 All right, thank you very much for the invitation. It's a pleasure to be here after four years. I was here for three months, four years ago, so it's nice to be back. What I'm gonna talk about today is, as the title says, relative field theories. And I thought I would, in the beginning, try to put you a little bit into, give you a bit of context also regarding the talks that are to come this upcoming week, because there will be several ideas that actually play a role in this talk that are closely related to other talks that we'll see this week. So I'll roughly have two sides of the story. Maybe this board is a bit too big, but it's okay. So I'm a mathematician. So as a mathematician wanting to study field theories, I like to have some sort of an axiomatic framework to be able to deal with, to hopefully kind of put this into a framework of context. And there are two different ones that will appear and that we will also see appearing this week, I guess. So on the one side, and this is the older one, is the functorial approach. Maybe let me call this approach to field theories, sometimes just called functorial field theories. So this is something you've probably all seen at some point. So this goes back to Atiya and Siegel and their axiomatizations of topological quantum field theories and conformal field theories. And roughly you should think of this more or less as describing some sort of a state space. And then on the other side, we have what I will write under the kind of combined word of factorization algebra. So this is the ideas that go into this in the example so that I will write down in a second. Of course, I've been around for decades, but they have somehow really been a very successful tool in all kinds of math and math approaches to mathematical physics, to quantum field theories in the recent years and somehow have been widely studied. So here you should think of them as being observables of a QFT or maybe I should say to be precise a perturbative QFT. And really using the framework of factorization algebras which I won't define here, really showing that this is really true is due to Owen William and who's here this week and Kevin Costello. So maybe to connect to some talks that we will see there will be a lot of talks that fit into this side here. So for example, we will see vertex algebras appearing. You should also think of them as somehow being on this observable side. We will see carol algebras appearing. So both of them I would like to write down. We'll see the talks with Emily Cliff, Owen William and Brian Williams. We will see N algebras appearing on Friday in the talk by Damia Colac. We already saw factorization homology this morning in David's talk and I assume it will appear again in Tim Veiling's talk on Wednesday and also in David Jordan's talk on Friday. And also maybe EN algebras which we've already seen just before I mean this goes into the factorization homology but just to write that down again. And for us factorization algebras for today for this talk are more like a tool and it won't really appear prominently. It's just the main tool that's running in the background that I won't really talk about that much. So what about the other side? The other side, so this was the original kind of approach defined by these two people and of course there have been many developments about this so let me just tell you abstractly what a functorial field theory is and what kind of extensions we will consider. So I would just like to give you the slogan that a functorial field theory is a functor from something I will call board N to something for now I will call backed. And I don't want to give you the precise definitions today I'm happy to talk about that in more detail. I just like to avoid the technicalities. But on this side, we will have something like category of n-dimensional co-boardisms. So the picture for this, maybe I'll draw it above because I'm in space, something like this. You should think of it this as a space time coming from this space to that space that's what a co-boardism is. And this co-boardisms we can put very quite a bit. So first of all, we can put some structures on it with some geometric structures. So for example, conformal structure, homomorphic structure, but also topological structures as we saw this morning. So I want to keep this a little bit ambiguous here in the examples that we will really look at that we will be looking at the topological situation like this morning. But more generally the story you can also write down in this non-topological situation. And another modification is that maybe here I don't want to just have a category, but some sort of a higher category. Maybe the index will be a k, might be a k category, infinity k category, something. But if you don't know what that is, don't worry about it too much for today's purposes. What about the other side? On the other side we have some sort of a category of linear things. So well, if it's just a category then this will be like vector spaces. Maybe more generally you would actually like to have something like a Hilbert space. But to make our lives easier for today we'll just take vector spaces. But you're free to change this and to add in extra things as you please, as you meet. So here we will put some sort of a possibly k category of, and usually they will be some c-linear objects. We'll see some examples later on. So just keep in mind vector spaces, c-linear vector spaces that's fine or some generalizations there of, some suitable ones. And again, I'm happy to talk about this in more detail if you wish. Okay, so, right. So we will also see things like that in a later talk this week, namely by Kuhl's at Sosa. Did I say that right? Yes, okay. In a variant of a homotopical quantum field theory. So there also, you have to vary this definition a little bit. But somehow what I wanna talk today is this relative field theory situation. So again, we have to sometimes the objects that we're interested in, the quantum field theories we're interested in, just don't fit into this rather rigid framework. And we want to relax this notion a little bit and modify it to make certain examples fit in. So maybe a problem you might have heard of is that sometimes you just deal with an anomalous field theory. So that can be in this framework, one way of describing this is to say, well, it doesn't really fit this strict functorial framework, but we have to modify it some way. And how to deal with this is somehow the purpose of these relative field theories. So then let's try to see what we can do. So now instead of having just a functor as above, and again, I'm a little bit agnostic about what my boardisms precisely are, choose your favorite boardism category. And we will have some target here. And for the purposes of including this K over there, I will add a little K here to indicate this K. But if you don't like higher categories, just forget about that. So here, now the problem is I wanna K plus one here, but again, it's not so necessary. Now, and now instead of looking at a functor, I want to look at a relative situation. So I want to put something here that I will call T. And I want some sort of a map from C to T. So this here you should think of as the anomaly, at least in case this theory is invertible, which means it assigns invertible vector spaces. So one dimensional vector spaces to everything. C, C is the trivial field theory that really assigns C to every object and the identity on C to every morphism. So you might ask, what's the difference between invertible and C? Well, an invertible is isomorphic to this one, but maybe the choice of isomorphism between the one dimensional vector space and C is not canonical. So that's a subtlety there. But yeah, right? I mean, if you have a one dimensional vector space, you can choose an isomorphism to C and that isomorphism we want to on purpose keep here. That's exactly where the anomaly comes from. Or we will usually call it the twist. Okay, so such a situation, I will call a relative field theory. So C, Z is relative to T. So sometimes this might be invertible, but sometimes not. There are also situations where it's not, for example, if you have the space of conformal blocks or so. But I have not told you how to interpret this picture. I have not given you a definition of what this means. Okay, so let's see. Is to disappear. So this of course is also not at all a new thing, writing it in this way. And somehow these anomaly theories have been dealt with in many different ways. And one thing that ties into this picture, this is one of the interpretations, is that of a defect field theory. And I mean, in a probably very different incarnation, but somehow related, we will see defect field theories appearing in Nekrosov's talk on Friday. And maybe the defect field theory is a better situation when I say, okay, here, I call this top field theory, I call this S. And in my relative situation, this happens to be C, but more generally I could choose also just any functor here and have some sort of a relative thing here. And now if S is really C, as in the situation I started out with, this is usually called something like a boundary field theory. So, okay, why can I think of such a defect field theory as implementing something like this? So let me choose two colors. Let me call T will now be blue. And S will be yellow. Is this visible? Also in the back. So now you should, you can think of a defect field theory. What's a defect field theory? Now you take a co-bordism, which has a co-dimension one defect. So a co-dimension one sub-manifold. Let me draw you a picture to make it should be white. And I color my co-bordism in two colors. Maybe something like this. And so now you can think of somehow S, if now all my co-bordisms look something like that, they might in particular also just be yellow or they might just be blue. They might just have no co-dimension one defect. So in the just yellow case, this will be the field theory S. And in the just blue case, this will be the theory T. And somehow you should think of the Z as somehow sitting on this co-dimension one thing, okay? So this is for N equals one, this is the picture that you can think of. So now we kind of define, so now if we give a definition of, well, we have a board with defects now to my plus one vector, something like this, then you can think of this as implementing this situation as being somehow a morphism from S to T or maybe we're a little agnostic here. It's not so clear which direction it goes. So maybe from C to S. So this is a way one can think about these things. This is not the approach that we will take today. Another approach that one can do is, while I defined a functorial field theory as just being a functor. And in mathematics, we have a nice notion of morphism between functors, namely a natural transformation. So another way we can discuss this is as natural transformations. And for technical reasons, I don't wanna go into details about, I have to add these words lax or oplex because we're really in the higher category world. So maybe I should say there is a paper called Relative Field Theories on the Archive by Frieden Telemann and their examples, usually are examples in this situation. And this approach has been kind of promoted by Stefan Stoets and Peter Teichner. And in this higher category setting, we need to make sense of this definition even and to make sense of this definition that was something I did together with Theo Johnson-Fried. Okay, so we'll come back a little bit very briefly to the side over there. But for the most part of the talk, I will just stay on this side. Just because here we have some tools that make it easier to deal with this side. But they're not completely unrelated and we'll see that. So there is some sort of a philosophy behind this, which I would like to briefly mention and which will be somehow the motivation. The motivation for the examples that I will give at the end. And the motivation is that, well, we can't always get our field theories into this rigid framework here. And there's also choices to be made even if we go to this relative situation, right? What's S, what's T, that kind of story. And so one philosophy behind this is that the field theory, maybe it doesn't exist somehow in this absolute sense, but it should exist relative to something coming from its observables. This is a philosophy, this is not a precise statement, but it will be the philosophy or the guiding motivation for the examples that we will look at, the simple ones. So I want to have some sort of a picture. It takes my boardisms. I have my C, my trivial field theory at the top. I put my observables as this relative thing and then my field theory will be something like this. Again, I want to be a little bit agnostic about the direction of this morphism and the direction will have some different meanings. If you go the other way, you could think of this as being some sort of a expectation value. To an observable, you assign a number. It's something like an expectation value maybe. The other way, you pick out an observable. So maybe it's more something like a vacuum vector, something like this. Can you help me understand how things match up? So on a closed n manifold observables, it will be like a vector space, not a number. And that's keeping, so by k plus one back then, board n, how are the numbers matched up? Okay, so if this is a k category, this will be a k plus one category. So you will only see the k part here for the observables. So on closed n manifolds, you will get a k and a top thing, so a vector space, yes. Okay. Exactly. The plus one here only comes into the play in z. So z, because it's exactly lax or oplax and not just a natural transformation, it really uses that you have extra morphisms here. So the vacuum here is not the c, but from the observables you choose, but for the c here, is it the vacuum or? Well, this is again, this is the trivial field theory. I mean, I have not really told you what this is. But yeah, if it's the field theory that assigns c to everything, the trivial object in here. So it's the vacuum. Well, but you pick out one. Maybe let me do an example. So the vacuum, I mean, this really just picks out c, the vector space c. You can call that the vacuum, but you won't see any relation to anything else. The relation to anything else, how the vacuum looks as a vector in somehow the state space, that really uses this morphism here. You can say, okay, this is just the vacuum. You can say that if you wish, but it doesn't have a meaning without like, it just really assigns the trivial vector space c to everything, you don't see it. Is it the creation of the interview from the back? I did not see what, here. Is it the creation of the interview? Creation of the interview. Okay, so I told you before that the factorization algebra side somehow describes these observables. So if I want to kind of realize this motivation, this philosophy, starting from a factorization algebra, I should be able to get something like what I have here. So this is one thing that Weier-Sturz and Teichner did, is that starting with a factorization algebra. And now G here is a geometry. Just want to keep to their notation. Then we get, now this is the boredism category for this geometry. And here, it's something like topological algebras. Here you have your trivial guy, and here you have this F, a twist coming from this F. They do also get something here, but this is not really a state space. This is more something like, we will see this in examples later in a very simple one. But you do get this twist. So really indeed, from the observables, you get this such a candidate for a twist here. And so this here is in the one categorical situation, in the extended situation, and in the topological situation. So here, now I have one of these N categories. It's actually an infinity N category. I have a topological situation. I take framings. But yes, indeed I get this twist factor. And here we have to put something that I don't want to explain in detail. The objects are in algebras. And again, somehow, yeah, let's keep it at that. And the main tool here is exactly the factorization homology that we saw before. So I said before we want to focus on the topological situation today. And the reason is that there are strong tools that we can use in that situation. So we have something called the Kubordism hypothesis, which we heard about in the last talk. And this helps. It gives us a simple way to go about constructing these things. So why does this help us? Well, if we're in this situation that I wrote down at the bottom of the board over there, and look now at a TFT, a topological field theory, which goes from this category that I had over there, fully extended, whatever that means. I have a functor to some nice target category. In our cases of interests, it will be those linear things over there. And such a TFT, fully extended TFT, is fully determined by its value at a point. So this value at a point, so I'm in a case here where my board isms, I go all the way down, I have n dimensional guys and n minus one dimensional guys and n minus two dimensional guys, and I go all the way down to point. So my point will be something in here. And so if I have the value of the point, this fully, I can recover my Z just from knowing what it is at a point. So this somehow tells you that you have a locality in some sense, no matter where I am, it's determined by what happens locally. And what is this value of the point? Moreover, this is any so-called n-dualizable object in C. And this n-dualizable, I did not tell you what this is, but this is, and this is somehow important, it's an algebraic condition and can be checked by hand. And may or may not be easier than just writing down Z itself. But in certain situations, it definitely is. So to give you an example, if we take C to be just vector spaces, then being one-dualizable is just a fancy word for saying I have V is finite dimensional. So there are dimensionality restrictions, essentially. These algebraic conditions translate to having some sort of a finite dimensionality conditions. So what is another example? And now I can start giving you some examples of possible things that we can put here. So one thing would be I can take something I will call algebra. So maybe I should tell you this here is an example for K equals one. This is an example for K equals two. This is a two category. So we have objects or algebras. Morphisms from an algebra to another algebra are AB bimodules. Composition is tensor product, relative tensor product. And then we have two morphisms are homomorphisms. So note, if I just take A and B to be K, the ground field here, C, if you wish, then a CC bimodule is just a vector space. It's just an object up here. And a homomorphism of those guys will just be a linear map. So that's the sense in which I mean this is a generalization of vect and I have ceiling your object somehow. Okay, what does two-dualizable mean here? So A is two-dualizable, if and only if. A is one finite dimensional and two it's semi-simple over C. So again, these are really strong finiteness conditions. And there's some other examples that I could give you, but yeah. There is a common generalization of this result. Call it over there, algem. This algem over there that I did not really tell you about. So objects are en-algebras, but here, n-dualizable, A is always n-dualizable. So here, this is an en-algebra and it's always n-dualizable. And that's the translation of this bottom arrow over there. This is something that Owen and I recently proved. Okay, why did I take this detour? I took this detour because I showed you now that the co-borderism hypothesis allows us to build TFTs quite efficiently. Namely, we only have to look at our target category and do some algebraic manipulations. We can translate our problem from a problem of understanding, well, writing down something to do with geometry into just checking certain algebraic conditions. And this definition on the right here allows us to do the same thing in this relative situation. On the left, there is also a sort of co-borderism hypothesis statement, which at the moment is more conductually. So there is something one can do here. But it's not proven in all details. So let's go back to the relative situation. And so I have my S at the top and some T at the bottom, I have some Z here. And now I have any target, it might be one of these, might be something else. So now the analogous situation to the statement over there, we're in this topological situation now. And the analogous statement to the statement up there is that this is fully determined by the value at a point again, which is in this left-hand situation, we have, this is more like a conjecture, that Z of the point now is what I will call for purposes of today, strongly relative dualizable. And on this side, and this here is a theorem proven by Theo Johnson Fried and myself from above. And this is based upon the usual co-borderism hypothesis. And this says that it's determined by Z of point, which is, and now here I will just call this relatively, and dualizable. And one can prove actually, that this implies that. So this strongly relative and dualizable implies, sorry, I should be more precise, this condition implies that condition. And this, I would just like to mention the name, this was pointed out to me by Manuel Arajo. So if we want to have a chance at building relative field theories, if I start on this side, there's not much harm, because this is somehow the minimal thing I can do in this topological situation, and then I can try to see if I can enhance it to one of these stronger ones, that maybe you would be more interested in. Okay, so I will unravel what this condition here means now in the low examples that I would like to show you. So in some sense, we translated our problem of finding these topological examples to algebraic manipulations. And what I thought was kind of neat was the types of algebraic objects that started appearing. Of course, this is a toy model or to kind of trying to test the definition if you want to actually go to more physically interesting examples and more geometric situations. So, and these examples I would like to say is something that Owen William and I worked out. All right, let's start with N equals one. So in N equals one, I want to get a situation like this, put my S at the top, my T at the bottom. And then I will choose just because I already introduced it, this algae here. So this should now be a two category here. This is the K plus one that I had above. So this is a one category, this is a two category. And I'll take this over there. So what will my Z of the point be? This is a morphism in algae. So a bimodule. Actually, this is true for both examples. So in situation A, I will choose a particularly simple one. I will just take an algebra and view it as a module from C to itself. So this is kind of a stupid bimodule, but it works. And so this is, and something I've been suppressing so far is that in the very beginning here up there, I said I need something called lax or oplax. And this is the only situation. N equals one is the only situation where this matters. So I didn't want to dwell on it, but now I have to add it in again. So it's either lax or oplax, relatively one dualizable. If and only if A is finitely presented and projective. And two, yeah, sorry, that's all. Finitely presented and projective over. And now I have the distinction in the lax case, I need C and in the oplax case, I need A. So lax corresponds to this situation. So what does that mean? Well, this here is actually just says that it's finite dimensional, right? So I'm kind of back at the finite dimensionality properties, but this one, this is always satisfied. A is always finite present and projective over itself. So that means in this relative situation, if I choose this case here, I always get something, not just in a finite dimensional situation. This might look a little bit arbitrary. But it turns out to be rather useful. Okay, so one idea that you might have, and actually this is true more generally, if I put any module here, it will be laxly relative one dualizable if this M is finite present and projective over this left side, and it will be oplaxly relatively dualizable if it has the same property over the right side. So you get this relative finite mess here. Okay, so now an idea is that if A is an algebra, I know I start with A, that's an algebra of observables, then choosing a laxly one dualizable M from A to C gives on S1. Well, I first have to evaluate at S1. That will be just the Horschelt homology. Then my M gives me a functor, a morphism, sorry, an algebra homomorphism, a homomorphism to the value over S1 at C, which is just C. And now here I can also get the canonical map from A going to its Horschelt homology. And this is something you can think of as the trace. So to every A, I assign a value in C. So if you think of A as being the observables, well, I kind of gave you a way of extracting a number and you can think of this as somehow expectation value. So just from starting from one of these, you get something like an expectation value. You just started with observables. Okay, and now my other example, and this other example will be motivated by this philosophy on the right. So now I start with a vector space, which I would like to think of as the states and look at the following bi-module. I look at V as a module for its endomorphisms and here I put C again. And so now I can check my conditions, my algebraic conditions. So this is relatively one dualizable. If and only if. Well, again, I have two conditions. The first one, the lax one, this is always. So what do I have to check? I have to check that V is finally presented in projective over its endomorphisms. That's always true. And in the second case, in the oplex case, I have, again, my finite dimensionality. So in the TFT world, we had the one dualizability, had this finite dimensionality. And now in the relative situation, relatively one dualizable, we have finite dimensionality in one situation, but if you look at it the other way around, actually we always get something. And this is, somehow for me, showcases this idea of well, it should always be a field theory relative to its observables. Again, this is a very baby toy example, of course. So let's see how to generalize this, how to categorify this. So now we wanna look at n equals two. So what will I put here? I will put here this mysterious algebra that I did not define. So now for n equals two. And we'll just stick to the vector spaces. So what's this? So objects are commutative algebras. Morphisms, one morphisms are bimodules again, which are in addition algebras. And now forgetting that it's a bimodule, I'm back to algebras. So now I can kind of stick in the thing that I had above. So now two morphisms. Are bimodules of algebras, which is compatible with the lower structure. And then three morphisms is just homomorphisms. So somehow it kind of souped up this example over there. So usually I will call these R and S. These are RS bimodules. I have two elements A and B here. And these will be AB bimodules. So how can we go about generalizing the examples above? Well, the first example, we just looked at an object as a module over itself. We can do the same thing again. Pick R commutative algebra. And I can look at R as a module for itself. More generally, maybe I can take pick A any R algebra. And then I can look A as a module for R coming from the right. And then this determines a 2D relative TFT by these 2D relative dualizability conditions. If and only if, we'll write down the general statement. A is finitely presented and projective over R. And 2A is separable over R. And so this is the bottom situation. So if you look at what happens up here, if now A equals R, this first situation will just be true. And the second case is also true. So again, this first guy gives you an example of a relative two-dimensional situation. More generally, we still have this final presented projective condition that we had before, which is finite dimensionality. And a new thing appears, the separability. And now for the last one, for B, this will be the generalization of this endomorphism's example that we had above. This board? No? Where? Did I just erase that? I guess I just erased that. So before we were looking at V as a morphism from its endomorphisms to C. So how do I categorify that? Well, I replace V by an algebra A. And I replace my endomorphisms by the center of A. So these are all the elements in A, which commute with everything else. Why this is an appropriate generalization? I would like to not go into it here. If you think of a derived situation, if you look at derived endomorphisms, then this will be exactly the derived center, the Huschelt-Cohomology. So that's the analogous thing that you should keep in mind. And now our theorem tells us that we can look at A. And now for certain reasons, I would prefer to put Z of A on the right. It doesn't really matter that much. This determines a 2D relative T of T. If and only if one A is finally presented projective over Z of A, and two A is separable over Z of A. And this has another name, namely this is called an azomaya algebra. So this is something that algebras are excited by. And classifying these things is something you can use Broward theory and stuff like that to do. Okay, so maybe one example of such an azomaya algebra is if you can take the differential operators on the affine line, however, in characteristic P. So that might or might not make you happy. But it's somehow a showcase that if we take an algebra, maybe we can go back to what we had before. If we started with an algebra here for 2-dualizability, we needed these rather strong conditions, finite dimensional over Z and semi-simple over Z. Now, if we're in this relative situation here, if this were over C, then we're exactly in that case. But now we're relaxed our conditions to being over C to being just over Z of A, just over its center. So this is somehow a relaxation of these conditions over there. Okay, I will stop here. Thank you. Given such an azomaya algebra, what kind of invariance does that T of T? That's a really good question. Tell me. It looks like just a primitive magic solubility with a primitive invariance. Yes, absolutely. Absolutely. It's kind of a parametrized version, right? Yeah, so an example that comes to mind, which is not maybe quite azomaya, but if you take a quantum group at the root of unity, then somehow it's like... The categorification. Yeah, so that's actually a hope that for the three-dimensional situation that will be exactly what comes out. Great question. Any other questions? Is there a structured version of this instead of frame-portism-categorically, if it takes some... Portism-categorically, some structure? Yeah, you can apply... So this theorem that gave you the relative endualizability that uses the usual Hucco-Bordism hypothesis. So you can use the version with structure of the Hucco-Bordism hypothesis. However, unraveling what that means in terms of these conditions is not so straightforward. That will have to depend on the example that you're in. Understanding that G-Action is kind of hard.