 Continue with Leonardo's second lecture. OK, thank you. So today, we will switch gears from the basic Lagrangian approach we had yesterday to this more abstract conformal booster viewpoint that I outlined in my colloquium yesterday. But to recap, let's remind ourselves of some basic notations. So these are my conventions for the n equal to 2 vector multiplet. And this is my convention for the n equal to 2 half hyper. And there is a alpha and alpha dot R spinor indices. I, that runs from 1 to 2, are SU2R indices. So you can see that the k genus of the vector multiplet form an SU2R doublet. You may wonder about the SU2R transformation properties here in the half hyper multiplet. They are a little bit hidden. And but essentially, it's the scale as that carried the SU2R representation. This is made most transparent if we graduate ourselves with the n equal to 2 full hyper, which is two copies of the half hyper. And then one would construct a doublet, which would be the combination of q and q tilde star. This complex conjugation is actually necessary to make manifest the transformation properties under SU2R. So in my notations, q and q tilde are both n equal 1 chiral multiplets. But in this n equal 1 notation, the SU2R transformation properties are a little bit hidden. And finally, I didn't spend much time on this yesterday. But in my conventions, the U1R assignment is minus 1 for the complex scalar in the vector multiplet. And it necessarily, it must be 0 for the complex scalar in the hyper multiplet. And these are conventions such that the super symmetry generators carry their, of course, SU2R doublet. You see this index i. And they carry r equal plus 1 half. And so since we go from phi to the lambda's with the q, this is r equal minus 1 half. And this is r equal to 0, whereas this would be r equal plus 1 half. Again, since I'm at it, let's also write once more symmetry algebra. And then, yes, in the discussion section, I took the opportunity to also introduce the S super conformal fermionic generators, which square onto the special conformal generators. So I don't have too much time to review this in enormous detail, but there is this name, 222 slash 2, again. If you want to learn about super algebras, Wikipedia is pretty great. Wikipedia entries are generally terrible for physics, but for a culture reason that I don't fully understand, mathematicians spend an enormous time editing Wikipedia. And so the mathematics and the Wikipedia are actually quite good. And so anyway, it doesn't matter. You can learn about it in Wikipedia, but the generators of the algebra are the momentum generators, the special conformal generator, the translation, the Lorentz generator, which I'm writing in by spinon notation. So this is 3 plus 3. And then there are the fermionic generator that I just wrote down. And finally, so there's q tilde s and s tilde. And finally, last but not least, there are the SU2R times u1R generators. OK, so that is a recap of conventions and basic algebraic facts. And from now on, I would like to switch to this more abstract conformal field theory viewpoint. So here we have elementary Lagrangian fields. And we are interested in, remember, we're interested in gauge theories. So actually, let me also review very briefly the story of n equal to 2 Lagrangian theories. The data are a choice of semi-simple gauge group, cannot contain one factor, and then a pseudo-real representation R of G, under which the half hypermultiple transform. And then the condition that the beta function mass vanish imposes restrictions on this data. And the only continuous data one is left with, if you insist, on conformality are the complexified gauge coupling for each of the simple factors of the gauge group. So in such a setup, we have elementary fields. The vector is in the adjoint. So this is in adjoint of G, whereas the half hyper means a representation R of G. And we are going to be interested in correlation functions of local gauge invariant operators. So is a local gauge invariant combination of the elementary fields. And so the local operators of the conformal field theory in this description are really made up of these elementary constituents. And that's the fallback option and intuition to which you can always go back to if you are confused by something. Just try to realize the abstract segment I'm going to make in terms of these fundamental ingredients. But the viewpoint I'm going to take is that these are the basic objects, and we do not necessarily need to think of each of these local operators, being a composite operator of something more fundamental. So as a way of transitioning, let's give examples of what these local operators are in a Lagrangian theory. And so we could be very concrete. We could take G to be SUN. And then we take R to be, well, we did a little hazard size yesterday with the vanishing of the beta function. We concluded that if we insist that the hypermultiply and the fundamental representation, we need to take NF fundamental with NF is equal to 2n. And so here is the full hypermultiply story. So I'm going to have q and q tilde, where q would be in the fundamental, and q tilde would be in the anti-fondamental of everything, really. So let's even fix indices. Let's a going from 1 to n be the gauge index. And so this would be an upper index a to the not the fundamental representation. And this would be a lower index a to the not the anti-fondamental representation. And in order to ensure vanishing of the 1-lubeta function, we need to take NF copy. So I'm going to add an additional index i that goes from 1 to NF, which is always equal to 2n, if you want conformality. And so these objects would be in the fundamental for SUN gauge. And this would be, well, really u and f, because they're also charged under the u1 flavored symmetry. OK. So that's our matter. And then our gauge fields, of course, would transform in the adjoint. So I could use a fundamental anti-fondamental notation where it's implicit that the trace vanishes or, alternatively, I could introduce an adjoint x and a that goes from 1 to n squared minus 1 and rewrite this object as a single adjoint. OK. So that's a simple example of a n equal to 2 super conformal field theory. And then we would construct local gauge invariant operators by contracting gauge indices. And so examples of these local OIs would be, for example, trace phi squared, trace phi to the k, et cetera, phi cube, et cetera. And or you could have objects that contain the q's. You could do q i, a, q tilde j, a, where the index a is contracted. You have always to contract your muscle contract gauging. So this object would carry some flavor representation in this i and j. OK. So we'll come back to this example to illustrate some of the features of what we do more abstractly. OK. Questions about this? OK. So I gave you yesterday a lightening review of the conformal booster philosophy. I really have to take for granted the basics of conformal field theory, such as, for example, the statement that in conformal field theory, there is an isomorphism between the space of states. So there is a Hilbert space of states h. And then there are local operators, like the ones I just described. And the space of local operators at a fixed point, which, for example, we could fix to be at the origin, is isomorphic to the space of the Hilbert space of states of the quantum theory. And how do you go back and forth? Well, you can construct a state by taking an operator and acting on it on the conformal invariant vacuum. That's clear enough. It's a little more subtle, but nevertheless possible. And one can make this rather precise to reconstruct uniquely the state starting from the operator. And intuitively, if you have a potential description, the way you are going to do it is to insert the operator at the origin, do a path integral, and then cut open the path integral on a sphere. Let's do this in four dimensions. So we use the deletion operator as our Hamiltonian. And so surfaces of constant radios, which are three spheres, are our special slices. And so in this quantization scheme, which is called radial quantization, we have to construct the wave functionals of the fields at this unit sphere. And if we insert an operator there, and we do a path integral, that defines a state. But the statement really can be made more axiomatic and more abstract without resorting to the path integral. And that is the state operator map in three minutes. OK, so I can perhaps leave those notations there. So it follows almost immediately from the assertion that there is a state operator map that you have a convergent operator proudness function, which I again briefly sketched in my talk yesterday. And the idea being that if you have two local operators, you can read off the state here, and then use the state operator map to construct a local operator back. And with a little bit of thinking, you can see that the convergent OP just follows from completeness of states in the Hilbert space. So these are all things that I'm going to pretty much assume. They are now very nice pedagogical views on the whole conformal. First of all, most of what I said is already in chapter two of the Sting Theory textbook by Polchinski, but a more modern presentation that applies to higher dimension can be found in many lecture notes. For example, I can recommend the ones by Slava Richkov, David Simmons, that thing. It's a well-established toy that I invite you to study. And so again, the idea is that we have this operator proudness function with some efficient functions here, which are fixed by conformal covariance. The sum runs over the entire set of local operator, which as I just said is isomorphic to the spec of states. And this expansion is absolutely convergent inside a correlation function till the insertion of the next operator. So if this is operator 1, this operator 2, and then I have some operator 3 here, you draw a big sphere, which does not encounter any additional insertions. And by the argument that I just sketched, you can replace these two operators by a single operator, say at the center of the sphere. OK? So clearly, we want to use as much as possible of the symmetries of the problem. And let's start with the pure conformal case. Already in the discussion section, I took advantage of you and already quickly sketched how the representation theory works. So I phrased the discussion yesterday at the level of states. And I asserted that the primary state is by definition a state which is annihilated by the special conformal generators. The operator version of the statement is the statement that the operator that corresponds to the state psi commutes with the operator at the origin commutes with the special conformal transformation. It's necessary to put this at the origin because if you're away from the origin, there is an orbital piece of the action of the special conformal generator, we will not bench. And then, given this definition, then we classify representations in terms of the conformal dimension delta, which I interchangeably also denote by the energy E, because I was just said that we quantize the theory in radial quantization. And then, let's treat it now to four dimensions by the two cartons of the Lorentz group. If we are in Euclidean space, we have SO4, which is the same thing as SU21 times SU22. And these are the two spins. Or, of course, the Lorentzian version is similar. You are familiar with, you can label Lorentz representations with two spins. So we are now this j, i, r, half integers. OK? And so what I really mean by the state is the highest weight state of the two SU2s. So it has definite conformal dimension. It's the highest weight state of the two SU2s. And it's definite carton quantum number under each of the SU2s. And then we will build the whole module by acting on this state with the arbitrary amount of raising operators, which are the p's, and arbitrary amounts of also the Lorentz raising generators, which would be, well, whatever, j minus 1, j minus 2, et cetera. So the action of p mu on operators is just by taking derivatives. And so this is just a restatement of the fact that the full conformal representation is built on a primary by taking arbitrary amount of derivatives and then rotating, in all possible ways, the Lorentz quantum numbers. OK, so for example, for a scalar operator of x, which, for example, the one that I had earlier, something like trace phi squared of x could be an example of that. We build the full module just by acting with derivatives in all possible ways. And states that have a non-zero number of derivative would be called conformal descendants of O. OK, so this operator I should put at the origin. And this is the full module. OK, so in general, for the generic choice of the quantum numbers of the operator, this module is irreducible. But, and this is a phenomenon that will become more involved in the super conformal case. So I want to give a five-minute discussion of the conformal case first. If you suitably tune these quantum numbers, the module becomes reducible. And then you get shorter representations if you impose suitable conditions. Another way of saying this is we are going to impose the condition of unitarity. So we want norms to be positive. And it turns out that if you tune the quantum number suitably, you will encounter that some of the descendants states have zero norm. And they are orthogonal to the entire module. And so in unitarity, if you are really instructed to mod out by these null states, and you get a shortened representation. So the unitarity bounds on conformal representations are that delta, or the same, which is really for me. So if I take the two spins to be both of them non-zero. So I'm going to sit down to four dimensions. And the two spins are both non-zero. Then unitarity imposes this condition. If one of them is zero, then unitarity imposes this condition. And of course, there is a completely analogous condition if this is true. And finally, if both spins are zero, unitarity imposes this condition. So the statement that I'm making is that, so how does this come about? These conditions come about by starting with a state of arbitrary them. Let's do the case with the spins are both non-zero. I start with the highest weight. And I act with p and start computing norms. And I must require that all the norms of the descent states are positive. The matrix of norms is positive definite. It turns out that this is only possible when this condition is obeyed. If you violate this condition, in fact, in this simple case, you will encounter a negative eigenvalue already at the first step. So if you start with an operator with spin of whatever it is of this type, and you act with the momentum, and you compute the norm, and impose that this is positive, this will instantly give you this condition. It's a simple exercise that you can do. How do you do it? Well, you just remember that in radial quantization, p dagger is equal to k. And so you can just compute this norm by the commutation relations of the, you were going to get pA, p dagger, which is the same as k. And so you can just compute this norm by commuting k and p using the commutation relation of the conformal algebra. And it's a simple exercise that you can do, and it will give you this unitarity bound. It also follows from this way of I introduce a unitarity bound that if the unitarity bound is saturated, you are in the critical case where there's a certain, OK, so this is, of course, a little bit schematic. So there are, this is a whole matrix, because there are choices of indices that you have to make. So it's the same that this matrix is positive semi-definite. And the statement that you are saturating the unitarity bound is the statement you're going to find, a certain linear combination of the states of this kind, which has zero norm. And one, if you go through this exercise, which I really invite you to do, I mean it's a little tricky, it's a little involved, but once in your life, you are going to discover that the combination that is acquiring zero norm is this one. It's a combination where we are contracting precisely one alpha and alpha dot index. Or, OK, so I'm trying to be too general. Let's do the simple case where we have a tensorial operator so that we can use vector indices running by spinor indices. So we have a tensor. Say, a symmetric list tensor. And then this condition is simply the condition that the divergence of this operator is zero. So you have discovered that there is a state in the module that has zero norm. And so since we're insisting that the representation should be unitary, we must mod out by it. And so this means that we're going to impose this as a relation obeyed by the operator. You recognize that this is simply the relation that says that this operator is a conserved current. So we now have a little interesting resulting that follows up from representation theory that if you know for a fact that the operator has this dimension, you immediately conclude that the operator has zero divergence. The cases that, of course, are of physical interest are, so let's do an example. So of course, simplest case is j mu. It's the same as j alpha alpha dot. So that means that j equal j, j1 equal j2 equal 1 half. And then the unitary bound tells me that e should be greater or equal than 3. And if e is equal to 3, then this is a conserved current, which is hopefully familiar for me. And the other currents are dimension 3. That's obvious at the classical level, but now the same here at the full-fledged quantum level. It's an if and only if the current remains conserved, it has dimension fixed to 3. And if it has dimension 3, it must be conserved. And the other, of course, case of paramount important would be the stress energy tensor. j mu nu or t alpha 1, alpha 2, alpha dot 1, alpha dot 2. This is j1 equal j2 equal to 1. And so the unitary bound tells me that for a spin to operator of this kind, he has to be greater or equal than 4. And if e is equal to 4, then this operator is conserved. Of course, you can consider objects with more indices. And those would be conserved currents of spin greater than 2. And it's a theorem derived under very general broad assumptions that the presence of highest spin conserved currents with a higher spin, I mean spin greater than 2, implies on abstract grounds that the theory is free. Or more precisely, it contains a subset that must be described by brief free fields. So you can entertain yourself, take your favorite free field theory, the theory of a free scalar in four dimensions. You can explicitly construct by linears of that scalar with sprinkle with various derivatives, which are conformal primaries, and that obey this conservation condition. And the statement is that there is a converts statement that if you find the conserved current of spin greater than 2, then you must identify a free subset in the theory. This is great from our general booster viewpoint because we understand free field theories perfectly. And so we shouldn't bother with them. And so we are going to impose, as a condition on the theories that we want to study, that there must be no conserved currents of spin greater than 2. And so we can impose from the get-go that we're just talking about interacting quantum field theory. So the point here is that if I think a theory abstractly in terms of this list of operators, in particular, in the list of conformal representation that the theory contains, I can detect whether it's free or not just by looking at the list. So the statement that the theory is free or not is a statement that I can immediately read off from the so-called conformal data. The conformal data would be the list of dimensions and the list of opaque coefficients, and in particular, just from dimensions and Lorentz quantum numbers and their opaque coefficients. Those are the conformal data that determine the theory completely. And it's a very simple statement about just looking at the representation that tells me that the theory is free or not. The other thing that we're going to impose is that the theory contains one, and precisely one, conserved current of spin 2, which will identify the stress energy tensor. This is, again, obvious in the Lagrangian viewpoint. You can just construct the stress tensor by your favorite procedure, either the nether procedure where then you have to improve it, or coupling it to a background metric, blah, blah, blah. It's clear that if you start with a Lagrangian, local Lagrangian, there's going to be a stress tensor. And it should also be clear that if the theory is interacting, the stress tensor is unique. In the free field limit, the theory actually has multiple conserved currents of spin 2 because each, let's say I have a theory of n scalar fields, each scalar field will have its own separate spin 2 conserved current. But the moment you turn on interaction, you genetically expect that most of these spin 2 objects acquire anomalous dimension. And remember my unitary theorem, the moment you have an anomalous dimension, you instantly lose conservation and vice versa. And in fact, if that's not the case, is as if you turn on interaction, you find still that a generic point in coupling space, you have two separate stress tensor. What that really means, that you're talking about a product theory. So we are going to restate attention into simple theories. There's a condition if you're analogous to, again, remember I had this analogy yesterday where it is a little bit as if we are trying to classify the algebra. And then clearly, sure, you can try the most general semi-simple case, but it's easier to impose from the get go that we're looking at a single reducible simple piece. And what about the spin 1 object? Well, the spin 1 object is, of course, is the hallmark that the theory has a global symmetry. That's, again, obvious from the Lagrangian viewpoint. If you have a global continuous symmetry, you use Nether's theorem to construct the conserved current. And we are going to assume on abstract grounds that that's always the case. That if, in some sense, this is part of our axiomatic definition, we're going to declare that we have a local conformal field theory if, first of all, it has a unique stress tensor. And second of all, if the view is invariant under a continuous group of global symmetries, there is an associated conserved current, which, of course, we have to transform in the adjoint of the flavored symmetry group. OK, so far, so good. This is the abstract version of Nether's theorem, which does not have a proof. Nobody really knows how to prove this fact from a reasonable set of axioms for local conformal field. You can do what Harle and Noguri do, which is, basically, formulate a framework where it's basically obvious that this is true. But a more conceptual understanding of this. In fact, it would be lovely to be able to prove that just the assumption of this to the local stress tensor implies that whenever the theory is invariant under a global continuous symmetry, then it has a conserved current. For us, this would be an assumption. OK, so I'm not going to do this in great detail, but it turns out that these are free fields. So we will not care about them. And then this is also a free field. Sorry, what I mean by this is, when you saturate this unitarity bounds, you find free fields. So a simple example for this would be a self-dual. The self-dual to form for the Maxwell field obeys this bound, because this is a self-dual form. So it has equal to 2. And then it has j, well, with the dots, j1 equal to 0, j2 equal to 1. And so it operates this bound. And it's just the same that there is a free Maxwell field, which, again, will not be of interest for us, because we are going to restrict our self-interacting field theories. And when you saturate this bound, of course, we are talking about a free scalar. A object which has equal to 1 in this condition is just a free scalar. And in this case, it's the only one where you need to work a little bit harder, because the null state that forces the unitarity bound happens at level 2. And so the condition that the representation will be unitary is a condition that box phi equal to 0, which is, of course, just a free question to ask. OK, so that's a lightening review of representation theory of the conformal group. It's rather simple, really. Most representations are generic. And then when you saturate one of these unitarity bounds, you need to remove just a little bit of null states, which is a single null states, and, of course, a little bit descendants. Clear enough? I just told you. And that's my main point, that I don't have to ever think in terms of elementary fields. I just look at the list of representations. And if a certain unitarity bound is obeyed, well, if the unitarities bound that correspond to free fields are obeyed, I'm going to say that there is a piece of the theory which is free. And I don't want to consider that. I mean, I can consider the case if I insist. Then it's going to be trivial. But since we don't want to get confused by free fields, we will just assume that no such representations appear. Now, it's the picture that was drawn yesterday of the space of the conformal manifold. So the statement that I'm making, a generic point of the conformal manifold, none of these free representations appear, but they might appear. And generically, they will appear at singular points. So yes, I had this conjecture that the conformal manifold of a general X2 theory arises by weak gauging in a certain limit of a gauge group. Well, when you switch off that gauge coupling, then clearly, you spit out a free vector multiplet. So the free vector multiplet and several of its composites will obey some of those unitarity bound. And well, at this special point, you will have, in fact, multiple sustenance or higher spin conserved currents. But the beauty of this approach is that we can really now characterize, or if you want to define, what the special point are by stating that, well, what is the special point? It's this point where you have enhanced higher spin conserved currents, where you gain, in the limit, a higher spin conserved currents. OK, so that was fast, but any other questions? I'm not going to repeat the same story for the n equal to superalgebra. OK, sorry. Before I do that, let me mention a few other things that we can state in great generality. So some of the most robust universal conformal data would be obtained from the three-point function of the conserved currents, if any, and of the stress tensor. And so in particular, the two-point function of the stress tensor, I'm not going to write it as the tensorial structure, but in some normalization with the appropriate conventions is proportional. So first of all, the normalization of the stress tensor is canonically fixed, because you want the sense to be a generator of, in particular, translation. So you cannot mess up with the overall normalization of the stress tensor that's got given. And then given that normalization, the two-point function of the stress tensor is interesting, and that is the so-called C central charge. For a reason that will become appear shortly, I'm going to call this C4D. From the three-point function of the stress tensor, which is rather complicated, amid several different tensorial structures, you can again recover C, but you can also recover another combination, which is the A central charge. Alternative interpretation of these numbers is, by thinking in terms of coupling the theory to a background metric and looking at the conformal anomaly, looking at the vial anomaly for the trace of the stress energy tensor, which is, of course, 0 in the limit in R4, if you don't have any background metric, by assumption. That's a conformal field theory. But then it will, OK, so I'm going to get this wrong, but it's C times Euler plus, I think, is this plus other terms, which can be removed by local counter terms. So the A and C anomaly coefficients are important data of the conformal field theory, which you can read off, which are really among the set of conformal data, because I can read them off from the OP coefficients of the stress tensor. And famously, the A anomaly is the object which is monotonic under RG flow, whereas C is not. And then from the normalization of a non-Abelian current, I wrote this expression yesterday. I'm not going to write it again, but in certain conventions that, again, you cannot really mess around with the normalization non-Abelian current, because you need the conserved charges to obey the Lie algebra, which has fixed normalization. And then you can read off the level k, which is another basic datum of your conformal field theory. So now we do n equals 2. So now my super primary is annihilated by S and S tilde. And it's going to be labeled by, well, the same thing as before, and then by the eigenvalues of big R and little R. Conceivably, there could be additional labels under whatever global symmetry the theory may have. By definition of a global symmetry, whatever symmetry generated that commutes with the full super conformal algebra are going to declare that they are a global or flavored symmetry. So a priori, of course, that's a completely separate set of indices that you can tensor with this. Now, I don't have time to write down the whole list of shortening conditions because it's too much. But the story is similar in spirit. So again, the idea will be that you have a generic module that you obtain from the highest way state by acting in all possible ways with q, q tilde does that the anti-competence that gives you p's automatically. And then whatever other raising or lowering operator where we want to call them of the art symmetry of the Lorentz group you have. But for special values of the quantum numbers, you develop null states when you saturate some unitarity bound. And then the module is such that you must mod out by these null states and you get a short-time representation. Sorry, similar in spirit. And the shortening conditions now, before we just had momenta, which means derivative. So the shortening conditions for concept currents were just the divergence of the current is 0. Now, the form of the shortening condition will be the statement that some combination of the raising operator, which are the q's, annihilate the primary state. So this is something that may be familiar to many of you. It's a BPS-type condition. It's a statement that the state or the operator, if you use a state operator map, is invariant under some combination of the supercharges. OK, so we're going to have two distinct BPS-type conditions, which by a universally followed convention introduced by Dora and Oswald are called B-type and C-type. And the B-type shortening conditions are, so we have a state, OK, so the B-type shortening condition will be of the form that the full supercharge q alpha annihilates the state for both choices of the vile-spinner index alpha. And when this is true, this automatically implies that the corresponding Lorentz-Quantum numbers must be 0. So in other terms, this is a gadget that can contain alpha dot indices. And then I can impose q alpha equal to 0 for both choices of alpha, clear enough? Whereas the C-type is a condition where I'm going to contract the alpha index with an alpha index. So this can have a bunch of alpha and alpha dot indices. And clearly, this is a weaker condition because this is two condition and this is a single one because it's just the contracted version of the story. OK, by contraction, sorry, should be cleared. I'm just contracting the SU2 indices with an epsilon. OK, so I've done this for q, but clearly, they're the completely analog condition for q tilde. And then we have to remember that the q's and the q tilde's have SU2R indices. And so in their full glory, the shortening condition that you can have are b1, b2, b tilde 1, b tilde 2, c1, c2, c tilde 1, and c tilde 2. So the b1 condition with the condition that says the q1 alpha annihilates the state. And then necessarily, j1 equal to 0 and j2 equals arbitrary. Clearly, similarly for b2, this is just a statement about the choice of SU2R index. And then this is the complete analogous condition but for the q tilde. And in this case, j2 will have to be 0 and j1 is arbitrary, and et cetera. c1 would be the condition, let me put c up for 1, u1 alpha contracted with alpha equal to 0, et cetera. Hopefully, this is clear. Actually, there's a small subtlety. This is the version of the c condition if j1 is different from 0. But OK, I should have said this here. So this is the condition for j1 different from 0. There is an analogous version, the correct version of the c condition if j1 is equal to 0, it's the square of the q that will annihilate the state. It's a level 2 condition. And then this object has no alpha indices. OK? So of course, I'm not deriving any of these ones. I'm simply stating that these are the only possible null states that can appear when you start computing norms of the Sandman states. It's the generalization of the condition that the divergence of the current was 0. It's this more elaborate set of conditions that you can impose. The b conditions are twice as strong as the c conditions. And so in some part of the literature, the b condition would be called shortening conditions and the c condition would be called semi-shortening, but whatever. Exactly. And now, of course, the question is given that these are the possible special condition we can have, now we can mix and match. And now that's where the story really becomes a little bit baroque because you now can start taking various multiple conditions at the same time and that gives rise to a whole zoo of possible short and super conformal multipliers. But it's also clear, however, that not anything goes. Because we must be compatible with the, you know, I impose two conditions on the primary state. If I commute them or anticommute them, I must find something consistent. And so the anticommutation of multiple conditions I impose will further constrain the quantum number of my primary because, well, remember, for example, by assumption the s's annihilate the state. So if I impose a certain q annihilate the state, the anticommutator of q with s must also vanish. The commutator of q with s contains Lorentz and Archimetic quantum number. So that means that I will find relations with these quantum numbers. And if I, to greedy, and I start imposing too many of these conditions, I could discover that these conditions are not compatible. And so there's no solution where I try to impose all of them. A simple condition that should be immediately obvious, if I try to impose simultaneous B1 and B tilde 1, that means that I'm trying to say that something is simultaneous annihilated by q1 alpha and q tilde 1 alpha dot. But of course, q1 and q tilde 1, given that this is an upper index, this is a lower index, commute to the momentum. And so this implies that the states annihilate by momentum, but the only transition by a state in quantum filter is the vacuum. And so this implies that this is just a vacuum. So that's not interesting. So apart from the vacuum, there's no way I can impose these two conditions simultaneously. And so that's assuming that we have just one vacuum. OK, so what shall we do? So there's a whole story that I really cannot really summarize the next 10 minutes. But I will focus on the maximally symmetric conditions. So we have, remember, 16 supercharges, which is 8 q's or q tilde's plus 8s and s tilde's. These are automatically 0 on the primary. And so I will enumerate the shortening condition, which are various overlaps of those conditions, the result in 8 of the q's annihilating the state. This is what you may reasonably want to call a 1 half pps condition. Because the s's are automatically 0. But if half of the q's are annihilating the state, that's 1 half pps. And so what are those? So well, I told you you cannot do b1 and b tilde 1, but you surely can do b1 and b tilde 2. So that's a 1 half pps condition that would set q1 alpha or q tilde alpha dot equals to 0. And by playing the little game I mentioned earlier, commuting this q and q tilde's with the s's, et cetera, you learn that this implies, well, first of all, as part of the definition of the b condition, you know that j1 is equal to 0. Well, it's not the definition, but you see. So why is it that this object must have j1 equal to 0? Well, because if I commute q with s, which automatically annihilates psi, I will find the Lorentz generators for j1. And those must annihilate the state. So this must be a singlet under the SU21 subalgebra. So this must be a scalar. It has 0 u1r and a priori non-zero big r. And these are the famous, in some circles, b hat r. So they're completely specified by the only non-benishing r symmetry, which is r. And finally, the last condition, you find that e is equal to 2r. Then the other thing you can do is to impose b1 and b2 simultaneously. Well, let me not rewrite it, but this could lead to the condition that e, this is what you would call a. So I'm going to get my conventions a little wrong. But let me first put this as a b hat r. This is going to be an epsilon r or epsilon bar. So let me think for a second whether it's epsilon or epsilon bar. So this is, I think, is what I would like to call an epsilon bar multiplet. And this will have positive little r symmetry, 0 big r, and e will be equal to little r. And then there's the conjugate representation, which is the epsilon r, which will be this one. That would be negative. And these are all scalars. And finally, although these are not what you normally call one-half VPS, because they are a combination of semi-shortening conditions, you can get to impose simultaneously all the C type conditions. Half are strong, but if you impose all four of them, you get something consistent. I mean, I apologize, but this is a universal notation that did not invent. But these are called the C hat 0, j1, j2 multiplets. And they have a priori arbitrary j1 and j2. Big r is equal to little r. Sorry, big r is equal to 0. Little r is equal to j2 minus j1. And the dimension is j1 plus j2 plus 2. So these three types of multiplets are the one where you get to set 4 of the q's to 0 on the primary state. And now, very quickly, since this of course looks a little bit esoteric, if you haven't seen it before, let's do a sanity check and let's go back to our Lagrangian example and let's give examples of this type of multiplets. So let's start here because it's the easiest. So these are objects which have dimension equal to the little arc symmetry. And you can hopefully quickly convince yourself that if you do not want to waste, so the dimension has to be the smallest possible for given little arc symmetry. And the only gadget that satisfies that condition in our set of characters is the phi field. And so these objects, which are clearly must be purely made of phi's. So in our conventions, phi has negative little arcs. So these objects would be trace phi to the k, whereas this one would be trace phi bar to the k. It's clear that in a Lagrangian example, there cannot be anything else to debate this condition because the moment you sprinkle in another field, you violate this condition. These objects, on the other hand, are the object that minimize the dimension for given big r. And, well, I'm taking for granted, you know how to count dimensions. So the dimension of phi is 1. The dimension of the geshino is 3 half. The dimension of a is, again, 1. But of course, if you want to make a geshion variant, you have to look at the field strength, et cetera. And, again, the dimension of the scalar is 1 here and the dimension of the spinor is 3 half. And, well, the only way you can actually win this game is if you consider objects which are made of q and q tilde's. Because the q and the q tilde's are the objects which are in the doublet of SU2R. And so particularly if I look at the highest weight, the highest weight will have, well, dimension 1 and art symmetry 1 half. And so it obeys this condition that the dimension is twice the art symmetry. And so these objects here would be gauging variant combinations of q and q tilde. So, for example, earlier in our example, so we had the q, we had qA, qT. I think I gave this as one of the elementary examples of a composite operator made of the q. This object is gauging variant because the color indices are contracted. It carries some representation under the flavored symmetry, which is, in fact, the adjoining representation of u and f. And what are these objects? The first statement I'm going to make is that if J1, well, with the exception, the only multi will be relevant for us is the one where the J1 and J2 are 0. Because for higher values on J1 and J2, this multiplet will contain higher spin concept currents. Concept currents are spin greater than 2. So for J1 or J2 greater than 0, there are higher spin concept currents inside this multiplet, which, as we discussed, are the hallmark of a free theory. So we can find these type of multiplets in these degenerate limits. But genetically, we won't allow them. The lowest one is the one that contains a stress tensor. So this is the super conformal stress tensor multiplet in a line equal to 2 theory. This multiplet, OK, it starts. So let's quickly draw the structure of the multiplet. It starts with this, how should I call it? It starts with an object of dimension 2, which is uncharged under everything. And then you build the multiplet acting with supercharged at some firmness that I will not bother with. And then at dimension 3, you're going to encounter the currents for the arc symmetry, both for the U1R and the SU2R. And then at dimension 4, you're going to encounter the stress energy tensor. It's a famous fact in supersymmetry, in many cases, where the arc symmetry currents and stress tensor are part of the same multiplet. This is an illustration of that. You may be more familiar with the nico 1 story. In the nico 2 story, the bottom component of the sensor multiplet is an uncharged scalar of dimension 2. So in our Lagrangian example, this would be something like a trace of phi, phi dagger, some number that I'm not going to be able to remember, qq dagger plus q tilde, q tilde dagger, something like that. It's a dimension 2 guy, which is not holomorphic. OK, so again, we will assume the existence of one and only one multiplet of this kind and none of this kind for j. How am I doing with time? Done. OK, so anyway, we made some progress.