 First of all, let me thank the organizers for giving me the opportunity to explain some of the work we did here. We did in the past. So I'll talk about basically one paper, but it's a long one, and it's called Infinite Chiral Symmetry in Four Dimensions. So the main reference that I need to give is just this archive number. And I should probably list my collaborators, Chris Beam, Madalena Lemos, Pedro Liendo, Wolfgang Palais, who you've seen last week at Leonardo's Study. So before I begin, I would like to also say sorry for a few things in advance. One of them is that there are too many queues. So one queue is not another queue. And if it's not clear from the context which queue I mean, sometimes you should please interrupt me and ask which queue I mean. So also for indices, there are not enough letters in the alphabet. So sometimes I will not be very consistent with the use of indices. I'll try to be as or at least explain it when I use a specific index what it stands for. But it's not going to be consistent throughout the talks that just too complicated. And then finally, a third apology is that at the beginning, I think I would like to discuss quite a bit in quite some detail the representation theory of the super conformal algebra that we'll be considering. And so this means that you will have a solid background, hopefully, if you pay attention in the next two hours in the representation theory of this algebra. It comes at the expense of talking about chiral algebras. I think it was a worthwhile trade-off or a worthwhile thing to do because if you understand the representation theory, it'll be much easier to understand both the chiral algebras themselves, but also lots and lots of works on, say, the bootstrap approach to four-dimensional N equals 2 super conformal field theories. That's like no matter which paper you open on this subject, they basically assume some familiarity with the representation theory of this algebra. So that's why I think it's good to spend some time on this. So what's the context we'll be working in? I'll be almost entirely looking at four-dimensional N equals 2 super conformal field theories. These often have conformal manner of, or, sorry, modulized spaces. So Ovecura will be looking at these theories at their super conformal fixed points, so not going out on the modulized space, either the Coulomb branch or the Higgs branch or whatever branch. In these theories, you have a lot of observables in these theories. I'll be focusing exclusively on correlation functions of local operators. These are, of course, gauge invariant local operators if your four-dimensional theory happens to be gauge theories. And last of all, for at least the first hour or two, I'll be taking an operator algebraic view point. So I'll take the correlation function sort of as abstract objects and, of course, the algebra that they generate through the operator product expansion as an abstract algebra. And in particular, I will almost never refer to a Lagrangian, at least in the first half of the talk. So the Lagrangian, for now, does not really play a significant role in my talks. This is a bit orthogonal to what you have been seeing in all the other lectures, basically theories for localization applies, needs some kind of Lagrangian description, some kind of path integral. In these lectures, I'll not need it, at least in the majority of lectures. I'll not need this description. So if you look at this set of observables in these kind of theories, the first thing you should realize is that these local operators they transform under the symmetry algebra. So basically, the associated states from representations of the symmetry algebra of four-dimensional n equals 2, super conformal theories. And this algebra is SU 2.2 slash 2. So what does this algebra have, well, as a maximal bosonic subgroup? It has the conformal algebra in four dimensions, or maximal bosonic subalgebra. And then it has an SU 2 r symmetry, and then it has a u1 r symmetry. So in contrast to Zohar, I'll be using capital R for the SU 2 labels, and little r for the u1 labels. I think Zohar has a capital 1 for the u1, and then S, I think, for the SU 2. I'm sorry for that. So what does this algebra consist of? Well, first of all, there's the conformal algebra, which I have already given a few important elements, or a brief summary of the conformal algebra on this board here. So the conformal algebra contains, as a particular subalgebra, the prankaré algebra, which has, of course, the boosts or rotations, the translations, which are generated by this vector field, special conformal transformations, which are special to conformal theories that are generated by this conformal vector field. And then there's, of course, deletations, which are generated, which correspond to scalings, i.e. to this conformal vector field. The commutators of the prankaré, well, most of the commutators are obvious. Here are a few to remind you of the commutators for conformal field theory. The translation operator is just a derivative. So it has weight 1, deletation weight 1, and the special conformal transform, the generator special conformal transformations has weight minus 1. And the commutator of a special conformal generator and the translations gives you some rotations which are not very important, and then the deletation operator. So who has seen this algebra before? Almost everybody, right? OK, very good. Who knows before plunging into the representation theory? Yes, who knows what? Who can tell me? Who is comfortable? Who would be comfortable telling me what a conformal primary is? If I would not ask you to actually do it, but would you feel comfortable or not conformal primary? Yeah, a few of the organizers. Yeah, OK, it was just an opinion poll to see what you guys know and don't know. That's fine. I can do in some detail the representation theory of the conformal algebra first, and then see how that fits into the representation theory of the super conformal algebra. This was actually part of my plan, so that's totally fine. Before I do the representation theory of the conformal algebra, let me briefly list what extra stuff is there in the super conformal algebra. In the supersymmetric version, you get, of course, a bunch of supercharges which transform in a doublet of SU2. So here, this capital I index is the SU2 index, and they have u1 charge 1 half, and the alpha here is, of course, spinner index. There are q tilde's with opposite u1R charge and the opposite spinorial representation. And then there are special conformal, or what we call conformal supercharges, s and s tilde. And then, of course, there's the SU2R symmetry, which is this guy, these three generators. And there's a little u1R. There are a few extra commentators I can write down here. So obviously, q tilde gives you the translations as usual. So just as the supercharges are like the square root of translations in this precise sense, the special conformal supercharges are the square roots in this precise sense of the special conformal transformations. And the q's with themself commute to 0, the s's with themself's commute to 0, same for the q tilde's and the s tilde's. Maybe one more thing that I can write down. So qs with themself's, this is a bit of a nasty anticommitator, delta ij, delta alpha beta, d plus, well, delta ij m alpha beta minus delta alpha beta r ij. Where this r ij is an object that I wrote down here because I'm going to need it later. So r ij is defined as follows. r11, r12, r21, r22 is the same as the matrix r over 2 plus r. r plus, r minus, and r over 2 minus, just so you know. OK, so that's the super conformal algebra. Keep that in mind for later. So all of this is actually in this paper. So this algebra, if you want to see it in full glory, please go to appendix A of that paper, and you'll find it there. I should have probably mentioned this before you had to scribble this down, sorry for that. So representation theory. Let me ask a brief interlude. So intermetzo, open parentheses, we'll close them in about 10, 15 minutes. The representation theory, well, not the full representation theory, we'll discuss some representation of the conformal algebra, namely those that are useful in physics. There are other representations that are not used in physics, because, for example, they could be unitary, but they would have unbounded energy. Energy unbounded from below. So those, for physical reasons, other representations are excluded. So before I start discussing the representations, let me say that in radio quantization, I work in radio quantization where you take your time to be sort of the radio coordinate, and then there exists the Hilbert space of states. This state is a Hilbert space defined on the spatial sphere, a d minus 1 sphere. And for this Hilbert space, there exists a state operator correspondence. And actually, all that we'll be needing is very little. It just says that for us, it'll mean that for every operator, I can define a state, which is just generated by inserting this operator at the origin and acting on the vacuum. So here, I want to add that this is an element of the Hilbert space on a three sphere and not some flat space, Hilbert space. And this omega symbol here is the vacuum. It's a unique state that I assume is unique and it's killed by all the generators in your conformal algebra. Actually, since we're in the intermezzo, I shouldn't talk about the super conformal algebra, but I can already tell you that this vacuum is also killed by all the generators of the super conformal algebra that are not in the conformal algebra. So this Hilbert space is your vector space that's going to furnish representations of the conformal algebra. So what does a representation look like? Well, every representation begins with a certain operator and that we're going to call a primary operator, which has the special feature that, well, first of all, it's an eigenstate of deletions. Sorry, this is incorrect at 0. So now let me do this a bit more precisely. I'll just do this. So the corresponding state is an eigenstate of deletions and the corresponding state is also killed by all the conformal supercharges. So remember that the special conformal generator has weight minus 1. So in this sense, this is the lowest weight state in your representation because if I try to lower the weight, I get 0. And so how do I generate higher weights? Well, that's easy. The translations have weight plus 1. So I'll just act on this guy with the translation operator. So let me take the state and act on it with P mu. Well, the state that I get then is just the derivative of the guy. And of course, I can go on. So I can act with another P mu and so on. And this generates my infinite dimensional representation of the conformal algebra. And if this is a primary, then these guys are called descendants. So notice that here I have the operator at the origin and derivatives of the operator at the origin. I'd like to sometimes think of the operator i of x inserted not at the origin for at some other point x as just generated from the operator at the origin by exponentiated translations. Sometimes I want to do that, which is, of course, fine. And if I take i of x and I act on the vacuum and I let it act on the vacuum, then of course this thing here on the right hand side disappears because the vacuum is scaled by P mu. But this means, in particular, that since the special conformal generator does not commute with translations, it means that k mu does not commute, does not give 0. So sorry, k mu does not commute with this operator inserted at x, unless x equals 0 and i is primary. If we now go back to these equations, there's a 0 here, there's a 0 here, there's a 0 here. These are 0s are very important. The moment you start translating this operator, this first and this second equation no longer hold. So I just wanted to stress that. Oh, and this is actually the only comment I wanted to make here. So this is, in a nutshell, very simple representation. These are the essentials of the representations of the conformal algebra. Let me do an example very briefly. So we'll do a free massless boson, let's say, in four dimensions phi of x. So it satisfies the Klein-Gordon equation, box phi of x equals 0. So in this theory, you can easily show that if you just take the Lagrangian and you see how it is conformally invariant and how the special conformal guys and the dilatations and all of them act on the fields, you'll find that phi of x is a primary. So I say here phi of x is a primary. It means that just in that sense, right? So I take phi of x, I put it at the origin, and the state that I generate in that way is as these properties. Even though there's an x here, I still call it a primary. Phi to the n of x for any n is actually also any positive n, normal order if you want, is also a primary. I can define another primary like T mu nu, which is d mu phi, d nu phi, and then some other stuff, eta mu nu, d rho phi, d rho phi, plus 2 over d minus 1, phi d mu d mu phi. So this, as it happens, is also, so it takes a lot of work here. If you want to check this, but it is actually a primary. You notice there are a lot of translation operators. So you take this guy, you insert it at the origin, you act with k, you have to sort of, you have to commute this k with all the translations. You generate all this junk here. You do some rotations and so on. But in the end, you'll find that this is a primary. And there are many, many other primaries in this theory, but these are just some simple ones. And descendants, well, it's easy to generate a descendant, because if a phi of x is a primary, then the derivative of phi is a descendant. And similarly, it would be the case that sometimes it's not entirely obvious that an operator is a descendant. So let me, can I lower this all the way down? Great. So sometimes it's not entirely obvious. A somewhat too trivial example is, for example, phi d mu phi. This is, of course, the same as 1 half d mu phi squared. And now you see that this is also a descendant. And if you're given a general sort of bunch of phi's and bunch of derivatives, some contracted, some not, it's not so easy to see whether it's a primary or a descendant. Or maybe it could be a combination of those. And then you sort of have to see how everything fits in a big representation of the conformal algebra to see what kind of combination of primaries and descendant a given operator is made out of. One very important reason that I wanted to talk about this representation is that sometimes something funny happens. So for example, I told you that phi is a primary. So OK, let me act with derivatives and generate descendant. Let me, in particular, act with two derivatives and contract them. So I act with the Laplacian. Then I don't get a new state. I get 0. Laplacian on phi is 0. So this is actually a more general phenomenon. Sometimes a descendant is null. So it has 0 norm. I haven't really talked about norms. And I don't want to. But believe me that sometimes there is a good notion of norm here. There's a proper Hilbert space. And in that norm, you just find that the Laplacian act on phi is 0 norm. It's null, and therefore decouples. So this happens, and you just ignore it. You basically cross it out from the list of operators. And then you can say that the representation is said to be short. Of course, descendants of a null are also null. So you just remove the entire conformal representation associated, generated by the null states. So you think that the representation is not necessarily the same as the problem? That's correct. If you know what a verma module is, that's exactly correct. But if you know what a verma module is, then you should. Then this is not so hard. So if I continue this example, example continued, box phi is 0. So this is a null descendant of the primary phi. And this operator T mu nu that I defined also has a null descendant. If I take one derivative and contract the indices, I also get 0. And this, of course, makes you realize that this T mu nu that I defined is nothing but the stress tensor in the theory. So the stress tensor in a conformal field theory sits in a short representation of the conformal algebra. It's a primary, and it has a null descendant. And similarly, this is not really my example. So maybe I should write it here. If I have a current J mu, it also has a null descendant. So currents also automatically sit in short representations of the conformal algebra. Yes? When you say that the stress tensor is a primary, are you assuming that D greater than 2? Yes. In D equals 2, my algebra is infinitely bigger than the one I wrote there. I can still define this subalgebra. And D equals 2, it would be also to R, cross SL to R. And with respect to this subalgebra, the stress tensor is actually a primary. I believe these are then called quasi-primaries. Well, that was the question. In D, oh yes, sorry. The question now is whether the stress tensor is always a primary. In D greater than 2, that's always the case. And since I'm in this context of 4D n equals 2 theories, let's say yes. In this context, it's absolutely always a primary. That's a good question. Yeah, it depends a bit on what your axioms are, I think. If you want to say that this guy generates the conformal transformations by integrating it. So this P is not just in random P, and that T is not a random T. But you say, well, that P is just the integral of the stress tensor. And similarly for everything else, then I think it's easy to show that the stress tensor is a primary. Because that was the question, how do you show that the stress tensor is always a primary? Alternatively, I think you can just go over the list of representations and find that. Or actually, then it's absolutely clear that it has to be a primary. Because if the stress tensor would be a descendant, it would be the derivative of something. And if I would integrate it, then I would not get something meaningful. More generally, I think you can just go over the representations and see which ones are allowed. And you'll find only one feasible stress tensor representation, which is this one. Also, these no states and no descendants, they generate constraints. You, for example, see that this can only happen if the dimension of phi in general dimensions is d minus 2 over 2. And this will also tell you that the dimension of t mu nu is actually equal to d. And this will also tell you that the dimension of j mu is equal to d minus 1. So I wasn't planning to go into the details of that. You can just show this using the fact that you want this to have zero norm. But to explain it requires me to explain the concept of norm. And I don't really want to do that. OK, any questions about the representation theory of the conformal algebra? No? Good. Then I can close my parentheses and the intermeto ends. And we can start talking about super conformal algebra. So for the super conformal algebra, the representation theory is fairly similar, at least when it comes to the physically relevant representations, in the sense that they're always generated by one special operator. So we'd say the representation theory of SU2 comma 2 slash 2, or any super conformal algebra, for that matter. So there is one special guy. I've denoted SU hat i for now, where this is some multi index. So it captures all the quantum numbers for now of the super conformal algebra. So let's see what the quantum numbers are. Well, there's a scaling dimension for the operator delta. There are Lorentz spins, j1 and j2. And then there's an SU2 r symmetry and a u1 r symmetry. So those are all the quantum numbers that you need to generate to describe a representation. And this special guy is called the super conformal primary. And in particular, it's a conformal primary. So if I insert it at the origin, then it is an eigenfunction of dilatations. It is killed by all the special conformal generators as before. And it is now also killed by all the S's. So S and S tilde, for any value of their indices, also kills the operator. This operator could be a fermion. So let me write it like this. And then how do I generate descendants of this guy? Well, I can, of course, act with p mu as before. But I'm not going to write that. So that is one way to generate the descendants. But now I can also act with the supercharges. So I have the q tilde and the q's. And they give me some new operator. So let me now do this in the vector space language. So I'll generate the state, and I act, and I get some new xj on the vacuum, or some yk. And these, in turn, are conformal primaries. They're not super conformal primaries, but they're conformal primaries. So I can act, again, with the p mu's on them to generate the tower. But I can also continue to act with the q's or with the q tilde's, q tilde's. And I generate some z's and some other stuff. And this, again, gives me a conformal primary, and I act with it. So this is what the representation, roughly speaking, looks like for the super conformal algebra. You have a super conformal primary, a bunch of conformal primaries that you can find by acting with the supercharges. And on each of those, there sits a full tower of conformal descendants. And just as the case, the special conformal generators here, I didn't say that in my parentheses. So basically what happens here is that if you act on the k with the k on the first descendant, well, k commutator p gives you the dilatation operator. So it can eat up the p and basically up to dilatations and some rotations. It basically moves you back up in the tower. So the p's move you down. In the case, roughly speaking, move you up. And in the same sense, the q's here move you down. And the s's, they can eat a q and spit out the dilatation operator and some rotation and r-symmetry stuff. But essentially, they move you back up in the chains. So the s's tell them move you back this way and the s's move you back that way. All of this I should add now is very schematic. This is not quite correct what I wrote here. For example, this z, if you act with a qi alpha and a qi tilde alpha and a qj alpha dot and you said i equal to j or you contract the ij indices, then you basically act with the vector operator. And this z will have the same quantum numbers as the super conformal primary, except that its dimension is, well, sorry, it's the same r-symmetry quantum numbers as the conformal primary. And so this z can mix with the first descendant, the first conformal descendant of p, of the super conformal primary. In other words, if you act with a p or with two q's, you can sometimes get an operator with exactly the same quantum numbers, and then generally they mix a little bit. And so you have a bunch of operators here with the same quantum numbers, and you sort of have to undo the mixing a bit and re-orthogonalize your basis to isolate here the conformal primary. So it's not quite true that if I just blindly act with q's and q tilde's, I get always conformal primaries. I sometimes need to subtract some descendants of the guys higher up. So that's just a little warning if you actually want to do this stuff. So this is what happens generically. And so a general representation, let me do some important representations, a general representation is denoted with this a in some notation. And then, of course, all the quantum numbers, so r, r, j1, j2. And it looks like this. It's basically a full house. So 1, 2, 3, 4. So I have 4th q tilde's, because both i and alpha dot range over four over two values. So I can go four steps this way. I can go four steps, 1, 2, 3, 4 steps that way. And then, of course, I can complete the thing. So this is basically what this kind of structure looks like for a generic long representation. And this notation here, I should add, comes from a paper by Dolan and Osborn. Also, a paper by Kenny Maldesen, who I'm in Mala and Raju, has considered the representation theory. And there's a new paper in town. And in their language, this thing would be called l, l bar, j1, j2, delta. Let me just highlight this paper here. So it's a paper by Kortava Tomas Dumitrescu, Ken Intrilligator. And these guys did the great job of classifying not just the representation of our algebra as u2, 2 slash 2, but of any superconformal algebra in any spacetime dimension. So this is now sort of the standard work to go to if you have questions about the representation theory of superconformal algebras. And they used this notation. Unfortunately, this paper came out fairly recently, I'm fairly old, so I used the old Dolan and Osborn notation, which maybe we should sort of retire and we should move to this more modern notation. So this is a long multiplet, but of course, interesting operators often sit in short multiplets, just like interesting operators like the stress tensor in a conformal theory sit in a short representation just of the conformal algebra. So you can, for example, ask, well, where does the stress tensor sit in a superconformal theory? It must sit in some representation of the superconformal algebra. And in fact, where it sits is in a multiplet that is commonly written as c hat 0, 0, or al al bar 0, 0, 0, 0. And this is a multiplet that looks like follows. It's short, so I don't have to write a gigantic diamond like this. There's a superconformal primary. Let me call it t. It has delta equals 2, j1, j2, 0, so it's just a scalar. And it's also a complete r-symmetry singlet. However, it's killed by some supercharges, and therefore, I can go only two steps down in the q tilde direction and q steps down in the q direction. It's not terribly important to know the operator set the dots, so let me just ignore them. If I go one step down, it's a commutator. I did not write down, but the q's have dilatation weight to half, and the q tilde is also so that the momentum can still have dilatation weight 1. So if this is delta equals 2, then this is delta equals 5 half fermions, and this is delta equals 3. And if I act with both a q and a q tilde, I get something that can have vector indices, j alpha alpha dot. And it's dimension 3. And already, when we discussed the conformal algebra, we realized that this was a conserved current. So this guy has a null descendant, and it's a conserved current. In fact, there's another conserved current. So there's one that's an r-symmetry singlet, s2r-symmetry singlet, and there's one that's an s2r-symmetry doublet. And these are nothing but the s2r currents. So these are, I can write it here, s2r cross, well, u1r cross s2r currents. So the r-symmetry currents sit in the same super conformal multiplet as the stress tensor. So I haven't yet gotten to the stress tensor because it sits even further down. So the multiplet looks like this. So the stress tensor is the lowest components in this ordering of this multiplet. And it has delta equals 4, and it's, of course, conserved. So it's in its own short representation of the conformal algebra. So this is what the stress tensor multiplet looks like. Does anybody know what sits here? Supercurrence, right? Yes. So the currents for the supersymmetry transformations, for the q and q tilde supersymmetry transformations. So these are delta equals 7 1⁄2. So this is what your stress tensor multiplet looks like. And so it's important for later to realize that the r-symmetry currents sit in the same multiplet as the stress tensor. Maybe for, yeah, I want to discuss briefly one more set of multiplets. So that would be set number 3, which are so-called b-hat r-multiplets, or in the newer notation b1, b-bar 1, 0, 0, r, 0. So these are also short. These are the multiplets, the Higgs branch multiplets that Zohar talked about, I believe, yesterday. So they are super conformal primary. It's killed by q alpha 1 and q tilde alpha 2. The multiplet, in general, looks like this. So you only have two supercharges to act on. So it's a bit like to act with. So it's something very short. And it's labeled by just the r-quantum number, the SU2r quantum number. So what else do I need to say? I need to say the scaling dimension is given in terms of this thing, SU2r quantum number, by 2 times r. And j1 equals j2 equals r has to be 0 for an operator to be the super conformal primary of this multiplet. So notice here that they sort of label a multiplet by the quantum numbers of the super conformal primary. This was implicit, but I hope it's clear. And maybe it's interesting to look at the first two examples. So when r is a half, then I have one complex boson with delta equals 1, and it's a complete scalar. It's an u1 r symmetry singlet, and it's a fundamental of the SU2r symmetry. And then I act with the q tilde. And in fact, I can only generate these guys. And as it happens, for r equals a half, this multiplet is even shorter than this one. It's ultra short. And there's no more non-null descendants. So does anybody know what this multiplet is? Sorry? Free hyper? Hyper-multiplet? I wanted to make contact with more Lagrangian notions. So this is just the hyper-multiplet, a free hyper-multiplet, basically. So this is a dimension 1 complex scalar, and this is a dimension 3 half fermion, well fermion, about tilde fermion. So in particular, since it is dimension 1, you'll find that box qi is 0, and these guys obey the Dirac equation. So if you have a free hyper-multiplet theory, a theory of free hyper-multiplets, you can alternatively say that this is just that your super conformal theory has a b hat 1 half representation. And there's one more that I need to discuss. It'll be useful later, which is the b hat 1 multiplet, or bb bar 0, 0, 1, 0, which begins with a delta equals 2 r-symmetry triplet guy generally denoted mu. Then there are two operators that are not so important. And then there is a dimension 3 vector operator, j alpha alpha dot. This, again, is a conserved current. So it's, again, a short representation of the conformal algebra. And therefore, if your theory has such multiplets, it automatically has conserved currents. These are not the SE2, and you want r-symmetry currents. Those sit in this multiplet. But as it happens, any other current that is not part of some higher spin symmetry has to sit in this multiplet. You can go over the list of all the representations of the superconformal algebra. And you'll find that there's no good candidate for a flavor symmetry current in an interacting theory, except this b hat 1 multiplet, which is why it's so important. And in fact, maybe we should dress this guy up, because of course, if it has a flavor symmetry and it's non-Abelian, then this transforms, this current transforms in the adjoint of any non-Abelian flavor symmetry that the theory may have. But the flavor symmetry, by definition, has to commute with the superconformal algebra. So in fact, the entire multiplet then must transform in the same representation of the flavor symmetry. So all of them sit in the adjoint of the flavor symmetry. Finally, let me say in words that there is one more representation that is very well-known, which is so-called epsilon multiplets. These are the ones that Sohar talks about. These are the ones that are completely chirals. So they're, for example, killed by all the q tilde supercharges, or they're killed by all the q supercharges. So in my picture, they look like a single line, because you can either go only to the right or only to the left. So these are the ones that are important for the Coulomb branch category. OK, that was the representation theory of the superconformal algebra to the extent that I thought it would be useful for you guys in general and for what is to come. Do people have questions about this? Where do the Higgs branch operators be? Where do the Higgs branch? That's an excellent question. Here, the Higgs branch chiral ring operators are in B-hat multiplets. They're the superconformal primary of B-hat multiplets. So this, if your theory has hypers, is a q. This is the scalar in your free hyper. It does the guy that gets a VEV in a free theory. If you have an SUM gauge theory with NF flavors, I think I had it somewhere written down carefully. But I think I don't have it anymore. I do have it. So if you have NF hypers for the Lagrangian people in the audience, which is all of you, perhaps in a representation R of a gauge algebra, let me write it like funny G. So suppose you have this kind of theory, then I can build a B-hat one multiplet like so mu a ij is nothing but q tilde i q j. Yeah, now I need some extra. So I need some gauge indices m bar n. So these are gauge delta m bar n. So I contract it to make it gauge invariant. And then, of course, they also have flavor symmetry, so where these are flavor. And then I can make it the adjoint of whatever flavor symmetry. So this is the adjoint. So this is how I built, and then I normally order this. This is how I built my B-hat one representation. It's just a normal other product of two q's in the hypermultiple. Oh, so I should I'm now here. OK, maybe I should add one more thing with my. So what we have here is an SU2R doublet. But I think Wolfger already mentioned this because I looked at his notes a bit. So this qi is a doublet, and it's doublet of complex scalars. So it looks like this. In terms of the usual q and q tilde, it's q, q tilde star. But then q tilde j, which I define to be epsilon ij q star, j by the pseudo-reality of the fundamental of SU2 is, of course, is again a B-hat 1 half. And it is q tilde minus q star. So when I write this q tilde j or q tilde i and this qj here, I have that in mind. So this very precisely defines how you generate this B-hat 1. And of course, you can sort of normal order them further. And that's an easy operation because it's all part of one Cairo ring. And then you generate all the other B-hat representations, or not all of them, but you generate other B-hat representations. So any more questions? OK. Good. Then there's lots of time left to talk about Cairo algebras. So yeah, I am sorry? This one? Sorry, yes. I wrote here something down and then I missed it. So yeah, there's qi and q tilde j. I just introduced that notation. I won't be needing it much, but I introduced it just to explain that here. So I hope you have a bit of an idea that if you write down a, I guess the message that I want to convey here is that if you write down an operator, or if you look at the list of operators in an n equals 2 super conformal field theory, every operator comes with a lot of friends in its representation, in particular. All the operators come in these sort of nice structures where you have super conformal primaries and descendants and then conformal descendants of those. And if you think about the stress tensor, say, in a 40 n equals 2 theory, you hopefully realize now that the stress tensor is not isolated and the stress tensor is sitting in the same super multiplet as, for example, the SU2R symmetry curves. So this, for example, implies all kinds of relations, not exactly identity, but connections between, for example, correlation functions of the SU2R symmetry currents and correlation functions of the stress tensor. And similarly, in fact, for this b hat 1, it's beautiful because, well, we did some analysis of the four-point function of b hat 1 multiplets. And this root, in principle, involves a four-point function of currents. And that's just from the conformal field theory perspective, even without supersymmetry. A four-point function of currents is a nasty thing to investigate. But by supersymmetry, this thing was just completely determined in terms of the four-point functions of the super conformal primary of this multiplet, which happened to be scalars in this case. And so we were just, we could look, we could investigate the four-point function of scalars and then draw conclusions by virtue of supersymmetry about the four-point function of currents. So that's one of the nice aspects of these sort of relatively big representations that super conformal algebras have. OK, Kyra algebras. Let's begin with the pedestrian approach. So Kyra algebras arise in four-dimensional n equals 2 super conformal field theories. When you restrict yourself to the cosmology of a certain funny super charge that I'll introduce later. So all of the structure that I'll outline now will have some cosmological interpretation. But I think it's useful for pedagogical purposes to just give you the results without referring to supersymmetry. And then afterwards, so then it'll be sort of a claim that I cannot prove, that whose proof is not obvious. And then maybe in the second half of this lecture I can give you the proof of the statements. So now if you ask yourself in the course of this explanation why everything is true, please hold that question because I'll be answering that later. So the pedestrian approach is relatively easy. We're going to consider an endpoint function of local operators. And we're going to dress it up in some ways. So I'll write it here. But if you copy these notes, please leave plenty. If you copy these on your notes, please leave plenty of space. So I'll write one operator, another operator. But this could be an endpoint function, another operator. So let's look at an endpoint function in the theory. And let me now do the following things. So first of all, restrict myself to a class of operators that I'll call SURE operators. These are operators that obey by definition the following relation between the quantum numbers. Their scaling dimension is given by the sum of J1 plus J2. So these are the two spins plus two times the SU2 are symmetry charge. So these operators will all have to be SURE operators. Before I proceed with explaining what kind of correlation function we're going to look at, let me give you a few properties of the SURE operators. They are always short. So let me say that they're always in short representations. They're short irreducible representations of the super conformal algebra. So for example, let's look at this B hat 1. This B hat 1, this mu guy, has delta equals 2. R equals 1 because it's a triplet. And it's a scalar J1 and J2 equals 0. So if you plug it in, and I've made no mistake, this guy is a SURE operator. Similarly, if you go to distress tensor multiplets, you find that there is a SURE operator here. And I leave it up to you to check. Let me add this. So this is SURE. In some sense, I'll make this a little bit more precise in a second. And this guy is in some sense also SURE. So it has J1 equals 1 half, J2 equals 1 half, and r symmetry triplets. So that's 2 here, 2 times 1. So this is 3, and delta is also equal to 3. So these SURE operators, they sit in various places in short multiplets of the super conformal algebra. They're not always super conformal primaries. And as it happens, the actual SURE operator is always in the highest weight of SU2R cross SU2 left cross SU2 Lorentz left and Lorentz right. So technically, what I mean is that the SURE operator is always the r symmetry, and theses are always sort of the top ones. So it's 1111. Similarly, for the Lorentz indices, they're always in sort of the top ones plus. And the other Lorentz indices are also always plus. So this is technically, so when I say that this guy is SURE, I actually mean sort of the top components with all the indices pointing in this direction. And they also, let me squeeze it in here, sorry for those taking notes, they also always have non-zero SU2R symmetry charge. That's not obvious from what I said so far, but it happens to be true. They always have non-zero r symmetry charge. So in particular, this is one or more indices, because those are the SU2 indices, and these are the Lorentz. So these are zero or more indices. And in fact, we'll need the entire r symmetry multiplets. So we'll start playing with these top indices, the r symmetry indices. But we will keep the bottom indices fixed. So I'll not be writing those. So let me write that down, the entire r symmetry multiplet of the SURE operator. So I2R plus, plus, plus, plus, plus dot, plus dot, plus dot. So these are the kind of operators we're going to consider. So now I can dress up this correlation function so I can say, well, let this guy have spin 1. This one be some bi-spinner, and this one, I don't know, be a scalar, say. So these operators, these guys are always fixed. And then the r symmetry indices, so let this guy be fundamental, this guy also fundamental, and this guy be, say, a triplet, right? Oh, sorry. Yeah, you see, so now I'm mixing up my indices up until now. So far, up until now, these capital I and J were always SU2R symmetry indices. So I think I can save myself. Let's make these capital. And then these also need to capitalize. I1, I2, I2, R. So that's better. So these are the operators I'm going to consider. I just by hand decide that I'm only going to consider these kind of operators. I will further consider these operators not inserted at arbitrary space time points. But instead, I'll consider the correlation function where all these operators are restricted to the two plane. And here I work in Euclidean signature, x1 equals x2 equals 0. And then we have x3 and x4 left. And we'll set z equals x3 plus ix4, as usual. And z bar equals x3 minus ix4. So then it is the case that I can denote the positions of these z1, z bar 1, z2, z bar 2. I can denote the positions of these operators, which is z and z. And then the next thing I'm going to do is I have this r symmetry, and this is floating around. And I'll do something peculiar to them. I'll contract them with a strange thing. So I'll write it out explicitly, position dependent vector, ui of z bar 1. So that's what I'll do. I'll take this, ui of z bar 1. I need more space. I need more space on either side. ui of z bar 1, uk of z bar 2, and then I don't have space. So I'll add the other ones here, url of z bar n, z bar. This is the thing I can do. This is weird. If you don't think this is weird, then you should look again, because I'm here. I have an su2 r symmetry index, and I'm doing something position dependent to these indices. Direction in r symmetry space in which this thing is pointing depends on where I insert the operator. This is strange. But it's also very nice, because my claim is now that this correlation function is actually meromorphic. In other words, z bar dependence completely drops out. So if I take any z bar derivative from 1 to n of this object, then I basically get 0. Of course, I get 0 up to contact points. The z bar derivative of 1 over z is a delta function, all of that stuff. But if these guys are not inserted at coincident points, then this is at proper 0. Yes. The bar on ui is a mistake. Yes, thank you. So the z bar dependence of this thing just completely drops out. There's no more z bar dependence in this thing. So that's my claim. And this is nice, because now I have a function that is just meromorphic in all of the other coordinates. And a meromorphic function, I can specify by just completely specifying its position, its singularities, and the location and severity of the singularities. For example, we have a single variable z, and you know that this function has a simple pole at z equals 1, a double pole with given residue, a double pole at some other point, and then a triple pole somewhere else. And you specify all the residues. And then you also know that it falls off at infinity. And then you've completely specified the correlation function, the function in that case. And in the same way, these correlation functions, the chiral correlation functions, are completely specified by just giving you the short distance singularities of all the short distance singularities, so all the different ways in which I can put two operators together. And if I just give you the short distance singularities, I know the full correlation function. So you see where you see hopefully this already gives you a hint of the power of this structure, where all now I need to do is look at the short distance singularities of this correlation function, which often I know something about because I can use the operator product expansion. And I just need a few terms in this operator product expansion here and for all the other short distance singularities. And then in principle, if you just give me a few terms, I can decide what the entire sort of twisted correlation function is, so this contracted correlation function is. Of course, in a general, if you would not do this kind of twist, you wouldn't have the meromorphicity. If I just give you the short distance singularity of a general function, it's in no way clear, in no sense clear, that I can determine the entire correlation function. So it's really meromorphicity that helps you in deciding the correlation function, in extracting sort of what happens far away from contact points just from the short distance singularities. Which other two directions? Yeah, we restrict it to the plane. We have to do that. I'll show you why later. So the other two directions, things break down. There's actually a good general comment. So your question was, what about the other two directions, like the x1 and x2 directions? Because of this homological argument that I'm about to present, the moment you try to move one of these points outside of the plane, all the other points, it will no longer be meromorphic, not in just that position. But it won't be meromorphic in any of the other positions, because you're sort of outside the homology. So you destroy the sureness, say, of this operator, you've destroyed everything. Yes? You claim that this correlation is meromorphic, so why do you assume? Yeah, I said that in words. Sorry, so the question is why there is no contact term. I wrote it's equal to 0 on the right-hand side. But yeah, that's just my laziness. So OK, I'll fix it, because there was a question. So 0 is just contact terms. But this, I mean, this is just in the same way that the z bar of 1 over z is a delta function up to some prefectors. So if I have a 1 over z singularity here, and I act with this bar derivative, I get some delta function. So those are the contact terms you get here on the right-hand side. So to make this abstract thing a little bit more concrete, let me consider a example. But I'll consider a trivial example for now, because it's the easiest. But I hope it's instructive enough to see what's going on. So in the free hyper-multiplot theory, which is a B hat 1 half representation, I have two sure operators to begin with, which are q and q tilde, which have scaling dimension 1, r equals 1 half, and r equals j1 equals j2 equals 0. So you can check that this guy is sure. So let's see how this meromorphicity comes about then. And the very simplest correlation function I can consider is, of course, the two-point function of this guy. Let me already insert this on the plane. So I'll take qi of zz bar, qj of ww bar. This is 1 over z minus w squared. And I have to contract these r symmetry indices. So this is epsilon ij. So I know it is two-point function because it's just a free theory. This is the Fourier transform of 1 over k squared. And that's, of course, the propagator that you know for a free massless particle. So the Fourier transform is basically 1 over x squared up to some normalization. And I picked my normalization such that this is true. Oh, actually, it's q tilde. So if you see how I defined the q and the q tilde before, you'll find that this is the correct two-point function. So how can this guy be meromorphic? Well, it's not so hard. I'll contract like so. I'll take ui of z bar, uj of w bar on both sides, ui z bar, uj w bar. The epsilon ij will give me u1 times u2, u1 of z bar and u2 of w bar. So that gives me w bar. And the other one gives me a z bar. So this is basically w bar minus z bar over z minus w squared. And this is minus 1. So it's chiral. The z bar dependence and the w bar dependence has completely dropped out. So this is your very simplest example of this kind of chirality in a period. So the free hypermultiple has other sure operators. For example, I can take q's. I can take a normal-ordered product involving any of the following elements. I can take q's. I can take q tilde's. I can take holomorphic derivatives and say more q's and more q tilde's and more holomorphic derivative and so on. And all of these, oh, well, I don't have to give an insertion point. And all of these operators are, as it happens, also sure. You see, if I normal-order two sure operators, I just add the quantum numbers in this case. So I get another sure operator. And a holomorphic derivative has j1 equals a half, j2 equals a half, and delta equals 1. So it also is allowed. Taking a holomorphic derivative transforms a sure operator into a sure operator. So this is the same as dz, actually, in the coordinate system in which, in the conventions in which that I'm using. So you can. Do you define room-ordering by constructing propagator? In this case, it doesn't matter. But yes, in the free hyper, let's say yes, we will discuss normal-ordering in the Kyra algebra sense later on in the lectures. But for now, normal-ordering can just be subtracting the singular bits. Here, normal-ordering is what you find close. What do we know here? Yeah, four-dimensional. But it's a free theory. So was there another question? They are called sure operators. That was the question. Because they contribute to a particular limit of the super conformal index, which is called the sure limit. And this is where sure polynomials appear. So it's a bit of a historical detour that allowed us to call them sure operators. So let me say some words for those people who know Kyra algebras. And at the same time, for those who don't know Kyra algebras, I'll try to explain what the words mean very briefly. Just in the case of the free hyper. So let me denote by q of z. Basically, the operator that I get, qi of z per. In the Kyra algebra, that means in the twisted correlators of the sure operators. In other words, in the kind of correlators that I wrote there. So in these kind of correlators, I can basically define this discontracted thing as an elementary field. Let me call it q of z. And similarly, of course, I can define q tilde of z in the same way. Then the fancy words are that we have a Kyra algebra. So a Kyra algebra for us at the level at which we are working is basically a set of these meromorphic operators whose correlation functions are completely meromorphic. So that's what we have here. A set of meromorphic correlation functions. It is generated by q of z and q tilde of z. And what I mean by generated is that in parentheses that all the operators in the Kyra algebra are either q, q tilde, or normal ordered versions. No more ordered products. Like for example, I can define an operator. Let me say q del q tilde. But in my definition, that is just the limit. It's just a normal order. This is just the limit as w goes to z of q of w del q tilde of z minus a singular. So that's what I mean by generated. It doesn't mean that this is the only operator in this set of correlation functions. But it means that all the other operators are just normal ordered products of the generators. And so here, there's a small sleight of hands because now I'm defining normal ordering in the Kyra algebra sense. Whereas there, it was basically four dimensional normal ordering. But for the free hyper, you can show that it's the same thing. And in a general quantum field theory, I'll just use the Kyra algebra normal ordering for now. Yeah, I think it's a proper Kyra algebra in the mathematical sense. This is more efficient use of where it's specifically formed by the math people. Yes? So I'm asking, is it exactly the same? I believe so, yes. I believe it's the same as. When I say Kyra algebra, it's what math people mean for the Kyra algebra. Notice, however, that it doesn't have a proper Hilbert space. One interesting, it has a good vector space of states. But here, for example, you see that the norm, the inner product here is not positive definite. But you can typically define that it's more of the right and vertex algebras. That's correct. I just wanted to make that qualification. It's a vertex operator algebra in the proper sense. Yeah. But that's another question. If you start to define formal and conformal laws, et cetera, what does it mean at the level of proof theory? If you try to find formalism, that's what you know very well. Is it one-to-one course or not? I think before you say conformal blocks, maybe you want to ask about the representation theory of this algebra. You want to ask about modules of this algebra, right? And that's a wholly different subject. Because at the moment, we seem to just have the Kyra algebra. We don't have any non-trivial modules. We just have the identity or the vacuum representation, basically. I'm asking you, I had this job question. What do you expect the correspondents to know? Yes, but it depends on which object you ask about. At the level of correlation function, it's obvious that the Kyra algebra correlation functions are just twisted versions of 40 correlation functions. At the level of OPE, there is some kind of sense in which the Kyra algebra OPE is some twisted version of the four-dimensional OPE. That's all completely fine. At the level of characters, you may want to ask about that, you can show that the graded character of the Kyra algebra is equal to a certain limit of the superconforma index in four-dimension. At the level of representations of the Kyra algebra non-trivial ones, so operators that are not Kyral but transform in representations of the algebra, it's much more subtle. Because we don't seem to have a place for them here. And in fact, the only way we can see that these representation non-trivial modules appear is by the insertion of supersymmetric defects at the origin. And those, I believe, to give rise to modules of the Kyra algebra. But I definitely don't have time to talk about the defects story. And so I was not finished with this sentence, so let me finish the sentence here. So a Kyra algebra is a set of meromorphic correlation function, a bunch of operators with meromorphic correlators. It has two generators in this case. And everything else I just get by this kind of normal ordering. And to fully specify the correlation functions, as I said before, all I need to give you is the short distance singularities of the generators. So let me give you those. Singular OPEs operate the product expansion. So qz, q tilde w we've already seen is minus 1 over z minus w. And let me use this tilde symbol, which means up to regular terms. So up to order z minus w to the 0. qz, qw is 0, and q tilde z, q tilde w is 0. So when I write 0, it's again up to regular terms. So it's not really 0, it's just regular. There's no short distance singularity. And so this suffices to compute in principle all the correlation functions of q and q tilde. Any number of q's, any number of q tilde's, because you know all the short distance singularity. And then by doing this normal ordering procedure, so you bring two operators close together, you get some singular terms, you subtract them all, and then whatever is left, you define to be the normal ordered product. In this way, you can generate all the normal orders, the correlation functions of the normal ordered products of these operators. And so this gives you, in principle, these three equations are now sufficient for you to compute the entire chiralgebra. So these fancy words here, chiralgebra, generators, and singular opiates are important words. And I hope that in a simple example, at least you sort of understand what they mean now. So in fact, this chiralgebra has a name. It's a chiralgebra of symplectic bosons with hq equals hq tilde of dimension 1 half in the chiralgebra. In beta, don't care much. Yeah. So for those who know chiralgebra, so a little bit, this is already a sign that chiralgebras are not unitary. Even though the four-dimensional theory upstairs was perfectly unitary, but the chiralgebra is not unitary. OK. Any questions? Why it's not unitary? Because it didn't have to be. If a correlation function, a matrix, let's say, one particular consequence, a necessary condition for unitarity, is that, for example, the matrix of two-point functions of primary operators is positive definite, because it's some kind of norm. And you can perfectly write some positive definite matrix, but then if you take care of the r-symmetry indices and you group it into r-symmetry representations and you contract it with the use like I did, then suddenly you introduce some minus signs. You get some sign flips, because you contract the use with some epsilon, and suddenly you get negative norm states in the chiralgebra. So that's why it doesn't have to be unitary. I mean, we're working in some complexified algebra here, right? So OK. I say those words, but let me just maybe come back to that later, because then it's much more easy to see why we're complexified things. So this was, in a nutshell, how you generate the chiralgebra. We did a simple chiralgebra. But of course, it's much more interesting to ask, what is the chiralgebra of specific theories? What other information can we learn from these chiralgebras? What other kind of general structures can we learn about chiralgebras in 40 theories and so on and so on? So that's going to be some of that I'll discuss later on. But first of all, I have to explain to you why this is the case, why you would get this chiral structure. So that was the pedestrian approach, and now we'll discuss chiralalgebras, the sort of proper approach. Well, the really proper approach would be in the theory section of our paper, but I'll do it fairly quickly so that I have some time for other things. So the reason for this chirality is that we work in the cohomology of a supercharge that we are traditionally denoting as funny q. I guess in our paper, it's more like funny q, which is defined to be q minus 1 plus s tilde 2 minus dot. And in our conventions, we work in radio quantization. We have that complex conjugate, or it's a mission conjugate. It's this one. So this funny supercharge has the following funny properties. First of all, it squares to 0. So we can indeed consider its cohomology without any issues. The anti-commutator of funny q with its complex conjugate gives me the dilatation operator to the rents rotation and 2 times r, s e to r symmetry, one of the three s e to r symmetry generators. And this is maybe an object or quantity on the right hand side that you've seen before because you recognized from before. Because remember, I said sure operators have delta equals j1 plus j2 plus 2r. So for sure operators, this thing here on the right hand side is 0. So this m plus plus, I haven't really given you all my conventions, but this m plus plus gives you j1. m plus dot plus dot gives you j2. This is, of course, 2r. And that's just the dilatation operator. Gives you the conformal weight. So we deduce from the fact that this thing is the anti-commutator of q with q dagger that necessarily for all states in a theory, in a unitary theory, this thing is on the right hand side is positive for all states, for all the operators or states, and that an operator is sure even only if it is killed by the q's. So here, you already see that this abstract sure condition has some useful, has some kind of natural ish interpretation from the viewpoint of the representation theory of the super conformal algebra. What else? Well, that's just one, two properties. The third one is that, OK, now, so this operator, we're going to work in the homology of this guy. If I insert a sure operator at the origin, it's q closed. So that's good. Now let's start moving it. Let's try to move it in the one direction. I find that it's non-zero. So if I just move it out in the one direction, I find that it's no longer q closed, so because the commutator of funny q with p1 is non-zero. The commutator with p2 is also non-zero. So I cannot move in those directions, but the commutator with pz is zero. So we can move this operator, which was sure we can move it without punishment in the z direction. Then it will still be q closed. So it follows that q acting on e to the z pz, my sure operator at the origin is still zero. So what about the z bar direction? If I just try to move it naively in the z bar direction, I find the same as trying to move it as when I tried to move it in the one and two direction, it's not q closed. But remember what we have. We have these sure operators with r symmetry indices contracted in some z bar-dependent way. So if I move an operator in the z bar direction, I also rotate it a little bit implicitly. Well, I rotate it a little bit by virtue of this use in r symmetry space. So the thing I need to look at happens to be the translation in the z bar direction. Or let me make that explicit. So let me look at the z bar of ui of z bar, oi of z z bar. So let's look at this. Well, this is just ui of z bar, oi. So this is just a z bar derivative of the operator, oi z z bar. But then, of course, the z bar derivative can also hit the u. And then I just get, in my case, it's o2 of z z bar. So what is this? Well, you can write this as a z bar derivative plus an r symmetry rotation, r minus, oi of z. And here we're working the complexified algebra. So this r minus is just a matrix with all, with 2 by 2 matrix with three 0s and one half diagonal element equal to 1. So that's why it gives you this. That's why it gives you precisely o2 back. And so the operator that we have to look at to see if we stay in the cosmology is not pz bar, but it's pz bar plus r minus. And this guy happens to be q closed. But furthermore, it is actually also q exact. So it's the q anti-commutator of some fermionic guy that is not particularly important for us. And so this is why the z bar dependence drops out. Because, oh, yeah, sorry, I want to keep this. So I need to lower one of the boards. Some blackboard juggling, sorry. So I've said that the z bar derivative of these twisted, translated, sure operators is q exact. And so that would give me the corality of not translated, but twisted, translated, true operators. So in the q-carnology, the z bar dependence drops out. Yeah, I think this is all I want to say. Is this clear? Is this, do you believe all this fact? Is it then clear that corality follows? I've stated them as facts. There are a lot of things to check that not just the anti-commutator, but also the fact before that they wrote that the sure operators are always highest weights of all the SU2s and all these things. So I stated a lot of things without fact. You can find a lot more details in our paper. But I hope that the logic is sort of clear. We under-colonology of the supercharge. We insert something that's q closed. We move it around. And in the z direction, we're still q closed. And z bar direction, we're q exactly. So what's the second part of the question? Oh, you're asking why 1, 2 direction, it's non-zero, but the third and fourth direction, it's special. Well, that's because I took particular Lorentz synthesis under my definition of funny q. There's a minus there and a minus dot. And that specifies a particular two plane. I can rotate the whole structure. And then the minus there and the minus dot there will be some other direction. In spinner space, then it will not be the x3, 4 plane, but some other two plane. But none of that is going to affect anything, of course. So this is just a choice of conventions where we get the x3, x4 plane. So there's no other Kyra algebra that you can find. In fact, we did a more comprehensive analysis also. This is sort of the only Kyra algebra structure up to conjugation that you can find in the SU2, 2 slash 2 algebra. So we were busy compiling a list of properties of q's. We had property 1, q squared equals 0, property 2 is anti-commutator, property 3, holomorphic translations, and 4, anti-holomorphic, twisted translations. Then property 5 is going to be that there's, in fact, it's not just the translations that are q closed in the holomorphic directions. There's an entire SL2z as a subalgebra of the superconforma algebra is q closed. So this is an SL2 generated by translations. So this one I already showed you that it's q closed. But it's also true that the corresponding L1 and L0 are q closed where 2L0 is d plus n plus plus plus n plus dot. And this in the Kyra algebra plays the role of my dilatation operator. So in particular, the eigenvalue of 2L0 of L0 gives me the dilatation weight of the twisted translated operator. So from this, I read off that ui z bar oi zz bar in the Kyra algebra has weight h equals delta plus j1 plus j2 over 2. So let's check for the free boson. We had symplectic bosons of weight 1 half. And indeed, j1 and j2 is 0 for them delta equals 1. So h equals half. So that's one example. And there is one more thing that I want to say, which is that there is a twisted SL2 hat z bar where the hat means twisted, which is q exact. This guy is generated by bz bar plus r minus l hat 1 equals kz bar minus r plus, and l hat 0 is minus 2r. So this is q exact. But I already told you up there that the thing on the right-hand side is q exact. So the thing on the right-hand side, the q dagger anti-commutator is, in fact, the l hat 0 of some twisted SL2 hat algebra, which is a subalgebra of SL2, 2,2. And in fact, it's now interesting to see if you can do these twists more generally. And then you realize that the SL2 hat is just a subalgebra, not of the full SL2, 2,2, but just of an SL2, 2 subalgebra of the full four-dimensional super conformal algebra. And so this means this is the n equals to 4 super conformal algebra in two dimensions. So this implies that kind of algebras, well, I'm not sure if kind of algebras exist at this level, but at this stage, but I can say the same construction should work for any super conformal algebra that has an SL2 slash 2 subalgebra. So you don't need this full four-dimensional super conformal algebra. The moment you have an SL2 slash 2, you can define this twisted guy. You can twist the correlation functions and you get something where some z bar dependence drops out. And then you can go over the list of super conformal algebras, and then you'll find that this should work in 0 comma 4 theories in d equals 2 or 2 comma 0 theories, which is maximal in d equals 6. And so this is just a consequence of the structure of the algebra. The representations and the sort of analysis of the Schuer operators, we did already in d equals 6. And we have a nice paper about the Kara algebras for the maximally supersymmetric theories in six dimensions. For 0 comma 4 theories, we have some work in progress, but we don't have a paper on these Kara algebras yet. So these were the sort of 1, 2, 3, 4, 5, 6, maybe, properties that I wanted to highlight of this operator. So did you say that for 4 d equal 2, this is the sort of unique choice for the Kara algebra construction? Yes. I want to say that, I mean, how general did you install that you can take all the linear combination of supercharges? Or did it start from a short moment? No, so the question is how unique, how do you know that this was really the unique choice of a Kara algebra in 4 d equals 2 theories? So of course, yeah, you have to make some choices because a similar analysis of sort of protected correlation functions should give you all the Kara rings, for example, or at least the ones that have lead to non-zero correlation functions. So I think I don't quite remember the details, but I think if you ask for an SL2 slash SL2, then you're basically, and one of them you can twist, then you're basically done. Then you basically find that there's a unique choice. In fact, there's a unique choice, but within that unique choice, there's sort of two funny cues that you can decide, but they give rise to the same caramology. So it's not completely obvious that you buy a unit SL2 slash stills. Yes, no, I don't know. You can try to compile a list of twists for SL2 slash 2, and I don't think we can claim to have done that. So yeah, maybe I should point out the funny property of our funny cue that it has an S in it. And what people of the caramology that people have often considered in supersymmetric theories was the caramology of supercharges, so the caramology of cues. Cues commute retranslation. So in that caramology, you generally find things that are position-independent. You find correlation functions that are just independent of positions. In our case, we have an S, a conformal supercharge there in the funny cue in the operator whose caramology we consider. And therefore, this guy does not commute with translations, and that's what sort of gives us our more interesting structure where you're in the caramology, but there's still some position dependence left. Are there any role in three dimensions? In three dimensions, there is a paper by Chris and Wolfgang and Leonardo about how a similar, and also Silvio Pufl and Frans, about how a similar construction works if you put operators on a line, and then you get something position-independent. So there's no caram algebra anymore. I still have a few minutes left. So I can briefly explain a few of the Schur operators. So we can also make contact with this long story of the representation theory that I considered before, that I discussed before. So for example, from the B hat 1 half multiplets, we already saw that Q and Q tilde are Schur. So I'm not going to repeat that story again. I already pointed out to you that from the B hat 1, the super conformal primary mu Aij is Schur, because it has delta equals 2, r equals 1, and j1 equals k2 equals r equals 0. Now we have an expression up there for the weight of whatever the twisted translated operator is going to have in the caram algebra, and the weight of this guy is going to be 1. So we get an operator that already knows as jA of z. Remember this A is an adjoint flavor index. So it has nothing to do with supersymmetry. It's just a flavor thing, which is ui uj mu Aij in the caram algebra. Or maybe I should change the subscript, because now we know that in the caram algebra really means in the Q caramology. So this is a dimension 1 operator that transforms in the adjoints of some flavor symmetry. And you shoot maybe some of you recognize that this is likely to be the generator of a Cosmoody symmetry. But for that to happen, it would have to have a singular OPE, which looks like this, plus regular terms. As it happens, this is true. So you can take the four-dimensional OPE of this guy, which by supersymmetry is related to the four-dimensional OPE of the flavor current with itself. That OPE you know, the OPE of a flavor current with itself in four dimensions has this kind of structure, where there is some identity operator and then there is the thing itself with the flavor symmetry current itself with the same structure constant. Then by supersymmetry, you know this OPE, then you twist it and up to Q exact terms you find precisely this. And there is a connection. So in 4D, I may not write the full OPE because it's a bit messy, but let me just say that the two-point function of currents is something like do I have the corrections? Maybe up to some factors of pi, but let me swipe that under the rug, x mu, x mu over x to the 8. So in four dimensions, the two-point function of currents is characterized by this number here, which is the level, or the flavor, well, we call it the flavor central charge. And it happens to be independent of exactly marginal coupling. It's just a constant that is associated to any 40 n equals 2 theory with flavor. It's some number of flavor central charge. And of course, this is a two-point function, so it must have positive norm, and the four-dimensional flavor central charge is positive. And as it happens after you take this two-point function of currents by supersymmetry, you're related to the two-point function of the super conformal primary, you do the twists, you find some extra factors, and you find that k2D is minus k4D over 2. And by looking at the three-point functions of currents in four dimensions, you can similarly deduce this term in the OP. So the funny part here is that if you have flavor symmetry in four dimensions, then you have a flavor symmetry current. Then by supersymmetry, you have this guy. It's also called the moment map. It's the super conformal primary of the flavor symmetry multiplet. Then that guy can twist, and this guy gives me the twisted super conformal primary of the flavor symmetry multiplet, and gives me a affine-Catsmoody algebra current in the Kyra algebra. No. Yes. Well, there's no uniterity here. It's negative already. So there's no, I don't know of any constraints that I would want to impose on k2D. So in June, I think he's excited to discuss whether symmetry is not associated with the compact. To any? Contact. Yeah, the group is, well, I'm not sure about the group. There's a flavor symmetry algebra, which is just this guy transforms in the adjoint of. It's usually associated to a compact group that I agree with. But this flavor central charge here is not necessarily an integer either. And so this guy can also be random. And this is just some affine-Catsmoody extension of the group. In fact, I can ask you a question. Just so, for example, this katsmoody will cut things like the water can start to get us over. Does it have sort of hesitation? Yes, excellent question. So the question is whether there is a fear. But there's some kind of, well, the question is if the Sugavara construction, the associated Verozora algebra, has any four dimensional interpretation. The answer will be that there is a canonical Verozora subalgebra of the Kyra algebra, which may or may not be the Sugavara one. And that Verozora subalgebra comes from the n equals 2 stress tensor. Let me do that now. OK, so the other question will then have to wait a little bit. I wanted to briefly give an answer to that question. But since I have five minutes left, I should make a choice and let me choose to answer the Verozora question because it's very important. There we go. So another SURE operator was the stress tensor multiplet. It also contributed a SURE operator, 0, 0, 0. We found that the SU2 are symmetry currents. The delta equals 3, r equals 1, j1 equals j2 equals 1, half is SURE. And now we can also decide on the computed dimension. It's delta plus j1 plus j2 over 2, which, if I'm not mistaken, is equal to 2. So there you go. We have, not from the four-dimensional stress tensor, but from the four-dimensional stress tensor multiplet, we have a dimension 2 operator in the Kyra algebra, which is completely neutral under all of the flavor symmetries. So let's define t of z to be ui, uj, jij plus plus dot. And then we get an OP where t of z, t of 0, is c over 2 z to the fourth plus 2t over z squared plus delta t over z, where this two-dimensional central charge is equal to minus 12 times the four-dimensional central charge. And the four-dimensional central charge is defined by the two-point function of the stress tensor multiplet as, well, basically just the norm of the stress tensor multiplet. So I'm not going to write out all the ways in which you have to construct this indices. There's a unique way to contract all the indices here, consistent with conformal invariance. And in some conventions, the overall coefficient is then c40. So let me state my conventions for a free hyper. The four-dimensional central charge is equal to minus 12th. Plus 12th, sorry. Of course, the four-dimensional central charge is positive. So the two-dimensional central charge is negative. So there is your VRZora algebra. There is a canonical VRZora algebra. This, of course, should recognize this as generating a VRZora algebra. And this VRZora algebra arises canonically from the stress tensor in four-dimensions. So if your four-dimensional theory has a stress tensor, and basically every local four-dimensional theory has one, then the kyroalgebra has a VRZora stress tensor. But it has this VRZora subalgebra. It takes a little bit of work now to show that this is really the stress tensor of the kyroalgebra in the sense that its OPE bit primaries is also of the correct form. We believe that to be the case, but we don't have complete evidence of this because it would amount to analyzing upstairs in the four-dimensional theory, the OPE of the SC2R symmetry current with all of the Schuer operators, and see that certain terms that are forbidden, if this were the proper stress tensor would not appear. And that was a bit of work. And it seems a bit, we weren't able to show this conclusively. But we strongly believe that this is really, I can really say that this is the stress tensor of the kyroalgebra. Is the damas system as many parts of it as VRZora? Yes. So when I say that this is the stress tensor, I mean that the OPE with the particular primaries of the form so that it generates conformal transformations. So in particular, there's no, I think, triple pole in this thing. I don't know. There's no. No. I think the correlation between the damas is fixed, but I can write the difference, stress tensor and probably the different weights. There is no unique action of the RSA1-litre damas system. Yes. But then I want to say that there is that. What you are saying that there is one particular RSA1 which comes from 4DZ? Yeah, well, yeah, in some sense. There's this 4D. There's those 4D generators, right? They descend to SL2Z. And so is this SL2Z, when it acts on the local operators in the Kyra algebra generated by this transistor or not? That isn't a question. And our claim is that it is, but it's not trivial. And indeed, there is, for example, for the Katsmoody algebra here, you can define also a dimension 2 operator. So in my last 30 seconds, let's look at JA of Z, JA of 0. Sorry, now let's look at JAJA normal ordered in the Kyra algebra. This is also a dimension 2 operator. And it's also neutral on the flavor symmetry because I contracted the indices. So what is the difference between this guy and this guy? And as it happens, sometimes they are identical, so their differences are null states. Sometimes this guy is an admixture of the stress tensor and some genuinely new operator, which came from something different upstairs. So it's sometimes hard to decide what this guy corresponds to in the four-dimensional theory. And after Zohar's lecture in the end of my lecture this afternoon, I'll consider this example in more detail. We'll try to figure out where this can come from in four dimensions. And then we'll see that naturally it's often some admixture of the stress tensor and another multiplet. And the fact that the other guy must have positive norm in four dimensions will give us constraints on these four-dimensional central charges and four-dimensional flavor central charge, which just follow from unitarity in this Kyra algebra construction. So in this way, I'll show that the Kyra algebra can give you some very general consistency conditions on these numbers, like the flavor central charge and the central charge, which just follow from this entire construction. So I think I should stop here. Thank you.