complexes over a field of character C zero, and if like we have an E and algebra in complexes, then A shifted by N minus one is going to be a Li or is an L infinity algebra. And presumably, I mean, or like that's part of the conjecture that this L infinity structure should be related to the combinatorics of Conan-Kreimer, like renormalization. So that's the first remark. A certain remark is that actually, like part of this has been proven in some work of, let's say Costello, Costello with Owen Colliam. Maybe also by some physicists like Steffel Orlands. But all this work, they actually deal with theories which are very close to free theories. So only for the perturbative setting. And in principle, I mean, the conjecture should be true for the formations of theories which are not free. So I'm sorry, is it the proper tourist structure so that they get the section? I mean, is it something that you just need, I mean, is it structured? Do you need to just define a section or is there something you have to check so that the check that we get comes to construct? Yeah, you have to construct an interesting eddy structure on this, yes. So you really have to construct it, so it's not a matter of checking, it's not a deal. No, it should come from, let's say, some kind of operator product expansion. But yeah, you have to construct this eddy structure, the action of the eddy operator. It's not something that you have and you have to check that it satisfies the axiom. You really have to construct it. Yeah, and the third thing is that there's a kind of interpretation of this in terms of deformation theory. So there is this slogan, which is now a theorem, that formal deformation problems are equivalent to DG-de-Agebras or L-infinity-Agebras as infinity categories. And the fact that we would have a, so there's translation invariant forms with that in the bundle of fields. This should be up to some shift. This should be the tangently algebra of some formal deformation problem. So the formation problem of this translation invariant QFTs. And the fact that there's an eddy structure means that this deformation problem should actually lift to some en deformation problem. So this should lift to some eddy deformation problem according to what Lurie actually proved. So Lurie proved that formal modulator problems parameterized by commutative algebras are equivalent to the algebras. And formal modulator problems parameterized by eddy algebras, which are kind of, well, that these are commutative, but not quite commutatively algebras, let's say, are also equivalent to eddy algebras with some shift. Okay, so I think there's also some recent work of Chris Elliott and Pavel Safloff about, so that there's some results saying that if you have a factorization algebra with translation symmetries, if you're taking, well, I don't have like the DRAM translation invariant part of this, you should get something which is very close to an EN algebra. So there's the radius also problem, there are several results which are going in this direction. Anyway, what I want to talk about today is a discretized version of this. And so I'll state everything, so in dimension two, even though everything works in any dimension, it's just easier to draw pictures on the board. So here is what I'm going to do. So we consider a square lattice in R2 and it's gonna be oriented. So it's an infinite graph if you want. And I'll give myself two vector spaces, H and V for horizontal and vertical. So the element of the horizontal space will be labels on the horizontal arrow of my graph and the vertical element of the vertical space will be the decoration for the vertical arrows. So one way to, and I'll give myself R, some matrix, so an endomorphism of H tensor V, and let me explain how to interpret this. So let's assume like it's not necessary but it's easier to explain how it is, what it is about. So let's take some, so let's assume that H is and V are finite dimensional and let's take a basis of this. So like EI for H and FJ for V. So then the matrix R gets matrix elements. So it's some basis element EI tensor FJ. And like I'm not writing sums, so it's gonna be R, I, J, K, L, E, K, tensor, F, L. So here's how the way to interpret this. So we have configurations like this. So locally around the vertex of our lattice, we're having, so we'll decorate edges. So we'll decorate horizontal edges with E's and vertical edges with F, so basis element of H and V. And R, so the coefficient R, I, J, K, L is the weights associated with this local configuration. Or if you want to think in terms of probability, if you're counting things with probability, or I, J, K, L is the probability that you have such a configuration locally around the vertex. So and the game we're going to play is to count configuration with weights. So like that's the state sum game. So here is a bunch of examples. So, but before this, so I'm going to do some kind of state sum computation with boundary conditions. So I have to tell you how to deal with this boundary condition. So I'm going to have a space of boundary states. So let me take some, so U will be some open, bounded, and even like just for simplicity, I'm taking convex, but it actually doesn't matter. We can also even only take both. I mean, any basis of the topology of R2 would work. So I'm taking such a thing. And so I have my lattice here and some, so U will be something like this. So I'm going to have boundary edges. So like let me write D plus H of U. These are going to be the incoming, that's for the plus, horizontal, that's for the H boundary edges. So that's the state set of such things. So what is it exactly? So that's an edge which is horizontal. That's pretty good what it is. And saying that it's incoming, it's just that E intersects U. And moreover, the source of E does not belong to U. Basically, like this is a horizontal incoming edge for instance. And like, well, you can define like D minus H U for incoming, outgoing, sorry, horizontal edges. And you also have D plus V U for vertical incoming edges and D minus V U is going to be obvious as well, okay? Right, so now we can say, so let me observe that there's a canonical bijection between horizontal incoming edges and horizontal outgoing edges of U. That's pretty clear. I mean, like since the region is bounded, like if you enter into the region at some points along the line, you will go out of it at some point as well. So there's a canonical bijection and a canonical one between these and as well with vertical ones. So we can define W of U. That's gonna be our space of boundary states. That is going to be endomorphisms of H tensors to the power D plus H of U, tensor V to the power D plus V of U. So what it means is that to every incoming horizontal edge, we associate an element of H and then to every outgoing horizontal edge, we're associated an element of H dual and the same. And the same for the vertical edges. Okay, so now we're ready to define the partition function. And the partition function associated with a given region U is going to be counting all possible configurations with weights with associated boundary condition and the boundary condition will be some specific element in here. So here is how it works. And instead of giving you a formal definition, let me give you a bunch of examples. So first example, let me take this. So let's take U being this. So I'll have decorations like, so I'll have EI, FJ, FL, sorry. EK, FM, FM. So all possible decorations will be this. So I'm gonna construct, so ZU will be an element of WU. So I'll write it as a map from K to W of U and it sends one, well, to ZU one. And that is defined to be, well, that's gonna be an element of this. So it corresponds, so it will be like a basis of such an element of this space is given by tensor product of all these things. So I have to describe your sum over all these things. So it's a sum over all the indices like I, J, K, L, M, N. And then what I'm going to do is to put the matrix R at every vertex and sum over all, the only remaining possibility is summing over all possible decorations in the middle here. So now I'm putting, so that's gonna be sum over S and that's gonna be R, I, J, M, S times R, S, L, K, N. K, N. Actually it's better to put indices on the left and the right and the top and the bottom than to be. Sorry? It's much better to put indices on the left and the right and the top and the bottom than the top. Yes, you're right. Anyway, in the end I'll get out, I won't use this notation anymore, so there's kind of a better notation for that, so. In tech you can put indices anywhere. And times like this big thing like E, EI, tensor, E, whatever, K, tensor, FJ, FL, FM, FN, yeah. So yes, you're right. There's a star here and J, L here, okay. Anyway, so overall what we're doing is just matrix multiplication, right, here. Summing over S and taking the product of these two if you can forget about all other indices, that's just matrix multiplication. And what we're doing is just matrix multiplication in two dimensions somehow. So what is the image of one, that's this matrix. So let me assign numbers to every vector space, so that's one, two, and three. And this is just R12. So that's not, these indices are not the same as these ones, so that's why I put a comma in here. So that just means R acting on the first two vector spaces and then acting by identity on the remaining one. And then I apply R applying, acting on one comma three. So it acts on the first and the last guy. So it's an element in endomorphism of H, tensor V, tensor V, where these are one, two, three, okay. So more generally I mean you can write matrix multiplication, let me show you one more just to resolve some ambiguity that can be in the notation. So if I'm taking like this bigger one, like Z of U1 in this case, so let me number this one, two, three, four, five. That's gonna be R23 for this first vertex here. Then R24 for this one, R25. And then R13, R14, R15. And you can observe that actually, like these two ones actually do commute. So like the order in which we do things is not completed at the time, but anyway since things do act on different space, actually there's some commutation relation that you have which kind of resolves this ambiguity. Okay, but not only you can have this partition function, you have a kind of conditional version of this. So like we'll play with U minus some other open subsets U prime instead of just some this absolute version U minus like empty sets. So here's how it works. Assume that you have like U and we have a subset U prime in it. We're gonna associate to this a map from W of U prime to W of U, a linear map. And here is how it works. And again let me give an example. So the bigger open subsets will be this big U here and the blue one will be U prime. So boundary states for U prime, that's gonna be endomorphism of H tensor V square. And it goes to W of U, which is endomorphism of H to the third, V to the third. And basically we're taking some A here. So A is an element that lives here. So somehow we're being blind. We don't know what happened inside U prime. We only know what happened at the boundary of this domain. And we're counting all possible configuration given boundary condition on this domain inside. So basically we're filling everything with R here. So like again if I number things like this, it sends A to like this big matrix. Like we're gonna have R34, R35, R36, like for this three guys here. Then there will be A. So A acts on three different guys. So they're gonna be A245 and then R26. And then the remaining R, so like R1415. And R16. And you can play this game a lot. So what we're interested in in the, actually the, what happens when U becomes bigger and bigger? So we're taking the, let me define formally W of R2 to be the co-limits for overall use of W of U. And once you get some kind of the global section of this, you get an action of Z mod two, of Z square, sorry. Just because everything we did is completely translation invariant along the lattice. And the conjecture, which I think is impoverished. I only heard you talking about this in talks, but I never seen something written about this, right? So it's that's if you're taking the chains for the group Z2 acting on this representation W of R2, that comes equipped with an action of the chains on the little viscope rod. And again, I mean, I studied everything for in dimension two, but that works in any dimension. So in class, there's pictures you're drawing. Remind me of these operator categories, like Clark-Farrock study, do you have any, there's not a connection right now? I don't, not that I know. It may be, but not that I know. Okay. This is kind of discreet version of PX and nothing else. Yeah. So. There's a notion of an operator category and a equivalence between some operator categories and the N-algebra's. And the operator categories look at least vaguely like this, discretized. Make sure you find. Okay. It might be a notch. Okay. As far as I know, these operator categories are related to E and operator. Is that correct? Yeah. But there are cells in higher dimensions. In this picture, we only have like vertices and arrows and no higher cells. I don't know if it's, maybe there's a way to understand this that way, but I don't know. Okay. Actually, one remark is that instead of, if instead of including one open subset in another one, you include a bunch of open subset, which are perverse disjoint, you get that W is actually a factorization algebra on R2. It's actually a factorization algebra on the lattice and actually you can push forward it to R2 via the inclusion of the lattice. So that worked pretty well. And the first strategy to actually prove, try to prove this trajectory was, I never succeeded to make it work, but it's very easy to using this Z2 action on this factorization algebra to show that it's equipped with some kind of discretized RAM differential. And like you can cook up a factorization algebra in complexes that is not exactly locally constant, but you can kind of guess that it's somehow asymptotically locally constants. So that would prove that global section would be equipped with an E2 structure, but I never succeeded to make this argument work. So here I'm gonna present a different strategy to prove it, which is actually more direct. So that, yeah. There's a kind of like block spin thing you could do, right? Yeah, that's the kind of thing also I was trying to do. But block spin works pretty well in dimension one, also sometimes in dimension two. And furthermore, it works with spin. And like when you have like, like the dimension of the vector space you have, like really matters. And like if you don't have really spin system somehow, like you have just when you increase, like when you try to do some kind of finalization, you have non-local effects. And I don't know how to deal with that. But it might just be my ignorance. But that was also one strategy. So in general, I mean, I would love to see a way to make these discrete things fit into some kind of BV formalism or like to work out the analog of what you and Kevin have done for these discrete theories. That would be great. I don't know how it works. Okay, but anyway, like for this, like the main result that we proved with Damian Loge is that like this conjecture is true. And like the goal for the rest of the talk is to explain the strategy for the proof. So first of all, I need some discrete version of the category of little disks in R2. So let me explain how this works. So I'm going to construct. So let me write this DD2 for discretized disks. And this is going to be a two-one category, pretty much strict actually, a really strict two-one category. So it's a two category in which two morphisms are actually invertible. So objects of this category will be finite, disjoint, unions of disks in R2. One morphisms, they're generated by inclusion of configuration of disks in R2 into bigger configurations plus translations using the Z2 action. So typically, that's an example of a one morphism in this category. I think you can assume that disk is efficiently big, contains a lot of integer points so it can just picture these integer points inside. Yeah, yeah, yeah. So I'm not writing the lattice here? Yeah. Yeah, yeah, yeah. That's correct. So like when disks are too small, things that actually happen, you don't care really what, I mean, like it still makes sense, but in the end we are taking some big call limits and we won't see what happened at small scale. So that's where, so of course, what we ask is that at every single step, like we never end up being in a configuration where disks do overlap, right? So that's really important, otherwise like everything, we won't get any E2 structure. And then the two morphisms are like these discrete homotopes. So the two morphisms will be completely generated by like you have a disk, you can translate it that way or that way or this way and that way. And we consider that there's a two isomorphisms between, two isomorphisms between these. Okay, so actually this category, this two one category happens to be like there's a braided monodic structure or E2 monodic structure on it. And instead of showing, exhibiting an E2 structure, I will give you two compatible E1 structure on this category. And I'm sure you can guess what it is. I mean, there's will be a vertical direction and a horizontal direction. So here's how it works. So we have like two partial orders on objects. So we're saying that's A is smaller than B that way. If B is at the right of A, like typically if you have, well, something like this, this is A and this is B, but also if there are like this. So B should be on the right. So there are projection on the first axis should be disjoint and B should be on right, right? And like you have a vertical order. I mean, like A smaller than B that way. We say this if A is below B. Okay. And the two monodic structure are very easy to write. We're going to use the action of Z2. So the horizontal tensor product of two guys will be the disjunction of A with B translated by like say the vector M0, where M is the smallest integer. So that's A is to the left of A of B plus M0, right? So like if B is to the left of A, you have to translate B to the right to make it disjoint from A and to the right of A. But if B is actually very much on the right and far away, you have to take a negative M to put it very close to A. And the same for the horizontal structure, for the vertical structure. It's gonna be A disjunction B plus zero N, where N is the smallest so that A is below B plus zero N. Okay. And you can check that this actually defines strict monodal structure. There's even, I mean, the higher coherencies are completely trivial. It's a strict monodal structure. Like the, actually the lemma is that, no, no, no, it's fine. Actually, I mean, the correct way to phrase this is to prove that there's a single object in two one categories. So actually, like this, there are many ways to write this monodal structure and this one makes some choice. I mean, you could move A instead of moving B, for instance. But there's a single object, which is more fundamental and that's one way of presenting this. So there's a better way to write this, but this one is machine monitoring. Yeah, so like these are strict monodal products. And moreover, they are compatible, but the compatibility, this one is only up to higher isomorphism. So the compatibility is up to homotopy. Even though in dimension two, I mean, there's no match room for homotopies, but, and like in higher dimension, it makes much more sense to present this using iterated sigal objects in categories. Okay, so like because these are monodal product, which are compatible, we have like two compatible E1 structure. So it shows that this gets an E2 monodal structure on this DD2 to one category. So in particular, E2 monodal structure to one category. So in particular, if we mod out by two isomorphism and go to the homotopy category, it's gonna be an actual braided monodal category. So the consequence is that the homotopy category of DD2 is braided monodal. And actually the braided monodal structure in this case is pretty easy to write. I mean, you have like two open subsets. You have like A disjoint union, B plus some M0. That's, let's say, that's a bigger one. So that's A and B plus M0. And like here, you want to go to B and A plus some M prime zero. It doesn't have to be the same M. Well, it shouldn't be the same M. And definitely, I mean, the way to go from one to another, well, it's to use this discreet path to go to put this on the right. And then you might have to translate back again. And up to this kind of discreet homotopy is like, if you want really, like that B goes over A when they turn around, there's only one way to do this. And that's gonna be your braiding in your category. So they actually, in an ad hoc way, we could define the braided model structure directly on the homotopy category. Okay. Right, so in, let me say also in that in dimension two, like working with DD2 or its homotopy category, it doesn't matter that much because they're actually equivalent. It's just like this fact that in configuration spaces in dimension two are K by one. So in the end, I mean, there's no higher, actual higher homotopies, but in higher dimension, it's not anymore true. So we should stick with this DDN category. So yeah, maybe a remark. In dimension N, we have like EN monoidal N1 category. And it's not equivalent to its homotopy category while it is in dimension two. Anyway, we'll see in the end that the specific examples that will solve the conjecture actually factors through the homotopy category. Yes? That's for a fine minute of the conjecture. So sort of the interesting thing is this, if you fix this R, right? That's kind of the model here. Yes. Do you know what the, that's got pointing locally in this W of U? What's the co-limit of, if I pick my R matrix, what's the co-limit of R? Like what does, do you have a guess of what element in? So the co-limit of W, what is WR2? Is that the question or? What is the R, what's the, what's the distinguished point in the co-limit? WR2, yeah. It has some meaning. This WR2, you want to describe this space? You consider not an all I, but the partition function. Oh, that's just a huge product of R's. I understand that. No, it's kind of a unit or not a unit. Is it good at the unit or is it a thing? Okay, you get some square of this, yeah. Immediately, if you picked R to be zero, you'd always get zero. If you picked R to be identity, you'd always get identity in there. Yeah. So you do this thing, but if you pick interesting R's, which is picking an interesting vertex model. Yeah. Do you have that sense of how to figure out, I assume this is like a change in. I think it's n or operations n equals zero. Yeah, it should be unit. It's going to be infinity algebra unit, nothing else. Yeah, yeah, yeah, it's going to, but typically if you take the arm matrix to be zero, in the end, this WR2 is going to be zero itself. Because like you're taking a collimate for zero map, I mean the collimate is going to be zero. So, and in general, if you take a non-invertible arm matrix. W is this vector in this vector space? Yes. WR2 is a vector space. Now you get vector space is the element, it should be unit for all. It's the unit in the C2. Yeah, yeah, it's the unit of the C2. But W is the name of the space, right? Maybe that's where it would. Yeah, I think I'm a little bit, well I'm interested like where the interesting part of the physics is. This is all like linear algebra. Yes. Yeah, happening with the partition function. So, the partition function is the unit for the E and algebra that you're constructing. Then the interesting thing is if I start deforming it to see how that. Sorry, I didn't get. If I start deforming it somehow to see how it changes or something. Is that correct? Yeah. You have some? Okay, sorry, I'll speak. Yeah, I have to say that I'm not sure I can say what is the interest. What's the actual physics content of this? But- No, this is kind of formal deformation theory, but it's like the normalization upside down. Yeah. Yeah. Anyway, so now the goal is like let me make a small digression for why, like how we can produce E and algebra from these categories. So, maybe, like there's a general way of constructing E and algebras from, let's say, lax E and monoidal factors. So, there's this general principle that's like if you have C and D to E and monoidal, like infinity one categories and let's get F a lax E and monoidal factor. Then the column of this factor is an E and algebra in D. So, one way like which is quite pedantic, but which will be actually useful to reformulate this is as follows. So, like we have this factor C goes to D, we have the terminal factor to the point category, and actually the co-limit of F is just a left-hand extension along this. So, like if we get the left-hand extension of F along this terminal morphism, what we get is it's a factor which send the like the unique object here to the co-limit of F there. And like the yoga of con extensions works if things are structured, namely with actions of operas. So, in this case, if F is E and monoidal, I mean this factor is obviously E and monoidal, this one will be lax E and monoidal. And like, I mean an E and algebra in D, that's tautological, it's the exact same thing as an E and monoidal factor, a lax E and monoidal factor from the point to D. So, that's a way of, I mean that's a very pedantic way of saying what I said before, just it's going to be useful for computing co-limits of factors in several steps. So, let me actually give an example of this, which is the one that we're interested in. So, assume that C is this category of discretized disk, we get some factor F here, and assume that let's say D is any actually in three one monoidal in three one category, but let's say it's complex, it's a category of complexes. So, like here I have the factor two points, and I'm going to go to point in two steps. So, first of all, I'm going to take the function here from DDN that goes to this category, which is essentially the classifying category of Z2. So, it has one objects, it's morphisms are just like generated by Z2, you could put, well, oh that's ZN here, and you could say that there are like homotopes, like instead of like things which commute strictly that would commute up to homotopy, but actually these things are equivalent, so that doesn't matter. And then there's this here. So, if you first compute, so when you compute co-limits, there's a kind of Fubini theorem that tells you that, like if you compute can extensions here, like you can compute it in two steps there. So, first of all, you can compute the can extension with respect to this function here. So, what it does, like if you forget about the ZN action, it's just this co-limit when you get bigger and bigger, and this is a filtered co-limit. So, taking either the co-limit or the homotopic co-limit, it doesn't matter. So, in the end, what you get from here, it's really like F of R2 or RN, and this thing is just like getting an object in this with an action of ZN. And then, when you take the co-limit along this, that's just taking homotopic co-invariant for this with respect to this action. And if you work with complexes, what you get is really change for the ZN action with values in this, F of RN, right? So, whenever you have an E-n molecular function from this discretized category to let's say complexes, every time, so the upshot of this is that change in ZN with values in F of RN, this is just another name for the co-limit over all use of F of U. This gets equipped with an action of the E-n operand. Okay? So, you're regarding complexes like U-pennedy algebras? Yeah, yeah, so complexes like this isn't, yeah, that's the infinity one category of complexes, yes. It's symmetric monoidal, so in particular, it's also E-n monoidal, yeah. Okay, so in the end, the last thing to check is that we do have such a function. And indeed, if you look, W is an example of an E-tun monoidal function from DD2 to complexes, and it's even a strict E-tun monoidal function in the following sense. That's actually, W does not depend on the higher homotopies, so first of all, we have DD2, we can go to its homotopy category. Well, again, I'm doing things in dimension two, so in this case, they're equivalent, so it doesn't matter so much, but in higher dimension, actually, it does matter, but with this specific example, it factors through the homotopy category. So it's a nice one category, and like the W, the construction that we have with W actually land in vector spaces, and the construction is completely monoidal, strictly with the two monoidal products, the vertical and the horizontal one, like W on the disjoint union of guys, you just declare that it's the tensor product of W of each guys, and all the computation we did is compatible with everything, and then you embed vector space into complexes, as complexes concentrated in degree zero, so that's an E-tun monoidal factor. So in the end, it gives a proof of the conjecture that chains on W or two, gets equipped with an E-2 algebra structure. So maybe before I stop, I'll give you a few remarks. So why exactly are you doing this for an equal to two? No, that's just for expositioning, it's just easier to explain things, but it works for n equals whatever. Yeah, for any n, it works. Like a bunch of remarks, the first one is that actually if n is greater or equal to three, because like this specific example factors of the homotopy category, the homotopy category is symmetric monoidal, then because like E n monoidal, whatever, for n greater or equal to three, for strict one categories, that's just symmetric monoidal. Vector space is symmetric monoidal, so actually this is a symmetric monoidal factor. So in this case, if n is greater or equal to three, then, and that was actually some, I didn't expect that. We get that this E n monoid, this E n algebra structure is actually an infinity algebra structure, which was quite surprising to me. Sorry? Yeah, it is because there's basically no room for homotopies in this specific case, like because of course if you take an H and V being complexes and R being some kind of monophysm of this H tensor V as complexes, then there will, it won't be true, but if your H and V are concentrated in degree zero, like there's no room for higher homotopies actually, for the commutativity and it will be infinity. So like the question, I'd like to see an example and a natural example where in higher dimension, like this would produce something which is not an infinity, but E three or whatever. Maybe like, I don't know, other question I have is that if the R matrix has special properties, like let's say, let's say assume it satisfies the quantum Young-Baxter equation, I have like, what does it imply in terms of this E and E two algebra structure? I actually have no idea. And also I would be very much interested in other examples like maybe non-topological ones or something like typically examples I would like to understand very well very much but I didn't get any success up to now, is that if you have like a half space and like you take the hyperbolic metric and you have a kind of graph which is with geodesic, that there are ways to write kind of discrete models for this and can we get some kind of Swiss cheese algebra or things like this? Okay, lots of questions, but like for the moment, no, not many answers. Okay, thank you very much. Yeah. Remarked? Yeah. There was also some similar, at first I don't believe this in the course of your story. And second, there was kind of similar trouble with inductive limits and similar stuff. And at some point, it was Young-Sergeant and Roper paper about categories, whatever. And we didn't want to go to foundations and we kind of moved by some flow, like assumptions of the intersect, and if they're not intersect, we don't just don't define anything to product it. Only some kind of. And so there's also kind of axiomatic part to define the infinity thing with some flows. And then you can approach somehow that it's equivalent to infinity algebra. In fact, it's presented today, which I thought was internal to be used and make me think it's kind of similar trouble maybe even in the mission one. That's the long and sense here. And second thing which I want to say that I realize that in a few young generations, probably don't know because I didn't really read the book that came in, I have to say, there's a kind of, for example, this question about the formation of QFT, yeah? Well, it's not free, but we start with like, conformed fields sooner than anything. I have, for many years, I have an answer and maybe I can share with you. It's really one line. Yeah, sure. Yeah, just because I think it was never able to write it in proper paper. I think it's just... This is a formula. Yeah, so suppose we get after the species gets some spacetime, manifold, on which we have a pleiadium theory and we get to each point, we get some vector space and for if you get a i, ctkp of xi, the intellectual points you get correlated, yeah? And then we get some opi for this guy, so when this space is kind of filtered, it's union of vx that's by dimensions, should be interesting, yeah? So, of course, a little picture of opi. Now, suppose I have, yeah, so it's mentioned, this thing is given by final intervals, the interval of this element of some action to the product of some sort, that was roughly a good story, but no final intervals get conformed to this theory and now I'm going to deform action, replace action by action plus, that is small parameter, that the s, and that the s will be interval of Lagrangian, yeah, it will be integral of certain field Lagrangian, that affects v dx, v dx, we get kind of top form, that's where this infills, it will be about the end of this complex of chains, or bushes, and so on. Now I want to write new correlators, and the formula, it's kind of magic formula is a formula, so for the new formula to be series in a bar, it should be usual stuff, plus and correction, for the right formula, it's sum of m, where it is in zero, one of n factorial, and make integral of what? Now what can make integral, maybe we want to have make integral of a correlation function product of f i x i, we've got Lagrangian, I'm going to write j, we want to get the m, d, d, n, this will be what, maybe we should take this integral, this integral is divergent, yeah, yeah. Now, now what we do, we integrate over, we make kind of integrated things, first we, suppose we have fixed all points except y n, and the engineering, we integrate over y n, which are from d with some distance epsilon, and x i and all previous, then if you guys can't come back, and don't consider artificial things, you get divergent integral, you remove the small balls, it depends on epsilon n. Now we take limit, and by limit, I'll take, take a constant term on a synthetic expansion, so this is a power's logarithm, just, it's a constant term, to be limiting appearances, you get function of the rest, integrate all previous points, in the future, in the future bit, and the claim it will be, the formula for the particular theory, and depends on the series of, the sinc center if you change the metric, you get, how would you get the e n, e n structure? No, no, no, it's not the n, if you get logarithm to n epsilon, and you get constant term, no, no, yes, it's the constant term, it's not invariant by relation six, but it's kind of, it depends on the definition of limit, it's going to be some, linear function or function switch, have continuous limit, should get the limit to zero, in a linear functional, okay, and it will be some kind of, choice of formal coordinates, in the neighborhood quantum field theory, and if you choose the different limit, you get identifications, and then we can extend to e n algebra, and if you change the number six, it's just plain deformation theory, yeah, it's sufficient. Do you have any other comments, question? I had two other questions, one thing you do is replace, vect by some other interesting, even like say a braided minoidal, category or something in certain settings, or just like g-vect, or g-aq, or something, do, have you thought about it? No, I'm kind of lacking examples, like, I don't know, I would consider kind of fake examples, I would be interested in, like these models with R, I mean, they naturally show up in statistical mechanics, I would be interested in seeing like, yeah, examples that show up naturally in, I mean, are these lattice theories that are supposed to correspond to the sort of three TFTs we learn in math? Yeah. I mean, have we tried those? No, no, no. Like with Germany-Felder, we have a different approach to a lattice gauge theory in dimension two, and like we recovered basically the McDowell-Witton renormalization stuff for this Younger's theory in dimension two, but like, it's completely independent of the shape of the lattice, while here, I mean, because of the z to action, we have to have a square lattice, for instance. Yeah, there are also questions like, instead of vertex model, there's this model that we've, I think we've seen them this morning, with what I call IRF model, interaction around the face. I don't know. I think it's sort of the same. I think it- You put tensors at my basis, I put a lot of zeros in this tensors, and you have the same stuff. The thing is, there's D2, I just don't remember, does it have like sort of an empty set in there? Yes, yes. Is that where the matrix of R is being hidden? Yeah, so like the map from empty guy to any open subset, that's gonna be putting R's everywhere. So that, yeah. That's the partition function, basically. Right, so the color of like the W part, ignoring the R, is easy to compute, but like where, so you get some big, you get some E2 algebra, but you would think about it being sort of the same thing for every trace of R, but the image of R on the color, it's sort of moving around, right? I think the W part is just this big combative endomorphism of some big vector space. Yeah. But like, no, no, R, it's gonna be a huge product of R's. It's not gonna be one single R. I understand that, but the big, the E2 algebra for the underlying chain complex, in this case, is vector space, it doesn't change that you change R, right? No, no, that's true. So what I'm asking is, as I vary R, do you know how that changes inside of there? They change R on just the single vertex, how that changes. So I thought in the column that you take R as the map on all the vector space, right? So if you change, I mean, the shape of the column doesn't change, but R, this column, it looks like sort of concrete. Like, if R is invertible, and if R is zero, you get a very different column, it's, yeah, so yeah. Yeah, yeah. Okay. This thing is filtered by pieces, by kind of like radius of domain, and the gross, like a dimension of filter capon goes to the spawn of size times dimension minus one. Yeah. And in conform field theory, you can see filtered by dimensions, it's the same gross, yeah, it's the same, yeah. Oh, we can see the volume of the volume of the ball, that volume of the ball to power N over, whatever, D over, D minus one over D. And like in conform field theory, these things goes like expanse of square root of energy, yeah. But it's a similar story, just inductively. In conform field theory, you go to smaller balls, and here you go to a larger ball. Yeah, you will. So I've got two torques where you get. You should really give me like a, tell me of vector space, and as I vary the army tricks, like I get a different set of numbers, what will click to you? No, I'm pretty far from being able to compute in variants of white surfaces, just from this. Yeah, but also just want to say that it's, these things could be zero on zero, it should be, there are universal Turing machines, which can put on this already two dimensions, this dials, and these things is kind of algorithmic, non-compatible, yeah, so the gross could be very, very slow.