 We'll come back everybody. So let's continue with the second talk of today's program. Luke Stahauer will tell us about invertible topological field theory with fermionic symmetries. Please. All right. Thanks for the introduction and thanks for the opportunity to speak here. Thanks to the organizers for this organizing this interesting topic. So I'm not sure I hope that the blackboard is sort of readable. I try and write very large and if the general opinion is that it's totally unreadable, let me know then I'll switch to my iPad instead. All right. So my talk is going to be about mainly about work in progress of my advisor and collaborators of Kekstotz-Tajche and their work is very is inspired very strongly by the famous work I would say by now of Frieden Hopkins and I'll tie it a little bit together in my own formulation and I will combine it with some work that's joined with Luke Stahauer. Okay. So the plan for the talk is the first half will be a blackboard talk and the second half will be slides on my iPad. And the first half I'll mostly explain one point of view on how to include fermionic symmetries, how to describe symmetries of fermionic systems in topological, maybe topological quantum field theories. And I will also go into, it means in a similar spirit to the last talk, I'll go into like these topological actions, I will call them topological partition functions, very similar to the last talk. And then in the second part I'll focus more on topological field theories themselves and I'll also discuss unitarity and if there's time I'll discuss spin statistics as well. So but for now my goal is going to be describing one mathematical framework for symmetries of systems with fermions. And there was already a very nice motivation in the last talk for invertible topological field theories coming from anomalies, we're mostly discussing anomalies and one other big amplification of invertible field theories is symmetry-protected topological phasers. So I'll start with a Nobel Prize, which should be well known, the Nobel Prize from 2016 on topological insulators. And I'll not really discuss what a topological insulator is at all. The only thing I want to do, oh I have to write large, large, large question, I want to extract the information of what kind of symmetry the system has. So if I ask a physicist they will tell me there's a tiny reversal symmetry. So this is the symmetry protecting the topological order and then there's the electromagnetic U1 charge symmetry. And they satisfy a couple of commutation relations. Like we have been told that time reversing symmetries should be anti-unitary. So you should have a relation that looks something like this. And in particular there should be also a relation that looks something like this. e to the i a q, so that's my U1 element corresponding to charge. So a some real parameter here. It doesn't quite commute with t because t is anti-unitary but there's a little sign change in the experiment. And then there's another thing that's important for me on existence and that's that t squared is not equal to 1 but it's equal to minus 1 to the n. And what do I write down when I was a mathematician? I want to know what the symmetry group is. My conclusion from this is that g is U1 that's the charge part. And I have a Z mod 4 coming from the time reversal part. But they are not quite a direct product but it's a semi-direct product. And the action of Z mod 4 on U1 is given by the surjection to Z mod 2 and Z mod 2 acts by inverse like that. And then there's one additional thing that you could demand in this situation if you like, which is called the spin charge relation. It says that there's a relation between spin and charge. And one way to formulate it is minus 1 to the q is minus 1 to the f. So physically it just says that the particle has on charge if and only if it is a fermion. And this leads me to if I assume this then I have to require that this minus 1 to the q which is in here is identified with the minus 1 to the f which is an element in here. And I have to quotient by a diagonal Z mod 2 group here. And this allows me to identify the symmetry group in this specific example. So what are some particular properties of this symmetry group? I want to formalize it in a definition. And the definition is the following. I define a fermion symmetry group, a fermion group to be a collection of a couple of things. So I have my compact V group which is like the symmetry itself. Then I have a continuous homomorphism which measures for me which symmetries are time-reversing and which are time-preserving. And I think I'll try to write Z mod 2 additively in this presentation. And please excuse me if I'm making mistakes with that. So for V Z mod 2 it's going to be 0 and 1. And then there's a final piece of data which is the choice of a central element which I call minus 1 to the f. So that's fermion parity. That's some element in the group. And it satisfies well clearly I want to have that squares to 1 if the notation makes any sense. And I also want to demand that it's time-preserving. So that's my definition of what the symmetry of a fermion system should be. We've seen already a very physical example. Here's a very mathematical example. If I start with a finite dimensional superalgebra. So Z2 graded algebra respecting the multiplication, the gradient is respecting a multiplication. And I want to think of this as the on-card being time-reversing and the even part being time-preserving. So it's not really super in a sense of fermions versus bosons. Then I define the following group. It's the collection of homogeneous and invertible elements inside A. And this comes with an obvious map to Z mod 2 given by theta of A is just the degree of A. And this is not compact because it contains all these scalars, scalar multiple of the identity. So I'll just take a quotient by only positive scalars just to get rid of the scalars to make it into a complex group. And now I define my minus 1 to the f to be the class of minus 1 in here. And this gives me an example of a fermionic group. Yes. Do you allow the central elements to be the identity, or do you want to have exact order? Oh, I really was afraid that I would get this question. Like maybe at some point in my life I will be able to make this decision, but at the moment it's very hard, but I think for this presentation I allow it to be one. But I mean, I agree that then it's not really very fermionic, but it's a very good question. So related to the example I just gave, you have this relevant theorem, I think, by Wolf, which says that there are only, or there are exactly 10 so-called super division algebras. So those are super algebras over the universe, for which all homogeneous elements are already invertible. And in particular, using the last construction, this gives us a condition, a sort of, this gives us a list of 10 very special fermionic groups. And this is related to so-called tenfold way in common matter physics. So this gives a collection of 10 internal symmetry groups that one could describe as internal symmetry groups of so-called 10 symmetry classes that occur for symmetry protected double-digital phase. So one example which goes back to the example that I just raised is if I take the super algeba to be the clippard algeba with two negative squares, then G of A is going to be a thing to minus. And in fact, you can show that it's isomorphic to the class A2 tuple algeba insulator example that I just started the lecture with. So these are fermionic groups. I want to go on to describing quantum field theories. Well, I don't know what a quantum field theory is, but at least starting with describing what fermionic symmetries should be, topological field theories. For that, I need first some notation. So I will denote group will be the grading involution of a fermionic group, which is the map that it's a polymorphism of a fermionic group that maps a group element to minus 1 to the f is if the element is off, or sorry, if the element is time reversing, sometimes I'll say off, since it's a z2 variable, I should say time reversing. So if it's time reversing, it's not to minus 1 to the f times itself, and otherwise it's just sent to itself. So that's some automorphism of the group of order two. And now this will be the main construction that I will use to describe quantum symmetries on spacetime. So if I have two fermionic groups, then their fermionic tensor product is defined as the following semi direct product. G semi direct H mod commons in two. I'm at a z2f. So every time I write z2f, it means 1 minus 1 to the f, group z mod 2, in which one of the two is permanent parity, in which H acts on G by first its gradient and then the z2x by the gradient homomorphism on G. And this looks very abstract, but you can make it super concrete by just writing out the formula, what this means. So very concretely, if I write two elements of this group, then the multiplication is just sort of, it's sort of tangent products over the z mod 2f with the coso sign. So it is, well, there's a sign, a coso sign, but it will just be G1, G2, or H1, H2. But then there's a sign, which depends on how much elements you, all the elements you switch with each other. Is there a good chunk? Okay, so this is like similar to what the tensor product would be of super algebras, that's the ID. And the resulting thing here, just like if you take the tensor product of two super algebras, is z2 cross z2 graded, like there are now sort of two gradings. I want to use them both. It decomposes into four parts now, even times all, and all times even, and even times even, and all times odd. And in particular, you have the sort of diagonal gradient, and I want to take the even part with respect to that. So very concretely, I could take H, G tensor H, and what's the even part, so zero for the kernel of sort of the matrix of two. Well, it's G0 tensor, so very schematic. It's this, and I just throw away all the parts that contain something in the form G0 times H1, or the other one. Okay, so now I get to what's, this is what's related to the last topic, I guess. If you're given an internal symmetry group, how do you make the fermionic group that of course that you should put on spacetime? So given an internal fermionic group, I will define the spacetime structure group. So this is where the Lorentz symmetry comes in as well, and let's fix some spacetime dimension. And what is it? It is the fermionic tensor product of the internal symmetry group with sort of the Euclidean plus fermionic version plus allowing reversing orientations of the Lorentz group, and then I take the even part of that. So I only allow symmetries that combine, for example, a time reversing symmetry with a symmetry that, a Lorentz symmetry that reverses the orientation of spacetime. Okay, so let's do maybe some examples. So a simple case is, well maybe a very general case. It's the case in which there are no time reversing symmetries. And then what you recover is you take the internal symmetry group and you just take the product with spin. And the only thing you have to do is you have to mod i about a common z-mount 2 right here. So in particular, we can get back to the situation. And this is the reason that I answered Fiat-Johnson-Crete's question, as I did. So in particular, if I take minus one to the f to be one, which is a little bit weird, I'm allowing it now in the presentation. So this is, I take it to be one inside G. Then the thing that I recover is going to be G-Mod, sort of G cross SOD. So that's going to be the structured group of a bosonic theory that a time-reversing symmetry. Maybe one case with time reversing symmetries. So let's take G to be in minus one, which is an extremely complicated way to say Z-Mod 4. But I write the Z-Mod 4 as one minus one to the f t minus one to the f t. So t squares to minus one to the f. And then you can check that HD is going to be, confusingly, it's going to be in plus B actually, even though you might have expected it to be minus. But that's the thing that should agree with the physics, if I'm not mistaken. And what's now the point of this whole part, the point is that I claim that what we should do is we should endow, if we have an internal symmetry, we should, I guess, I should say, I want to couple the symmetry to a background gauge field. So endow the spacetime with an HD structure. I don't really have time to explain exactly when an H structure is on a manifold, which I apologize. But at least if H is thin, then it's a spin structure. If it's by SO, it's something like an orientation. And I'm very topological field theory focused here, I should say. So if you're not doing topological field theory, you should probably endow differential HD structures. So you also include gauge connections and stuff like that. But I will discuss topological field theories for you. And in fact, I will start with not even doing topological field theories, but just homomorphism sort of out of boredism groups. So just partition functions for simplicity. And those will be related to the invertible topological field theories with permanent symmetries. And for that, since I want to also consider non-unitary invertible field theories, I will need a little bit more complicated version of a boredism group. If you know any type of boredism groups, this might be one that you're not very used to. And if you're from the Max Planck Institute in Bonn, then you know this as the S K K group. If you grew up somewhere else, you might know it as Reinhardt boredism, I think, or vector field boredism if I'm not mistaken. But what is it? I take closed H-manifolds. And I, so those form a monoid under this joint union, then I take the growth group associated to that, portioned out by the S K K relations. And I decided to risk my career and make a picture of the S K K relations on the Blackboard. So there we go. So the relations goes follow. It's a four termination. So I take two manifolds like this, and I could take two sort of hyper surfaces that are diffeomorphic here. And then I'm allowed to switch them. So this is equivalent to, and now I just flip the two pictures around. And now it should look slightly similar if my drawing skills are reasonable. See, I flipped the upper parts. So it looks something like this, I guess. This is the S K relation. And this will give me some of the group under this joint union. And then what is going to be a sort of invertible field theory? Well, I don't want to do topological field theories just yet, but a invertible topological partition function, maybe I should have called it a topological action, because it's closely related to that notion that we had in the last talk. And what is it? The partition function is a homomorphism from the S K K group. There you go. To C cross. So this can represent some possibly non-unitary invertible topological field theory. And we get the partition function for that. And maybe this is the right time to remark and comparison to the last presentation that I'm considering discrete invertible field theories here. So I'm not doing the physically correct thing of considering this Anderson dual and doing all the nice things that we've seen in the last presentation. So in particular, things like terms, silence terms and stuff like that will not be contained in here. So that's unfortunate. But let's do an example at least, an example that is different from examples that we've seen in the last presentation. So what I could do is, I mean, this is, I'm going to fix the complex number. This is sort of my phase angle, if you will, though it's not quite that. And my partition function, my manifold is just going to be pulled up to the boiler characteristics. So these are just boiler theories that were also sort of briefly mentioned in the last talk. So this is one example. And in fact, in general, this is not unitary invertible efficient function. Okay, so I'm going to tell you a couple of facts. So you might know another borders group. And in fact, this STK group, there is a map from the STK group to that border. The border group is usually written like this. So that's the monoid of HD manifolds of the border. And this is not always an isomorphism, but it's always subjective. So here's one interesting example, which is the case that I mentioned to and the structure of this O in that case, the exact, I'm making a mistake, sequences like this, it turns out that the degree to the dimension to borders and group is isomorphic to see not to. And this extension is actually not split. So this is a non split extension. But this is sort of the most general non split that can happen in this kind of invertible theories. So you can prove the following, which classifies basically invertible theories, if you know the, if you know the unitary ones, but we'll discuss that later. So in the even case, you can show that the SKK group is isomorphic to a pullback of the ordinary borders and group. It's a little diagram like this. It's this pullback. I can map the borders and group always to Zmont to by taking the other characteristic modulo modulo two, that's always the borders are invariant because it's an integral over the topstiff Whitney class. And here it's just the Zmont to map. So this is indeed even. What happens for the odd? I don't quite know. It doesn't seem to be well known. But something has been proven by Matthias Trek and others. This is again a theorem of Kretschfeld's Titian. For the odd, there is a exact sequence. It's always surjective on the right, as I just told you, but it's not always surjective in the beginning. And in fact, this map is exactly injective if HP plus one manifold, so in one dimension higher, has even more characteristic sounds like a great condition. But in practice, it's actually really hard to determine whether every every manifold has even other characteristics. That's a super hard question. And I'm also not aware of any non split examples. So that would be interesting to know. Incredible. This was exactly how the matter is. I now wanted to switch to the second part of my presentation. So now go to share my screen. So the next part is going to be more about actually topological field theories. I will not just consider the invertible case, but also some non invertible things. I hope that's not too confusing. So let's get to it. So I will discuss like the main two things I want to discuss about topological field theories are reflection positivity and the space statistics here. So for that, I will start with first explaining what reflection structures are on manifolds, a little bit reflection structures on topological field theories. And next, I will get to reflection positive topological field figures and touch upon the relation with dagger filters. And if there's time, I hope there will be time. And I will also discuss the statistics theorem for topological field theories, in particular for invertible topological field theories. Oh, and I should say that when I say topological field theory here, I always mean the TSC style functorial definition of topological field theory, mostly just the non extended version. And if you're fancy, I sometimes do once extended. And mostly just the classical definition of topological field. So first, what are invertible topological field theories? I already mentioned that word a lot of times. So basically invertible topological field theories are one to one correspondence with these invertible topological partition functions that I wrote down. One very mathematical way to say that is that if you have an invertible partition function, like I just like I wrote on the board, then you can with appropriate target category, you can extend it further and further downwards to make a extended topological field theory out of it. So what does that mean in practice? So for example, just the ordinary TSC definition, you can just consider topological field theories with as a target, the group boy of super lines. So you just consider one dimensional z two graded vector spaces. And so in other words, it's just a normal topological field theory, but the super vector space is very important. We need those, we need those fermionic state spaces as well. And to make this a one to one correspondence. And yeah, so it's the usual thing that you demand for invertible things need all state spaces are one dimensional all partition functions are non zero things like that. And you can also do something like that once extended and one example of such an appropriate target would be the category of super algebras, which objects are super algebras, one morphisms are by modules, and two morphisms are interfiners. But for most of the talk, we'll just focus on this on the second point, just normal topological field theories, you know, to see the sense. Okay, so to motivate reflection structures and reflection positivity a little bit, I wanted to mention this little proposition about what do topological field theories with fermionic symmetry in dimension one look like. And this proposition here is pretty is not very hard to prove, and it takes a bit of work, but it's not very hard to prove once you have the borders of my boxes, you know, but the main point of this that I want to show this proposition is to show you the general structure of such such a one dimensional topological field theory. So, as you might have expected, you get some kind of representation. And it's a representation of this only at the time of preserving symmetries. But what did the time reversing symmetries do? Well, you get all these weird by linear forms and how so you get a whole collection of non degenerate by reforms. Oh, I forgot to include here. It's very important that it should be discreet. So let's assume that it's fine symmetry here. So you get all these non degenerate by linear forms and they satisfy this this, yeah, bunch of relations that relates different by the new forms to each other together with these representations. So that tells us somehow how the representations are adjoined to each other or something like that in this unreleased by the new forms and they have an even worse equation with this expression. And it's just both mathematically bad, and it's physically not really clear what this means. But once you assume reflection positivity, and I don't think I already mentioned that the reflection positivity is just the the Euclidean version of uniterity. So it's just if you have a Lorentzian theory, which is unitary, we rotate it and people usually call the Euclidean version reflection positive. So what do those look like? Well, it's much easier and it's much it's much closer to what you expect from the physics. So one dimensional reflection positive topological field theory, they're all isomorphic to basically your representation on a complex Hilbert space. And it's topological field theory. So it's kind of dimensional. But it's not quite a complex representation. So the time preserving elements act unitary, but the time reversing elements don't even act complex linearly, but they do act anti unitary. And that's exactly what you would expect from things like for example, witness here. So this would make more sense to if this is I would guess. Okay, so this is a motivation to define reflection positive theories. So let's get to I will briefly go through reflection structural manifolds. So given an HD manifold, I didn't define what an HD structure is. But I claim you can define its reflection, which is so it's a kind of generalization of orientation reversal in the case that manifold only has an orientation. And in particular an HD structure does. Yeah, I didn't say what it was, but it also includes an HD principle bundle. And I just wanted to highlight what the reflection does to this HD principle bundle. So coming back to the part I did on the blackboard, I defined this, this even part of the Spermian tensor product as the space time structure HD, responding to the internal symmetry group. But I mean, I define it as the even part of this. So by definition of HD, there's a short exact sequence of this form. And what I can do if I have a principle HD bundle, I can always enlarge it to make it twice as big by just taking a product like this with each hat. And then I have P sitting inside it. And I sort of take the other product, take not P, but I take the other sheet in this, in this enlarged P. And this will give me P bar, which is the HD structure. Sorry, it's the principle HD bundle corresponding to the reflective manifold. And you can continue playing with this. And eventually, you can show that this reflection actually extends to a whole Z2 action on the bordered category. I seem to have included a G here. I'm sorry. So I just mean HD here. But by HD of G, I just mean that it's the HD corresponding to the internal symmetry group G. So that's a little notation issue. Anyhow, so it acts on this whole bordered category, like that. And that's a symmetric model action. And now what is a TFT? Well, I always want to consider topological field theories with target super vector spaces, because I'm also doing fermions. So a topological field theory with internal fermionic symmetry G is a symmetric model function like this. What is a reflection structure on this? So a reflection structure is a way to make this into a Z2 equilibrium function under this Z2 action. And that actually turns out to be data, so it's Z2 equilibrium data for these two actions. So extremely, I mean, this is all very abstract. So very concretely what this means is we just have a bunch of isomorphisms like this. So get where M is like a spatial manifold. And they satisfy a bunch of properties compatibility with district unions and grading and things like that. Okay, so that's a reflection TFT. Now I want to get to reflection positivity. But first I want to make a brief detour to the connection with that dagger categories here. So what's a dagger category? Dagger categories have been used in several reformulations of quantum mechanical systems. And maybe it's not super surprising that they might also come up in this situation. So what's a dagger category? It's a category which is equipped with a contra variant functor from from itself to itself, which squares to the identity. And it is also the identity on objects. And you can extend this definition if you work hard enough to be compatible with symmetric motive structures which we need to make sense of topological theories. And the main example you should keep in mind here, which is also main motivation I guess for defining dagger categories, is the category of vector spaces with Hermitian inner products. So here I take S Herm, which is I want to take a complex super vector spaces. And they're equipped with Hermitian inner products. So they're like Hilbert space inner products, but they don't have to be positive definite. That's my definition of Hermitian vector space. It doesn't have to be positive definite. So in particular, there's the collection of super Hilbert spaces is contained in that. That's another example. But now it turns out that you can also with some work, you can make the borders of the category in the dagger category. But then you need to make some choices. And what are those choices? Those choices are Hermitian structures. And what does an Hermitian structure? So it's a little bit abstract definition again. So why does this make sense? Hermitian structure on a spatial manifold is a diphthermorphism which is compatible with the HD structure from the reflected manifold to the dual objects in the borders of the category. The borders of the category has a dual. And it satisfies this relation. And why does this relation make sense? Why does this make sense? If you would not write n here, but if you would write a vector space v, and it would be a map from v bar to v star, then you can check for yourself that if you have such a map like this, that's exactly the same as a Hermitian inner product on the vector space. Sorry, it's good to know that as well. All right, so what's the statement? So the statement is that you need to choose some Hermitian structures to make the borders of the category. So suppose that h is a choice of Hermitian structures on every object of the borders of the category simultaneously, which is compatible with this strong union. Then using this, you can make the borders of the category into a symmetric amount of dagger category. And the crucial formula is this very trivial looking expression, which is just basically what you would do to a matrix algebra. If you wanted to find the Hermitian adjoint, you would do bar transpose. So I'm doing bar transpose here. And then on objects, we have chosen a Hermitian structure, which allows us to identify Bono, which allows us to identify m bar star with m, so that it becomes the identity on objects after this identification with h m. So then the dagger is the identity on objects, but it could be not the identity on morphisms. Okay. Now additionally, if you have a topological field theory, which is associated with re-variant with respect to the reflections, then you will, using this choice, you will get a symmetric monoidal dagger function from this dagger category of borders to super emission vector spaces. So I should explain some notation here. So this, I don't know if I, this, the symmetric monoidal dagger category that follows from this construction, but I put a little upper h here to remind myself that this construction depends on the choice of Hermitian structures, possibly very strongly. Yes. And this will land not actually in super vector spaces, but in super emission vector spaces. As it turns out, I will go into that in a moment. And why is this nice? Well, I mean, it's nice because now our state spaces are suddenly, they have been approached, like you would expect from physics. And also because now we can require them to be positive definite, which would be something like reasonably reasonable to ask, because then there would be Hilbert spaces being finite dimensional. So that's the next step. If you have a topological field theory, which is C2 vector variance, then we say that it's reflection positive with respect to a choice of Hermitian structures. If the induced Hermitian inner products that I just talked about on all state spaces are all positive definite. So we had this induced symmetric monoidal dagger from here. And we require it to land in super Hilbert spaces, not just in super Hermitian vector spaces. And here I wrote a little equation to tell you that it's, it's pretty concrete. If you want to know using all the data that I just gave you, what's the, what the Hermitian inner product is on all the, on all the state spaces, it goes like this. So we want to identify the dual with the bar, that's the Hermitian structure on the on the vector space in the super vector space. And the first thing we do is remove the star inside using the fact that Z is a monoidal functor. And then we had this very special choice of Hermitian structures on every object, HM, and we apply Z to that to get here. And then we use the Z2 equivalence of the functor to find the inner theorem. This gives us an inner product. We require it to be positive definite. And note that, yeah, again, that this all depends on this, on this choice. The choice is really important. If I would compose somehow HM with some function F. And it would still be a Hermitian structure. And I would use that for let's say that F is like an automorphism M. And this will still be a Hermitian structure and say just for the sake of argument that C of F is maybe like minus the identity or something like that. Then the resulting, the resulting Hermitian inner product on the state space will be different by a sign. It will just be minus the thing from before. And therefore, if you start with something positive definite, and you change the Hermitian structure to something different, it will actually be negative definite. So you cannot just demand it for all Hermitian structures. You have to pick a choice of Hermitian structures. And this, you can also see this in, oh, I thought that something else. Okay, so this is first. Okay, so we get back to the invertible case. So what happens in the invertible case? So first thing I should say is that there turns out to be a somewhat canonical choice of Hermitian structure on the boredom category for any internal fermionic symmetry group as I gave. And from now on, we'll use that. And then you can prove this theorem, which I mean, I'm not sure who I should cite here. So I cited everyone. I guess the original statement is due to free op gates. They have the great reformulation of Unicura for non-extended field theories and also actual types now using these dagger categories. So what's the statement? So we start with this invertible topological partition function, which is the same as an invertible topological field theory. Then there's a distinction again between odd and even. In odd, it's a little bit nicer. So for odd dimensions, partition function has a reflection positive lift exactly when it factors through a homomorphism from the boredom group to U1. So that's very nice. And it's also unique. So there can be many different reflection structures on a different topological field theory. But somehow, if it exists, then the reflection positive lift is unique. And as a consequence, you get an isomorphism, something like this, which is closer to something you would find in the physics literature. Often they're closer to homomorphisms from the actual boredom group. And maybe I could mention here again what I already mentioned before, that I'm doing the street invertible topological field theories here. So I'm not getting this Anderson dual thing that we had in the last presentation. This is a little bit of a simpler situation than that. And for the even, it's a similar situation, but we only have this additional example of Euler theories that you take. You took this laptop and you took it to the power of order characteristic. And that gives you a theory, invertible topological field theory. And if you take that, if you take that to be a real number, then it turns out to be still in this definition still reflection positive. So you have this extra weird factor of our positive hang around. But otherwise it's very similar. Okay, then I hope that now is the example. Yeah, exactly. So this is a subtle example that I wanted to idolize. It's nice to ponder about. So what's the difference between the internal symmetry group Z mod 2 and the internal symmetry group Z mod 2? I mean mathematicians, I guess would say the same. But for me, this one contains minus one to the F as a non trivial element. And here, minus one to the F is equal to one. So that makes them different. But otherwise they're extremely similar as internal symmetry groups. So what happens? So if you run the story that I just told you, the two internal symmetry groups, maybe as expected by the physics will give you the space time structure groups, spin and SOD cross Z mod 2 respectively. But the funny thing is in dimension one, they're actually equal. They're just equal on the nose. They're both Z mod 2. Additionally, you also have this isomorphism. They're both Z mod 2. However, there is one subtlety, which is that in one of the cases, the generator is the anti periodic circle and the other is periodic circle. What do I mean by that? So there are not many connected one dimensional manifolds with these structure groups. And there are just two. You have the circle with periodic boundary conditions and anti-periodic boundary conditions. And maybe you know that for the spin case, it is actually the anti-periodic boundary condition that's bounding. But for the SOD cross Z mod 2, it's the other way around. It's a little bit unexpected, maybe if you've never seen it before. And as a consequence, if the theorem that I put on the last page is true, then, well, the only, in both cases, you said have a single non-trivial reflection positive preferable topological field theory. But I mean, it should be a homomorphism from the border group. But in one case, it should map the periodic circle to minus one. On the other case, it should map the anti-periodic circle to minus one. So it's the other theory that's reflection positive if you take the other symmetry group. And there's only one way, in my opinion, that you can make these two different notions of reflection positivity happen. Namely, you have to choose a different Hermitian structure on the borders of the category. So there's a different Hermitian structure associated to it in which you have to put an additional spin flip on one of the two points. Okay, so that's a subtle example. So let's see, I'm a little bit out of time right now. And I don't want to rush too much. So let's see. I think I'll just mention this in a few words. So I just wanted to say that it's not very well known, but you can make spin statistics also very precise for topological field theories. And I don't know if due to who this is, but I learned this from a nice paper by Theodos Preet. And basically, the arguments, the argument goes as follows. So physically, the spin statistics theorem just says that if a particle has happened to your spin, then it's a fermion on the other way around. And it's actually a theorem for, as far as I understand, for unitary reflection positive theories, or it should be. But otherwise, it's a definition you can require topological theory to satisfy this. And how do you make this physics definition precise? Well, you have to make the observation that basically, a representation of a spin group has integral spin, if and only if, well, if it's like factors to SOD. So if the element of the kernel to the amount to SOD acts trivially. And then you can make a definition that's roughly the following form and has several generalizations, but this is the simplest one, that if you have a super representation of spin, then it satisfies the statistics exactly when this element is exactly the gradient operator. And this formulation has several generalizations to topological field theories and including internal symmetry groups. And let's see what I want to do. Maybe I just want to at least highlight that it's known. I'm not sure how alone it is, but you can show which of the invertible topological field theories satisfies spin statistics. So all of the reflection positive ones do, all of them in even dimensions do, even the non-reflection positive ones. And again, in other dimensions, there's a weird condition that it depends on the Euler characteristic of age manifolds. And I don't know how to say it's more simply than this, unfortunately. Okay. Yeah. And then I had some results finally, and I'm working on together with Lucas Miller in which we explored two-dimensional extended topological field theories with target super algebras. And we've classified all topological field theories with internal fermionic symmetry group satisfying spin statistics, at least in time-preserving case. And we're working on the case without time-preserving and still in progress. So this is not really a conclusion, but this is more of like a happy ramble. Let's get the physics closer to the math. When physicists talk about unitary and statistics, we can actually say what it is. And that's fun, I think. So thank you for your attention. Yeah. Thank you very much, Luc, for this very interesting talk.