 And the reason for it is somehow this rigidity of small representations of supersymmetry. Okay, so as a remark, if we define, well, we're in a hurry, so let's just say, in our example, this invariant has a well-known meaning, chi of h is just the Euler characteristic of the manifold M. Which of course is well-known not to depend on a Romanian metric, right? This is one of the simplest topological invariants that you could attach to M and we recover it by this kind of elaborate talk about supersymmetry algebras. Okay, oh, sorry. Yeah, of course what I meant to write, thank you. Yeah, two zero energy states but sitting in different pieces of the Hilbert space. Okay, so let's now talk about, so in the notes you'll find some discussion of a few sort of more non-trivial examples of this structure. But so in view of the time, I better go on right away to the more interesting structure. So let's talk about quantum field theory. So up till now I was talking about quantum mechanics and you can think of quantum mechanics as being kind of one-dimensional quantum field theory. Now let me talk about d-dimensional quantum field theory. So d-dimensional quantum field theory, which I'll just formulate. So in general you have a d-dimensional quantum field theory you could imagine trying to formulate it on some general d-dimensional manifold. Let me just formulate it in d-dimensional Minkowski space. So r to the d minus one comma one. So then just as before we get a Hilbert space. In fact, this is actually an example of the structure we had before. So d-dimensional quantum field theory is an example of quantum mechanics. But it has a lot more structure. So the Hilbert space will come to us now even before talking about supersymmetry. The Hilbert space comes to us as a representation of the group of isometries of r to the d minus one comma one. So I'll just talk about part of that. Iso d minus one comma one. So what is that? That's SO d minus one comma one. So linear transformations plus the translations. And they intertwine together in a semi-direct product. So this is just the rotations and boosts. So preserving the metric in Minkowski space together with the translations. And these generators actually won't talk about too much explicitly. But these generators we'll talk about so let's give them a name. So the translation generators. The generators of the Lie algebra here. I'll call them p zero for the translation in the zero direction. That's the time like direction. Sometimes we also call that h the Hamiltonian. And then p one up to p d for the translations in the other directions. P d minus one sorry. So d is the dimension of space time. So the case we talked about before if you like was the case of iso zero comma one where this group was trivial. And r this was just r to the one with a single generator h. Okay so now we're doing some jazzed up version of that. Okay so let's ask now what kind of representations of this group are we gonna find? What kind of unitary representations of this group are we gonna find? And now we're getting into the world where so in quantum mechanics you can easily write down explicitly examples of Hilbert spaces. In non-trivial quantum field theories it's already quite hard. And I certainly won't try to do it here to write down explicit examples of these representations. So all I'm gonna do is tell you some sort of formal properties of the representation that'll be enough to get us going. Okay so let's just think about how we would organize a representation of this group. Okay so before I was organizing them in terms of diagonalizing the Hamiltonian that was because the Hamiltonian was central. Now the Hamiltonian is not central anymore right? So I'm not gonna try to organize things by diagonalizing that. Instead what I gotta do is find something central. There's a Casimir operator. So an operator in the universal enveloping algebra which is take p zero squared minus p one squared minus up to p to d minus one squared. So you know essentially because the inner product in the Kowski space is invariant under this group this operator in the universal enveloping algebra or Lie algebra will be in the center. So that's a Casimir operator. So it acts as a scalar in every irreducible representation. So we can use that to try to organize the representations. So let me call that we could call it row or it'll be convenient actually to call it m squared. So we're only gonna consider representations where this guy acts as a positive number. So I'm gonna call it m squared. Okay now so here's a fact about the representation theory of this group. So for any constant m greater than zero and any finite dimensional for any irreducible representation of the group spin d minus one that's the double cover universal cover of SO d minus one. There exists an irreducible representation, a unitary irreducible representation unitary irreducible representation v m comma s of my group g. So I'm saying what's the data that I have to give you to choose a representation before the data that I was giving was just the value of the energy. Now the data that I'm giving is two things. One, a positive real number and the second thing is a representation of this group spin d minus one. So how should you think about this representation? Well so first of all so how you should think of it physically is you should think of it as the representation that represents states of one particle and that particle is supposed to have mass, rest mass m and it's spin. So in general spin is given by a representation exactly of this group, the rotations of space which in this case is gonna be well SO d minus one or spin d minus one. So that's how you're supposed to think about this representation. So in the Hilbert space there's gonna be a bunch of different representations that represent states that just have one particle. Yeah so maybe already I should draw a little picture of what the Hilbert space is gonna look like. I'll come back in a second and tell you a little bit about how to construct this representation. But maybe I should first draw the picture of what the Hilbert space is gonna look like. So in a quantum field theory with a mass gap which for a minute is all I'll consider what the Hilbert space is gonna look like. So now I'm gonna organize it according to this this invariant m which for one particle states you think of as the mass of the particle. And so the picture is gonna be at mass zero there's the vacuum. Then there's gonna be a bunch of states that occurred discreetly and so these are the one particle I'll call them one particle representations. So each of them is gonna be isomorphic to some VMS. So this is telling you roughly the various kinds of particles that exist in the theory. So there's one particle with mass say m one, one particle with mass m two, one particle with mass m three and all have different spins. They're given by these representations S. And then at some point there's gonna set in there's gonna be a continuum starting at two m one and this part comes from the multi particle states. So roughly the picture is if I have a multi particle state then those two states think of it as being literally just two particles think of it just a nice classical picture you literally have two particles. Those two particles can have any kind of relative momentum to each other. And depending on what the relative momentum is the eigenvalue of this Casimir operator changes the least it could be is two m one if they're sitting at rest. So if you have two particles that both have mass m one and they're sitting at rest then you get a state where the Casimir is just two times m one. But they can also have relative energy they can be moving relative to one another and so you actually get a continuum above two m one. So the picture of the Hilbert space is gonna be like this we're gonna be trying to study these one particle representations and avoiding this continuum. Okay, so that's the picture of what the Hilbert space looks like. Let's just say a word about how these representations are constructed. So actually the easiest way to understand the construction is to first imagine you had this representation. So suppose you have this representation and let's look inside of it at just the states at rest. So inside of this representation we can look at all the states such that p zero just acts as m and all the p i's act as zero. Sorry, yep. p zero x with eigenvalue m all the p i's act as zero for i not equal to zero. So that's what mass gap means. Mass gap means that there's a gap between the vacuum and the first excited state. Yeah, that's right. Having a mass gap means it doesn't have any massless particles. All right, great question. So Yang-Mills theory is supposed to have a mass gap. On the other hand, Yang-Mills theory as it's formulated in the Lagrangian framework seems to have gluons which are massless, right? Yeah, so the point is that Lagrangian, how can I say it? The sort of high energy description of the theory is indeed in terms of massless particles. They're not asymptotic states. So maybe the easiest way to say it is that the theory, if you look at it in Lagrangian looks like it has massless particles, the actual theory does not have massless particles. So when you actually solve for what the infrared physics is, it's not a trivial thing, but that theory in fact actually does not have massless particles. Okay, so yeah, so here I wanna understand what is this representation VMS, particles with mass M and spin S. And so the way I understand it is to look at the subspace inside of the representation consisting of particles which are at rest, so they have momentum, if you like, this is like saying the momentum of the particle is M, 0, 0, 0, 0. So we look at this subspace, that subspace is a representation just of the group G rest, which is spin D minus one plus the translations, R to the D. So this whole representation is gonna be very infinite dimensional because it describes, it has states corresponding to particles with every possible momentum. On the other hand, when you look at this particular subspace, this is gonna be a nice finite dimensional space, which is a representation of this group G rest. And so the picture is that, so this V rest, M comma S, is just isomorphic to S as a representation of spin D minus one. And what you can actually do, so you can actually build the representation essentially in reverse. So you can build the representation, the whole VMS as a representation of G by this procedure called induction, starting from the rest as a representation of G rest. So the moral of the story is that this representation, it looks like a big complicated representation because it contains states of all possible momenta, but everything is actually determined just by a nice finite dimensional subspace, which is the space of states at rest. So in order to understand the unitary representations that appear in this quantum field theory, it really boils down to the representation theory of this relatively nice group, the compact group times the translations. Okay, so now in the last few minutes, I wanna describe for you as much as I can the supersymmetric extension of this. So let's talk now about supersymmetric quantum field theory. So now as before, we're gonna have a Hilbert space, which is graded and which is acted on by a Lieb super algebra. And now the idea is that this is extending. So before we just took the Hamiltonian and threw in some odd generators. Now we're gonna take the whole ISO D minus one comma one and throw in some odd generators on top of that. And we're gonna study the representation theory now of that super algebra. So this is a thing where it's now very hard to say anything uniform. So it really depends in detail on which exact dimension and which exact super symmetry algebra you study. So let's just talk about one example, which is the example I'm gonna, it's gonna be relevant for the next lecture, which is n equals two comma two supersymmetry in two dimensions. And so here, let me just write out the algebra that's relevant. Do I have about five minutes, is that right? So we'll do a kind of condensed treatment. So let me tell you what the super algebra is. So I'll just give you like a list of its generators. So for a zero, the even part, we have the generators of SO 11, the rotation and boosts. Well, just boosts in this case. Then we have the translations. So P zero, I'll call H as usual. And in this case, we just have one other translation. We just have one new space dimension, one space dimension. So I just call that one P. And then we're gonna have two new generators that I'll call Z and Z bar. Those are called the central charges. So we're extending, we're actually, in this case, we're actually extending even the even part a little bit. And then in the odd part, we're gonna have just four odd generators, Q plus, Q bar plus, Q minus, and Q bar minus. So before I had two Qs, I just had two Q and Q bar. In this example, we have four Qs. And I'll just write what are the brackets? The brackets of the Qs come out to be like this. So Q plus with Q bar plus is P plus. So before it was just the Qs bracket to the Hamiltonian. Now there's a more interesting, oh yeah, sorry, I'll say anything. Yeah, thanks. So P plus minus is H plus or minus P. So they're kind of the light cone translations, but explicitly it's H plus or minus P. Okay, I gotta go to the next board, I guess, for the rest. So now the bracket of Q plus with Q minus, is Z and the bracket of Q bar plus with Q bar minus is Z bar. So here the bracket of some odd generators gives us translations. The bracket of other odd generators gives us these new elements Z and Z bar. And Z and Z bar, I should say, are central. So they commute with everybody. Okay, so now I think I'll just tell you what's the kind of basic fact about the representation theory of this algebra. By analogy with what I told you in the non supersymmetric case. So now for any M that's a real number and Z a complex number and obeying an inequality that the mass is greater than or equal to the absolute value of the central charge. Then there's gonna be two irreducible representations. VI, M comma Z of A. And so how do we think of it? No, no, no, Z bar is a complex conjugate of Z. My convention, I'm talking about a unitary representation and the unitary representation Z bar will act as the complex conjugate of Z. Some people might have said there's one complex generator called Z. So the name of these irreducible representations is they're one particle states, one particle representation where the particle has mass M and now it has this funny new invariant which didn't occur in non supersymmetric theories. So in non supersymmetric theories we had the mass and the spin. Now we're gonna have this other number, the central charge and these representations come again in two kinds. I'll call them short if the mass is exactly equal to the absolute value of the central charge and long if the mass is strictly greater than the absolute value of the central charge. So this, we don't have time to do it although it's in the notes. This bound comes from a kind of hodge theory kind of argument much like what we did for the super particle, right? The fact that the Hamiltonian was the bracket of two Qs meant that the energy had to be non negative and that something special happens when that bound is saturated. Here we have the exact same kind of situation. You do a kind of hodge theory thing which is in the notes and you get that the representation, first of all, M is bounded below not by zero but by central charge and the representation is smaller if the mass is equal to the absolute value of the central charge. So I'll just say one other thing and then stop. Sorry. So as before, the representations Vi, the mass is absolute value of Z and Z, these representations are rigid.