 OK, tako bitten, to je po verbenju, da v bistvu pomečenih tudi in zelo, zelo, ni ne bo SmartCraft, masli možemo pričvega kimi, zelo, arrike, tudi zelo, naredišno idemo nekih spojftakov, kot tako nekaj vsočim, ni odmah je odpovedal, kaj kaj bi se zelo, pa tudi se našel kako mi je popril, preseperatora algev we get is interesting by enumerating some examples, so I told you that the boring examples of freefield theories where we get some symplatic bosons obeying this kind of OP where in some representation representation R of some group G, those would be the half hyperths, and then we are going to have a b c c gos system, these are adjoint indices of the group, they correspond to the vector, and so you may think then to find the vertex operator algebra that corresponds to a Lagrangian n equal to 2 theory with group, gauge group G and R matter, I mean half hypermultimate representation R, you just need to take the tensor product of these two free vertex operator algebra and construct composite operator which are singlets under the action of G. That's of course a reasonable first guess, but upon further thinking it's clear that cannot be the right answer. So simply restricting to gauge singlets will generate some voa, let me call it zero, that clearly corresponds to the voa which is obtained just from starting from the free Lagrangian. If I switch off the gauge coupling, remember there is a gauge coupling tau and I switch off the gauge coupling, I literally will just find the tensor product of the free hypermultiplet and the free vector multiplet, and all that I have to do is to impose the Gauss law constraint which amounts to restricting to gauge singlets. So this is the correct answer in the free limit, but as I emphasize multiple times, the free limit corresponds to one of these singular casps, and it's boring because we have this just tensor product of free theories, we want to do something more interesting, and so we really want to describe the vertex operator algebra in the middle of this conformal manifold, and there are a first set of good news is that all you need to do is to consider the infinitesimal one loop correction to this result, and what that is going to do is to lift off a bunch of the operators that you have in this free approximation to the vertex operator algebra, by the mechanism that I quickly reviewed yesterday, several of these sure operators multiplet will combine together to form a long multiplet, which is then able to acquire an anomalous dimension as a move away from this free field point, and then away from the street zero coupling point, I should forget about those operators because they are not protected. And so the moment I move away in infinitesimal distance away, so as I turn on the coupling, or coupling on zero, there will be a smaller VOA, strictly smaller VOA, which is the one that I really truly want to identify with my interacting theory in the middle of the conformal manifold, and one can argue that this is sufficient to look at the infinitesimal tronkation of the operators, and moreover the other good news is that then any continuous deformation inside the conformal manifold, which is epsilon away from these casps, the vertex operator algebra is invariant under any such continuous deformation. In other terms, the state of space as well as all the OP coefficients are independent of exactly marginal couplings, the moment you are careful to take into account this continuous jump that happens at each of the casps. So in that sense, the vertex operator algebra associated with the 40-th theory captures, if I have a Lagrangian theory with a marginal coupling, this captures protected information, which is truly invariant under any deformations of the theory, including marginal couplings. And there is a very elegant, concise prescription to find this restriction of the naive VOA associated with the freefield theory, to the one interacting theory, which consists in the following. First of all, we are going to take this naive tensor product of VOAs, and the correct VOA will be obtained by, let me put it this way. We are going to compute a homology, a homological prescription, where there will be some differential, which I'm going to call QBRST. I hope the story is clear. Now I'm purely in the two-dimensional world, so I'm not talking anymore about the square Q that I had earlier, which is a four-dimensional object. Now I'm purely in the two-dimensional world, and I assert that it's possible to take into account this continuous jump that happens away from the cusp by imposing that the states that remain are a subset of this naive tensor product, which is the homology classes of this operator, vcGhostSystemA. So let me explain a notation. Well, first of all, CA is clear. It's the ghost that corresponds to the four-dimensional gauge, you know, with a joint index A. And then by assumption, we know that the matter field transform in some representation are of the group that I'm gauging. And so I told you earlier that very generally, any time that you have a continuous global symmetry that induces that statement is sent to the fact that there is a affine, this is an affine cut smoothing current, which is purely made of Q and Q tilde. And in fact, given that Q and Q tilde have dimension one-half, you can instantly write it schematically. This would take the form Q, Q tilde, again, with the matrix of the representation of G in the representation R. And this is just the ghost current, which is made of BC. You can also write it down. It corresponds to the, again, to the same continuous transformation G, but now acting in the adjojn representation on the BC ghost themselves. And it turns out that this object is nilpotent precisely when the level of the matter current equals to minus two, the dual-coxet number of G. And if you translate in two-dimensional notation what I told you in the first lecture, this is precisely the condition for the vanishing of the four-dimensional beta function. So, if this gauging procedure is allowed in four-dimensional, meaning that it preserves conformality, then I can find nilpotent BRST operator purely written in terms of these two-dimensional data. And I'm claiming that pass to the homology of these BRST operator removes all the sure operator that recombine into long operators and leaves me with a truly protected operator in the middle of the conformal manifold. OK, so you will understand that then this gives a prescription, a very concrete prescription to calculate the VOA in all Lagrangian examples. And in fact, it's more general than that. If you have one of these mysterious building block, generalized matter theory that I talked about in the first lecture, and you know the VOA that corresponds to it, let me assume that I know it, and then I do a gauging procedure such that I gauge some subgroup of the global symmetry in such a way that this condition is obeyed, you see this condition is phrased in a purely universal fashion, that does not depend on having a Lagrangian description, then the exact same procedure will work, where now I will replace the free VOA for the symplatic bosons with this strongly coupled building block, which is the VOA of the generalized matter super conformal filter. OK, so this gives a powerful way to construct VOAs in many examples, although admittedly computing this homology can be a little complicated, and so what we often do, we come up with a, we do a level by level calculation starting from h equals small and then going up, we convince ourselves that we have a plausible guess for the VOA, and then we subject it to a variety of checks, and then formulate a conjecture, and then mathematicians prove it. That's a typical workflow in this kind of examples. OK, so let me now give you a few, let me give you a couple of simple Lagrangian examples, and then I will make a list of VOAs for several more non Lagrangian examples, where this kind of approach is not available, we cannot, this is in some sense, it's a preferably constructive approach. We know the free theory, we compute this very psychomology, maybe hard to compute it, but it's clear that there is an answer, that often you can guess. In the non Lagrangian case is more of a, again, a guesswork based on plausible assumptions. It's often the case that you know some data of this mysterious Lagrangian, you know, central charges, you know the global symmetry, you know a little bit of information about the Higgs branch, and put this information together, it's often the case that there's one compelling guess for the associative VOA, which you can then send subject to a variety of checks. OK, so let's do the simplest example in the Lagrangian word, which is this, we take g equal s u 2 and n f equal to 4, as I indicated yesterday, what you really should do is to think in terms of half hypermultiplets, which are in the so 8, so the naive flavor symmetry would be u 4, but it truly enhances to s o 8, and then the index a is the, this is the flavor, and this is the, this is the gauge group, and this is the four dimensional field, but without further ado, we have learned that that descents to a, to a simplect boson, and so the matter part of the theory gives me sympletic bosons that obey this OP, and then, and then I have B and C ghosts in the adjoint of s u 2. OK, so no, so you can do this exercise now, you can compute this comology, this exercise is actually not so easy, and what you discover is a rather elegant result, the result in voa, the voa of this s u 2 super q c d is nothing but the so 8 current algebra and level minus 2. I mean, of course, the level you could have computed instantly by using this formula, we know the level of the four dimensional flavor current, so we can just compute the level, and it's the other statement that I want to make, so you remember that the flavor currents are in the B-hat 1 multiplet, and we also know very well, because in Lagrangian theory we can, we can study this in without any problem, we also have detailed knowledge of the Higgs branch, in fact, I left that as an hazard size yesterday, did anybody do it? I will tell you the answer later, and I explained yesterday how one can describe on very general grounds, one can describe the Higgs branch of a four dimensional super conformal field theory as a hyper killer cone, it has to be a cone because as an action of the dilation symmetry, and for our purposes, I also explained that the best description is useful for us is to think of it as a simplatic, as a holomorphic, I explained this in the discussion section, perhaps not all of you were there, as a holomorphic, simplatic variety, which is a cone, almost simplatic cone, and then we can look at the, so this is hem Higgs, we can look at the halgibrolomorphic function of the Higgs branch, and we identified this with the Higgs chiral ring, which is generated by the Schur operator equal to 2r, so which is the ring of Schur operator equal to 2r, and I also explained that we can describe this ring in terms of a finite set of generator obe in some relations. So the generators of the Higgs branch are then in the language, in the four dimensional language, are then the highest weight of this behead, behead r multiplets, such that by taking products of them, I generate the entire set of operator obe this condition. Is this clear? So we have detailed knowledge, not just in this simple example, in many examples of the Higgs branch, as a variety, so we know the generator, we know the relations, we know the simplatic structure. In Lagrangian example, like this one, we can compute them by the hyperkeler quotient, and in non-Lagrangian examples knowledge of this Higgs branch data comes from more indirect consideration, maybe you can embed them in string theory, maybe you can get them by various duality, so we have, anyway, the Higgs branch data is very robust data, so it's something that we can often get either very directly by doing the hyperkeler quotient or indirectly by some other considerations. And the structural fact that I want to emphasize is the following little theorem, as you can prove using the construction I've given you so far, that generators of the Higgs branch, first of all they are Schur operators, so they must appear in the chiral algebra, but they are necessarily generators of the chiral algebra. Okay, so let me emphasize this little fact, which is important. The generators, the Higgs branch chiral ring, which is also the same thing as homomorphic function of the Higgs branch, under our map kind become what are called in the mathematical literature strong generators of the VOA, and let me, you have always known all your life what this generator is, but let me explain it to you, you know, all knowledge is reminiscence, as Plato said. So what is a generator, is the thing, if you, a VOA is uniquely characterized, if you find the generated VOA, which is the very large, which is pretty much any VOA you've ever seen in your life, is characterized by listing a finite set of operators, such that if you take normal order product of homomorphic derivative of them, you generate the entire set of operators. It's a bit of a mouthful, but it's, it's a triviality. Examples, Virazoro has a single generator, known as T of Z. All operators of the theory are obviously obtained by taking arbitrarily many products of T sprinkled with homomorphic derivative, clearly enough. For a fine cut smoothie is the, the generators are just the current, and then it's the same story. You're now gonna take arbitrarily many products of homomorphic derivative of the currents, and that is the entire state space of the VOA. You may be familiar with W algebras. For example, the vanilla type of, the vanilla type of Wn algebras will have generators T, which we also can call W2, W3 all the way up to Wn, which will have homomorphic dimensions 2, 3, up to N. And those taking normal order products of homomorphic derivative of them, you get the full state space, et cetera. Okay, so generators of the Higgs branch current ring become generators of the VOA. And of course we know that at the very least, okay, so perhaps I want to keep the suspense a little more, but, okay, let me, let me not give, I don't have time for this kind of, this kind of, of theatrics. So let me just give you the answer for what the Higgs branch of this theory is. The Higgs branch of, of SU2 super QCD. Okay, so let me write and then I'll explain it. It's the same thing as the closure of the minimal important orbit for SO8. And, okay, so that is playing the notation, it would take me a little too much time, but let me explain this in a concrete way. So this means that I take some formal generators x, let's call them xm, where m is an adjoint index of SO8. And I take the symmetric algebra of, of this complexity phi, the algebra of SO8. And I impose a single quadratic, a set of quadratic relations. So this object is in the adjoint of SO8. What are these objects? These are nothing but the moment maps. These are the dimension to, these, these, these are the things that correspond to the objects with e equal to 2 and r equal to 1. Right, the objects I have indicated earlier as mu. These are the objects which are the highest weights of, sorry, equal, the highest weight of SU2 which are thrown in the adjoint of the flavored symmetry SO8. Those are the ones that correspond, those are, those are elements of the Higgs branch, and the claims that they generate the, the full Higgs branch. But there is a relation that happens in the quadratic order. And the relations, so if I take a product of two such objects, given that critical mu, they are bosonic objects, I'm going to find, I have to take the, the symmetric product of the adjoint of SO8 and symmetrize. And the adjoint of SO8 is the 28, so, sorry, is it the 28? Sorry, it's the 8 times 7 divided by 2 is 28. Okay, yes, we're good. So I, I need to symmetrize 28 times 28, and you can check that that gives you the singlet plus the 35 vector plus the 35 spinner plus the 35 conjugate spinner plus 300. That's just a fat. And so the quadratic, so let me call this whole set of operator, let me call it I2. With this definition, then I'm going to impose that x tensor x restricted to I2 vanishes. Okay, so this may not appear immediately intuitive, if you haven't seen it before, but it's a standard construction. Okay, it's a standard construction because it's extremely general. It generalizes to, to any lealgebra, and for any lealgebra you do the same thing, you take the adjoint of g, any simple lealgebra, the adjoint of g symmetrize, and you define this I2 ideal as the thing such that I2 plus the red sum of the representation whose dinking labels are twice the one of the adjoint gives me the symmetrize prior to the adjoint. You can do this very generally, and for SOA that's how it looks. Let's do another example. Let's take g to be SU2. Well, then the adjoint of SU2 is 3, what is 3 times 3? Symmetrized. The same thing is 9, and you will recall that 9 is equal to 1 plus 3 plus 5, right? The triplet is the anti-symmetrization, and, well, I hope I'm not saying anything. This is really elementary now. So the symmetric product of two triplets gives me the 1 plus the 5. The 5 is the object whose dinking labels are twice the dinking label of the adjoint. Clear enough? The dinking label of the adjoint is 1. The 5 is dinking label 2. In this case, this is what I would call I2. OK? And so this I2 can clearly define what any Lie algebra is. It's called the Joseph ideal. And if I take the symmetric algebra of the Lie algebra g and I impose that the I2 vanishes, I find the variety, which is the minimal delpotent orbit of the algebra, and has a very simple physical interpretation, this is the modular space of instantons of charge 1 for that symmetric group. The modular space of SOA instantons with charge 1 is described by this variety. And so this is something, in the case of super QCD, this is something that you can really check by hand by doing the hypercaler quotient. If you start with the description in terms of the Qs and you do the hypercaler quotient, you will find that this is the answer for the variety. OK. So where does this lead me? Well, I asserted, and it's easy provable, that generator of the Higgs branch kaj la ring are generator of the vertex operator algebra. And so I know for a fact that the vertex operator algebra at the very least must contain this SOA at a fine currents. And the nontrivial part of the statement I was making earlier, so I said earlier, so I know, I certainly know, that the full vertex operator algebra SU2 super QCD must at least contain SO8, and I can easily compute the level which is minus 2. And the nontrivial part of the statement is that it doesn't contain anything else. OK. So that comes from a complicated homological calculation. But if you didn't know, if all you knew is that this symmetry has favored symmetry SO8, a level minus 2 goes, well, what is the minimalistic guess I can make is this one? A minimalistic guess in this business often turn out to be correct. OK. Questions about this? Now, there is, sorry. Sorry, I have a question. Yes. So the fact that that vertex operator algebra saturates the central charge that you want is not enough to conclude that there is nothing else. OK. Great. You are anticipating what I was about to say. So you may have objected that for sure I know that there has to be at least an additional generator or an additional operator t. Remember, I had this whole story earlier. I explained that very generally we always have a t that descends from the artsymmetric current of the four-dimensional theory. And so there must be a t. But in this particular Kd, I simply identify t with the Sugavara stress tensor of the affine current algebra. The reason for it, and this is a nontrivial calculation you can do, you can compute the 2D central charge using the formula I gave you earlier. The 2D central charge uniquely fits in term of the C anomaly. And it also happens to be the case that this value is numerically equal to the Sugavara dimension, the Sugavara central charge is the dimension of g times k divided by k plus a check for in affine current symmetry level k. And it agrees. So we know that in this case we could try to get away without introducing a separate stress tensor. But it's still not a compelling argument, or at least not a complete argument, because if my VOA had been unitary, this would have been a good argument. You check saturation of Sugavara central charge and then you know that you're done. You cannot add additional stuff. But my VOA is not unitary, so how do I know? Perhaps I could add an additional stuff that contribute to some substantial additional operator, which is orthogonal to this one. The story is a little bit more involved. But the conclusion is correct, and that's what I'm going to explain next. One needs a more elaborate unitary argument based on four-dimension unitary that will indeed lead to the conclusion that if the two-dimensional, in the case you have an affine current algebra, if the central charge computed this way, agrees to the Sugavara central charge, then you know for a fact that the stress tensor is the Sugavara stress tensor. So, yes. Sorry. They appear, everything appears, pretty much. Any kind of reasonable symplatic variety, a symplatic cone can be obtained in this huge group of theories. Okay, so, before I do that, let me generalize the story to a nice little sequence of four-dimensional theories. So, these are names, let me also put H0. These are names that are associated to kodaira singularities. Not all of them, but some of them have these names. And if I consider an f-theory singularity, which you can think of some d7 nonperturbative d7 brain setup in type 2b, and I probe it with one d3 brain, the field theory that lives on d3 brain reduces the low energy to a superconformal field theory. So, this is a simple nice sequence of superconformal field theories that have one-dimensional Coulomb branch, because you have a single d3 brain. Okay, so this is, you know, appealing to the knowledge of those of you who know a little bit of these things. If you don't, I'm just take my statements. The fact that d3 brain is inside of the seven brain immediately tells me the hex branch. All right, so remember that d3 brain can be understood as an instanton inside of the seven brain, and so the instanton moduli space of, so h0 is no flavored symmetry, h1 corresponds to flavored symmetry, su2, this is su3, and these are the obvious things, e6, e7, e8. I find superconformal field theories. Let me just name the superconformal field theories by the names of the associated kodaira singularity. These are rank 1 superconformal field theory, rank 1 because the Coulomb branch is dimension 1, with this amount of flavored symmetry. And the hex branch is the one instanton moduli space of gf. Okay, the case that I just studied, the Lagrangian case, is this one. In this case you don't need f theory, this is a perturbative. Something you can see in perturbative string theory, and so this is a case that has a weak coupling limit, and it's the example of superqcd su2 with four flavors. The other examples are non-perturbative. So with the exception of this theory that has no hex branch, the others have, the hex branch is the one instanton moduli space of the corresponding flavored symmetry groups. So that is a very nice simple generalization of the story here. What is the VOA? Well, it turns out that this one is the V erasoroalgebra at level minus 20, at central charge minus 22 over 5. This is su2, a level minus 4 thirds. This is su3, a level minus 3 half. This is the one that we just did. This is su8, a level minus 2. This is the 6 at a level minus 3, is 7 a level minus 4, and e8, a level, because you know it, but I wanted somebody to say minus 5. Okay, so how do we know this? Well, first of all, minimalistic guess. Again, we know that this must be subalgebras by the argument that I gave. We know the level, so at the very least, this is contained in the VOA, and the fact is, exalt the VOA requires further arguing. So now I'm trying to give further arguments for it by thinking a little bit in terms of the constraints of four-dimensional unit art. Okay, but before I do that, let me make the following. You know, what is this whole business good for? Well, at the very least, you should be mildly impressed by the fact that if you believe this correspondence, I now get to compute a huge amount of correlation functions in, you know, coefficients, dimensions, et cetera, for some strongly coupled theories. A priori, you would not have had easy access to, because everything downstairs, these are just a fine casue of the algebra at some funny level, everything downstairs is perfectly computable, and so I can lift up that information and learn about this nice BPS set of the four-dimensional theory. And this is something that is actually really useful, because, for example, for this E6 theory, we use this information as input to fix a large set of data in the OP of four external operators, those we fix using this scalar algebra, and the unfixed stuff, which corresponds to long multiplets, we fed into this numerical booster machinery that I discovered in Michael Locchio. And then using that numerical booster machinery, we fixed, or more precisely put bounds, but we believe that they are saturated on the non-protected stuff. But this program couldn't really have got off the ground if we had not known this huge amount of protected information, because that was the only way to tell the boostrap that we are speaking about this particular theory with the E6 symmetry. Okay, so there's practical implications, but let's look at now and more structural properties, and let me go back to this grading that I had earlier. So a priori there's a third quantum number, little r, but I'm going to forget it, because in all of this example, all the operators have little r equal to zero, so it's not important. And remember, we also had this relation that little h was r plus l, from which, of course, we conclude that r is smaller or equal than little h. Or operators where r is equal to h, these are operators that in 40, these are Higgs branch chiral operators. So the subspace v11 is the space spanned by the affine currents ga. But if I want to think in terms of states, I put a vacuum here. H is equal to 1, and I also know for a fact that this descended from the BHET1 multiplet that had r equal to 1. Okay, so in this simple case, there is a clear assignment of little r. Is this clear? Little r could have been zero or one-half, but I'm assuring that that's an impossibility because there's no four-dimensional multiplet with those charge assignments. So we know for a fact, also more concretely, we know exactly where the affine current came from. It came from the four-dimensional moment map. So I have an ambiguous assignment of this r-quanton number. But now let me look at j minus 1a, j minus 1b, and things here are more tricky. Well, first of all, it's clear enough that the h assignment is 2. I just count 1 plus 1. But what is the r assignment? Well, the r assignment a priori can be 2, or it can be 1. Yes? I don't know the r assignment a priori. And it cannot be zero because this is not a thing that exists. But now I'm going to play a nice little game, and first of all, let's first consider the case in which we project these objects to some non-singlet representation is not the singlet. So what is v21? V21 is an operator with r-charge 1 and dimension 2. But let me claim that the only possible state with these quantum numbers is the stress tensor. The only multiplet in four-dimension, if I assume that I have a unique stress tensor and there are no higher spin concert current, blah, blah, blah, that will lead to this charge assignment, is the four-dimensional arc-symmetric current. But the stress tensor is clearly a singlet. So in the non-singlet sector, I know for a fact that this is v22. Why is that powerful? Because I gave you earlier a little rule for computation of the norms. So if I know r, I know h, and in this case, actually it's very easy to decide what sigma should be. Sigma contains a minus sign. That is the minus sign we saw earlier in the flip of the level. I now have an inequality to satisfy that I can impose on this gadget. And the inequality is actually non-trivial. I can compute the inner part of these j's just using the commutation relations in the affine-cats-mudic current. And I find that this inequality leads to a bound on k. K has to be smaller or equal than some number that depends on g, on the flavored symmetry. When this inequality is violated, I find a contradiction with four-dimensional unitary. And, well, happily enough, the bounds are precisely saturated by this sequence of algebras. So the bound for SOA tends to k has to be smaller or equal to minus 2. But how do we interpret saturation of the bound? The only way that I can have saturation of the bound is if this state is null. And so this actually means, and you can check, that SOA is a level minus 2 if you go ahead and compute the restriction of this gadget onto the 35v representation. This is null. And what does this mean? It means that the upstairs theory, the four-dimensional theory, must have a relation among the Higgs-Brand generators, which was one of the relations that defined the SOA minimally important orbit. And that, of course, everything hangs together. It's only at this particular level that I can hope to reproduce the four-dimensional Higgs-Brand, because it's only at that particular level that the multiplication of my affine current as an outstate, and then outstate seen as the presence of a Higgs-Brand relation. So there's a very nice interplay between the four-dimensional geometry and representation theory of these vertex operator algebra negative levels. And you will recall that that they, well, okay. So that's half of the story to get the correct Higgs-Brand. And the other half of this, sorry, this sets to zero really all these gadgets, 35v, 35s and 35c. To get the correct result, now I need to argue that the singlet, there's also a relation in the singlet in the Higgs-Brand. And that requires a little bit of further argument, because if I now project this representation on the singlets, of course, that's precisely the case where I can have a contamination of the stress tensor. So what we can conclude on general grounds is, so now let's do the non-singlet case, the singlet case, sorry. So if I take now this to be the singlet, in fact, let me precisely map it using the gilding form. This is what is sometimes called S. It's the unnormalized Sugavara stress tensor. Well, now it's trickier. So a priori this could be an object, which I'm going to call mu squared, which lives in v21, sorry v22, plus alpha times t, which lives in v21. Okay, so t is the map of the arcimetry current. Remember that story. And mu squared is what comes in the image of the product of mu, of the operator that I called mu before in four dimensions. Okay, this is a Higgs branch operator. It has, is in b hat 2, and it has r equal to 2. And this is something else. So a priori I have these two components. But now it's a trivial hazard size, again using the fact that I know all the discriminational relations in the V8 to actually compute alpha. Alpha turns out to be something like dg. Let me actually copy it to be precise. Turns out to be k divided times cg divided by c. And so actually here you truly have an overlay. You see, that's where this business of assignies are grading becomes interesting. You may have thought that naively, given this is the problem of two things with r equal to 1, has r equal to 2, but not, not really. It has a piece with r equal to 2 and a piece with r equal to 1, but I know exactly how they come about. And now with this sophisticated, more sophisticated knowledge, I can go ahead and impose positivity of the norms. And I'm going to discover that positivity of the morph imposes inequality on the, sorry, it's the other way around, inequality on the two-dimensional central charge that say two-dimensional central charge is greater or equal than the Sugavala central charge as a consequence of four-dimensional unitarity. And when it is saturated, it means that I have a null state and what it is null state is mu squared. When this inequality is saturated, the two-two component that corresponds to the square of the moment map in the single representation is set to zero, but that is of course precisely the missing relation, the missing quadratic relation that sets to zero, the product of the SOA moment maps in the single representation. And so lo and behold precisely when these bounds are saturated, we can conclude that we correctly reproduce the four-dimensional Higgs branch and we're happy. Okay, so the story becomes very rich with these unitarity bounds because you see, here I was using very special properties of this very universal type of operators, but the moment I go to level three or to level four, I don't have that in my disposal and then the game becomes a lot more intricate, but nevertheless with more work you can play it and it leads to sort of very interesting conclusions. For example, one relatively easy thing to establish is a universal bound on C. By playing a similar unitarity game you find that there's universal bound on C alone. C to D is molar or equal to minus 22 over 5. This translates into a statement that C for D is greater or equal to 1130 and sorry, this assumes that there are no higher spin conserve currents. So you got a very cool result, that the four-dimensional central charge in a interacting four-dimensional theory has to be bigger than this number. When you have this number, it means that the corresponding Verozoro algeva in 2D is precise, the Verozoro algeva of the Li Young minimal molar, which is the 2.5 minimal molar. And so it's rather surprising you get to put a bound on the simplest interacting four-dimensional super conformal field theory. But then you can get going and using a more elaborate reasoning, which again amounts to careful examination of these graded spaces and norms, et cetera. We haven't finished it, but we are pretty confident about the following little hope for theorem. That the constraints of 4D unitarity and plus additional boost of consent that I don't have time to discuss. Using this, and if I up to now everything was complete universal, I had to make no additional assumptions about what further the vertex superalgeva could contain apart from these universal subalgebras. But if I further assume that the entire kala algeva is just Verozoro, this is a strong assumption, of course. If I assume that the entire kala algeva is just Verozoro in some central chart C, then I get to make stronger statements, because level by level I know that the entire space must be made of Verozoro and then imposing orthogonality is rather stringent. And then you show that the only allowed values of C are the ones that correspond to the 2,2M plus 1 Verozoro minimal models. Any other value would lead to a negative norm in four dimensions. Okay, so this is the beginning, hopefully, these baby steps into a classification program. So the main lesson here is that the kind of vertex operatoralgeva we get by this 4D to the map are very special. I illustrated in some example that truly are very special. You get very specific negative levels where you have a ton of null states, which are needed to reproduce the four-dimensional geometry. And they are special really because if you are away from these very specific fine-tuned levels, you are in danger of violating the constraints of four-dimensional unitarity. So if one could really characterize and axiomatize the property, the additional properties VOA must have in order to be same from four dimension, you could perhaps hope that this is the beginning of a classification program of at least those VOAs. Now the question of whether two different 4D theories could map to the same VOAs, of course, open, but there are no counter examples at the moment. How am I doing with time? Like zero minutes. Zero? No, how much time do we need? No, I mean, the story is open-ended. I can go on forever or stop now, but... And we can summarize sort of the big picture. So I at least give you a glimpse of how four-dimensional unitarity enter the story and how one can reconstruct the geometry of the four-dimensional theory from this representation theory of the VOA. Let me mention one last thing. So clearly the VOA has a lot to do with the Higgs branch, but it contains more information. And so this is part of our ongoing effort is to try to characterize precisely which additional information, apart from the Higgs branch, you need to reconstruct the VOA. That's one direction. And a simple first step in that direction is if I hand to you the VOA abstractly defined in terms of a set of generator and their singularity coefficients, can you reconstruct the four-dimensional Higgs branch? This is a rather non-trivial question because the Higgs branch is there, but it's clearly somewhat hidden. The reason it's hidden is precisely this funny business of this R-grading. The Higgs operators are these ones. But I impress upon you that precisely extracting the leading term where they want to have H equal to R is in general non-trivial. If you had knowledge of this quantum number assignment then you could do it instantly, but in general you don't have. And the Sugavara construction is already the simplest of infinitely many such ambiguities that can occur. And in all examples that we bother to look at, there is a very interesting and very universal way to recover the Higgs branch. And I'm going to end with that. So it's still conjectural, but surely true. So given a VOA there is a purely algebraic construction that associate to it a, let's call it, that's VOA, let's call it VOAV and let's call this XV. And I'm going to spring this in a minute, but the compelling conjecture that we have is that this is nothing but the Higgs branch. Okay, so how do we construct this variety? So let's assume that the VOA is given by some set of generators. I gave example before, some finite set. Then we can consider polynomials in the formal variables as we, we drop the Z dependence and we consider polynomials in formal variables, one variable for each generator. So example, in the case of Virazoro, we would just consider polynomial in a single variable t. And then we mod out by relations which are obtained by the presence of null states. Let me give an example. Let's do the Virazoro minimal model at of 2,5. Then you will remember that this, I don't remember the coefficient by heart, but there is a null state of this form in this theory. And L minus 2 square is just, is nothing but t squared in my way of defining this correspondence. But L minus 4 is twice the derivative of t. And the idea that I'm dropping the Z dependence really means I'm working modular derivatives. So I actually want to throw away derivatives which means that I should really set that I should really consider polynomials in t modded out by the relation that t square is 0. Okay? That's, and you can generalize this game. You consider the nulls and you look at relations of this type induced by the fact that the null can contain derivatives of the generators. And you set those to 0. Now, of course, this is not a geometric space because it's just a fatten point. It's just t square is 0. And so the associative variety is then this space here further divided by nullpotence. Is the construction clear? I sketch it here. So in this particular case, if I further divide by nullpotence, I'm left with nothing. In other terms, the associative variety is a point. And that nicely nicely agrees with the fact that the Higgs branch of this H0 theory is trivial. There's no Higgs branch. And you can keep going. Check this story in all this example. It always works in this example and many more. But let me end with that sentence. Hopefully it will inspire some of you. This state, so I always universally have the same state. So I always universally have the stress tensor, because it always comes from the arc current. But if this conjecture is correct, you know, the stress tensor is clearly not the Higgs branch generator, it is the arc current. So it means, if this conjecture is correct, that there always must be an outstate of the form nth power of the stress tensor is equal to a holomorphic derivative of something else, which means that in this quotient the stress tensor must be nilpotent. And the existence of such an outstate allows me to compute the torus partition function by inserting this outstate and turning the statement that the insertion of this outstate should equal to zero into a modular differential equation that should be obeyed by the torus partition function. And so we get the rather curious statement that the vacuum character of this carnal algebra must obey a modularly covariant differential equation. In fact, we have discovered that this is universally true. These are vector-valid modular forms where the other solutions correspond to non-trivial modules of the VOA. And so this is very much still ongoing work. The physical interpretation of this modularity and this over additional representation which presumably correspond to the insertion of surface defects in the transverse plane is very much. Ongoing work. In fact, there was a paper today by the Dushensko and Fluder. I'll stop here.