 Thank you for having me here. It's a wonderful workshop. This is work that I've been doing in conjunction with Tom Melia. It's in a writing stage right now. Tom's an audience. And it's basically going to be an ingredients talk, ingredients potentially, hopefully of use for truncation. I'll say that I'm also, I have no plots in this talk, but I'm currently working on some with a student, Jed Thompson at Stanford, but I won't say anything about this at the moment. So I think we've seen a lot of motivation for Hamiltonian truncation in this workshop. And I think everyone knows the basic problem, but perhaps it's worthwhile just to say a few things again. So in general, we are sort of assuming that you have some initial theory, some UV point, and then you deform it and you have some flow to some IR point, some theory down here that has an associated Hilbert space h1 and h0. And the sort of goal and hope is to be able to understand what this looks like in terms of this or anywhere along this flow. So there's two questions here, one of in principle and one of in practice. So in principle, you might be asking, you have some state which belongs to the Hilbert space along this flow, and you want to expand it in terms of states inside the Hilbert space h0. And you can ask in principle, is this valid to do, which is a question about completeness. And in general, we will assume this to be true with an assumption on the fact that the deformation here to the Hamiltonian here is some local interaction. Then Rg and things like the C theorem will sort of guarantee that this can be done. There's another in principle question here. So I will be talking about free theories, and so I'll just be going through some essentially Fox space construction. But you need to make sure that you're then having the complete Hilbert space here, and you might worry about things like super selection sectors and whatnot. But that will be, I'm not going to worry about that at the moment. So instantons, extended objects, whether or not there's any to somehow be included up here, or if they can be captured by the free field states themselves. Not to question, I don't really know the answer to, I believe it probably is true, but I'm not exactly sure. So the impractice question is, we've removed this question mark, assume that we can do this. And it's, how well does it work? All right, and that's really what this workshop is about. So, and this is not a new question, not new in the past few years, it's been around since the 80s, maybe even before that. And so you might ask, what's new in 2018? For it, why are we here talking about it? And there's been proposals on different sorts of bases that you can use up here at H0, which might be more efficient at capturing that data. So in the talks earlier today, you saw the use of conformal bases, organized at the UVCFT. And that's what I will also discuss. In earlier talks at this workshop, you've seen matrix product states. And that's very interesting, it's new to me. And I will have, both of these seem somehow more efficient at capturing the low energy dynamics. And for me, I have sort of a question about the matrix product states of if they're somehow complementary to this picture of the continuum CFT states, but that's a sort of an aside. So I will be taking the approach advocated in the previous talks that you kind of want to know the conformal data at the UV fixed point. And to do that, that really means you need an explicit construction of the states in the Hilbert space. And then there's scaling dimensions. And then from here, you can also compute the overlaps and the three-point coefficients and so forth. So the organizing principle here, what we're really saying is that space-time symmetry is the key. And this is the physicist's approach to everything since we learned what the word symmetry is. So this will really be an exercise in group theory, and in particular group theory of either the Poincare group or the conformal group. And since we're starting at UV CFT, I'll be using this, but I like to phrase things a lot of times in terms of here. And just to remind everyone, this is the isometry group of Minkowski space. And it's just the Lorentz group with translations. And the semi-direct product is here because translations don't commute with rotations or boosts. And the conformal group adds in another set. So these are your p-muse. Conformal group adds in a k-mu and a dilatation operator. And this actually then closes into a nice group, SO2d. So it has a little bit more rigid structure to it. Are you working up the level of the group or algebra? Yeah, that's a good question. So I'll discuss primarily algebraically, but since we're in the free field limit, this will be the double cover of the group is all perfectly. Okay, so a little just a background of how I got into this. I come from an EFT background. And there we like to write operators and the Lagrangian. And so you have some sort of kinetic term plus a sum on local operators. And you want some maybe operator basis for this. And these operators, you kind of want to get the minimum set. It's a minimum set. Multiple operators could describe the same physics. So what we really mean is in EFTs, we're interested in in computing the S matrix, which can be follows from Dyson's formula, a time-ordered exponential of the Hamiltonian. And so that sort of dictates if you have a Hamiltonian, which follows from Lagrangian built of scalar operators, it will guarantee the Lorentz invariance of this S matrix. So throughout this talk, you'll see me end up focusing a few times at times on the scattering states in the theory. But they are just some sort of subsector of the whole full Hilbert space. So the sort of what am I going to present to you? I'm going to present to you an explicit construction of the free field operator spectrum or equivalently the Hilbert space in four dimensions. I'm going to be using spinners in momentum space. So we'll have very much a flavor of the talk before us, which was formulated in momentum space. In fact, as fortunate enough to have zoo hair come to Yale and give a talk on Hamiltonian truncation, which is what turned me on to this. And so the map provided a very nice introduction to what's sort of needed there on the ingredients. Why am I using spinners? Well, spinners trivialize the massless equation of motion, but it's going to also give me one other thing, which is going to allow me to get polarization. So in the talks earlier today, these basis functions were constructed purely out of the momentum, which is a limitation then to just scalar particles. And so with spinners, you can, it's the only ingredient needed to capture polarizations in four and three dimensions, so you can capture everything with it. But the basic idea is still the same. We're just going to get polynomials and spinners. But I think that there's going to be a very nice organization to this calculation, which perhaps shed some light on the Hilbert space and how you might go about speeding up some of these computations that you need to do for truncation when you're evaluating matrix elements. Okay, so. Do you mean that even if I'm interested in scalars, this method could still be? Yeah, absolutely. I mean, the momentum is just then lambda, lambda, essentially. So, yeah. All right. So, yeah, it's not only going to get polarizations, I'm going to be able to get every massless particle primary operator's related description of the result here. And there's going to be a geometric intuition of it, which I'll explain now, is that almost all this can be anticipated from phase space. So, I'm going to work in four dimensions for the most of this talk and drop down to three and two later. So, in four dimensions, any physical quantity that you have will involving n particles has in its phase space delta functions for enforcing the masslessness and then a total momentum conserving delta function. It enforces it to some total momentum of the state. Okay, now, depending on what sort of coordinates you want to use, if you are working sort of an equal time commutation, you're using p0, p1, p2, p3, and there should be theta functions here. And of course, you have p0 squared or p0 is equal to the square root of p squared. And so this defines some cone and light cone coordinates. You're going to have p plus is equal to p transverse squared over p minus, so it's some parabola, some sort of geometry here. And then this aspect here defines your momentum components adding up to some total momentum and some linear set of equations. And so it defines some simplex. And to me, this is a rather intricate geometry to look at. So if you move over to spinners, so for a massless particle in four dimensions and let p a a dot, well, this is general. So any vector you can put into a two by two matrix, you have the light cone components on the diagonal and then transverse components on the off diagonal. And if it's massless, we all know that this means that this is going to be a rank one matrix, which means that I can write it in terms of the other product of two spinners, where this for real momenta is just the complex conjugate of this. So the value of this is it makes it manifest that this is a rank one matrix. So p squared equals zero is automatic. And now this whole set over here reduces to a single equation, which why don't we just write it out? It's that so I'll let me give the total momentum go to its center of mass frame, in which case it's on the diagonal. And this delta function forces you on to this sort of geometry. So just the lambda one one, some vector in the summing and then lambda one lambda two star, the two squared and then the complex conjugate of this. So this is to me now geometrically a little bit more apparent, which is that you have some vector which sums to the mass, another vector which sums to the mass, and then they are orthogonal. So you basically have two complex spheres, which are orthogonal to each other. So this is defined some manifold. And there's a very simple group theory definition of this manifold, which is that the total momentum now being equal to the sum on lambda a lambda tilde a dot i. This is invariant on a UN action, which rotates the lambdas. And so you have some UN action and these two higgs it down to a UN minus two. So you took if you we can go over to a real case if that's easier to think about you give a VEV to one radius that breaks on down to on minus one, then you have a second one which breaks on minus one down to on minus two. So this manifold is described by UN over UN minus two, which it's called the Stiefel manifold of two planes and n dimensions. It's over the complexes. And the whole point of saying this here is that physical quantities only have support on this manifold. That's what the delta functions are telling you. So there are harmonics on this as in the mode decomposition on this manifold tracking or give the sort of physical quantities in the Hilbert space states. And I'll show this more explicitly, but this is sort of the probably simple way of remembering kind of what's going to appear. Okay, so you are saying that you should worry about the harmonic analysis on this. Exactly, exactly. So this is a free theory, which means I get to construct the Hilbert space as a Fox space, meaning if I let H1 denote the one particle Hilbert space, the n particle Hilbert space is just given by tensor products of the one particle Hilbert space. And you might need to be careful about symmetrization or anti-symmetrization based on statistics, but okay. So H1 is going to be a very key ingredient, so let's spend a little bit of time with it. So what what is the one particle Hilbert space? It means that you're specifying the one particle representations of either the conformal group or the prank array group. Turns out that those two are the same for massless particles and four dimensions. And let's have a little bit of a look at that. So again, I have a spinner lambda a and it's conjugate lambda tilde a dot. And I want to tell you why these are so nice. So there's probably a variety of stories you've all read about this, and this is just another one of those I suppose. So if I go to some, I'm going to frame this as I pick some momentum and I kind of ask what else can I have? And the value of spinners is that they allow you to get everything else. You make the full use of the spinner. And so if I pick some sort of, I go to some frame for some massless particle, something like this. So, and this leads to lambda, the lambda which satisfies this is essentially square root of K. And it's very easy to determine in terms of spinners what group is preserved by this under this. So under for G and SL2C, the Lorentz group in four dimensions, K transforms as GKG dagger and little group by definition is what preserves this momentum. And it's very easy to see that these matrices are given by 1, 1, alpha, 0, and then a potential phase thing here. So this is a complex number. And this here represents translations in two dimensions and this represents rotations. There's a semi-direct product between these because they don't commute. So the little group, the well-known statement is the Euclidean group in two dimensions. Now in field theory, we want to say that you have a particle and it has a total momentum and then you're going to ask what other degrees of freedom can I give it for fixed momentum. So for fixed momentum, we by choice choose to only have a finite number of degrees of freedom in order to maintain a nice particle interpretation. So that means that you're going to specify some representation of E2, induce it into the Poincare group or conformal group, but this in general has infinite dimensional representations due to the same reason why field theories have infinite dimensional representations. This is a continuous thing here. And so what we want is something which only transforms under U1 and that's what's very nicely given by the spinners, which is that in this frame, the rest of the spinner which sits here is the modular space here. Let me call it lambda 1, the first component. Under an E2 transformation, lambda 1 goes to just E to the i theta lambda 1. In particular, it only transforms under the U1 which is exactly the requirement that we want. So this is the reason why spinners are so nice and it means that if you're going to then pick a representation of U1 that you want to do some holicity state, of course that's the interpretation of it, is that lambda 1 to the h interpolates that holicity state. So basically polynomials in lambda are just going to allow you to get any sort of representation that you want. And in particular, it's not just lambda 1 to the h which interpolates this, or let me now just go back and phrase it back in terms of this. It's that I can also add any power of momenta on top of this and this will also contain an object which transforms the same way when restricted down to the little group. So there's a notion that I'm using here which is that we are inducing a representation from a subgroup into a bigger group of some representation of h and you can reciprocally view this as a sort of restriction from g to h as some representation of g. So if some representation of g, I'm going to use this again throughout the talk. So some representation of g if it restricts down to h and contains the representation you want here, then it necessarily is included when you induce that representation into g. What is g and h in your construction? Yeah, so g here is going to be Poincare group and we'll see that the conformal group comes along for free. And then h is this e2 little group and I specifically just want the representations of u1. So when I specify a u1 charge and go up, it means I'm going to get every single one of these terms. That whole thing is actually very familiar. Perhaps it's phrasing is not familiar to everyone. So the sort of simplest thing that you can sort of think of for an example of this is to take a two sphere. So on a sphere and consider functions on the sphere which any scalar function on the sphere can be decomposed into spherical harmonics on the sphere. These are the modes. And so here what we've done is we've induced from SO2 to SO3 the singlet representation. Singlet of SO2 meaning this was a scalar function and then the modes would show up. So these are all the spin integer spin modes which when restricted down to SO2 they all have a singlet in them. Like a vector, a three vector has a component which transforms as a two vector in a singlet. So now just in terms of polynomials I can start looking at my holicity states. I'm confused. These polynomials that you wrote, these polynomials in lambda, lambda a to the h times lambda on the bar to the n. I guess they correspond to some local operators. Yeah, yeah exactly. So if this is, it's the momentum of the particle. So this corresponds to adding d mu on top of, if we pick the holicity zero one and pick the first one here it's going to correspond. I'll give a local operator interpretation in just a minute. Yeah exactly. So I can sort of draw this two-dimensional picture of I have some sort of leading operator upon which I'm adding momentum on top. Obviously these are playing a distinguished role in this game. And so we see in fact that with just these polynomials I can get every u1 holicity representation that I want. And so in fact this itself is the regular representation of u1 showing up which is consistent with this formula and it's the generic case of n equals one that u1 breaks down to nothing. And I can capture all this object in terms of just so every polynomial in lambda and lambda tilde plays a role here. So this just corresponds to the basis of some polynomial ring. And for those of you who don't really know rings and I so forth it's not that fancy at the moment just view this as a box that I'm pulling polynomials out of. Okay so Slava just asked for the field interpretation of this. So what this corresponds to is what you would normally call like the field strength a left handed fermion a scalar a right handed fermion a right handed field strength gravitinos gravitons and all the way down. And what you can think of is and sort of some you're used to opening up a field theory textbook and seeing it written in terms of raising and lowering off modes. So this is going to be an integral over the momentum lambda there's some volume factor it's important at the moment. And then the lambda lambda is exactly what you will get for the wave function p mu epsilon new commutator or anti symmetric. And if I this is e to the minus i lambda lambda tilde a dot a dot. So if you think about Taylor expanding this operator and evaluating out the origin these are all just the modes of that. Okay but what's with the stress energy? We're going to get to the stress energy tensor is a two particle state which will be built out of this and we'll talk about that. So does it mean that in field theory I will not need more than the first like the five rows of this table because because you will never have anything fundamental with more than two inches? Well if you want long range interactions yes. This furnishes a representation of the Lorentz group obviously it furnishes a representation of momentum as lambda lambda tilde the Lorentz group in here is given by rotation generators which are traceless. These are the obvious derivatives associated with it. And then what comes along for free when you're looking at this is that obviously these have a well-defined total degree in the polynomial or any term in this thing here. So the total degree is just given by the number of overall lambdas plus the number of overall lambda tilde's and so I'm going to add in a factor of two just for sake at the moment. And then I want to point out something here which is that these are one all quadratic operators in terms of lambdas and derivatives that's an important role here. And if you are thinking about this these all act on lambdas and return back lambdas so they just act on that ring and return objects back in it. I could have equally formulated this in terms of a ring in terms of derivatives which plays a role as well. So there should be another object here. This is not surprising to anyone in this room which is the special conformal generator. And this whole set here closes into the conformal algebra. That's why this factor is here. It tells you that the scaling has mass dimension one. That's all in four dimensions. So yeah. Okay. Now they only act horizontally in your picture. That's right. They only act horizontally within that picture up there which goes on to also prove to you that every massless particle in four dimensions is uniquely an irreducible conformal representation as well. So if I want to the other nice thing that you notice about these basic terms here is that they are all holomorphic. So it's quite easy to see that they are annihilated by K as in their primary operators not surprising to anyone. But I want to like a simple way of isolating these sort of primary operators. And the way I've drawn this is that momentum keeps coming off to the side here. So the space of primary operators if I call that ring R1 and then subscript P can algebraically be captured by just modding out by momentum. So this gives a mathematical definition to the physics question besides momentum what else can we observe? What are the other states in the space where now we've translated in English this to modulo momentum. That's its proper definition. So this here just means take polynomials modulo momentum. And when you do this this isolates out the just these components the one lambda squared etc. Okay the last thing I want to point out before moving on is that in addition to these sets of generators here there's another set of quadratic generators which commutes with all of them and that's something which labels the holicity. And so if the total holicity here is always determined by the number of lambdas minus the number of lambda tildes. All right so mathematically lambda dot d minus lambda tilde dot d tilde. This is a U1 generator. And the important thing here is that this object was invariant under this U1. So it's not surprising then when I mod out by the invariant theory thing here I'm fully aiding it into sort of the space of U1 representation. All right now I've hammered to death this one particle ring because oh yeah. Representation of the lambda tildes related to the oscillator representation. Yeah that's actually I will get to that that plays a very yeah. It's basically the same thing right? It's basically the same thing yeah. Except maybe piece quick. So yeah so what ball is saying is that basically any polynomial ring is essentially a fox space that I'm just considering objects if it's a ring polynomials in x and then x square this is just like adding one extra energy level it's our way of doing fox spaces. So and then in particular if you look at this and notice that lambda d delta ab it's very much a fox space. Commutes to it. And that plays a very important role in terms of the role of these as quadratic generators and that's a story that I will get to maybe if I have time. But yes okay so after beating the death one particle I hope what I'm doing is sort of clear I'm trying to get away from I want to go as algebraically as possible just to get very simple like algorithms that you can use to pull these out and everything that I've said here will generalize nicely to n particles. So for n particles what do you do? You have a spinner for each one or equivalently a momentum in a holicity state and you could of course use as a basis for n particle states just the sort of direct product basis plain wave basis and so on. But that's what's been used in the past and that's not what this we've seen that we can get sort of better convergence so what I'm going to do is I'm going to couple these particles together and decompose them back into conformal representations. So I couple them to some total momentum p mu or p a a dot and then ask the same question as I asked back there which is besides momentum what else can I see and that gives you the answer of finding all conformal primaries. So I will say what that looks like now is as follows is that I'm considering polynomials in lambda and lambda tilde i runs from one to n the claim is that this modding out by momentum will just leave you with the space of primaries i.e. the polynomials will be annihilated by k. So to get a feel on that what this how that whole decomposition looks like what you're doing when you take any polynomial ring is its basis so you're taking the objects here and just repeating them it's polynomials which means taking a symmetric product of them this should be okay that's okay and here I had two separate ones which themselves have two indices so I will split this ring this sum and I'm not going to use direct sums all the time here because it's lazy for me so that study maybe the two particle question first because I'm confused by what does this mean operation if I take two particles yeah understand what you mean construct like all derivatives and these are the descendants now we want to subtract them away so on a certain level I see a wave function where there is a derivative and there is something else now I have to pick something else which is a primary what is the I cannot just take any arbitrary combination which is not the descendant and declare that this is a primary there's just going to be one very good yeah so so it's not specified by this algebraic so this specifies pick any representative in this equivalence class and that's your primary but you don't have to specify any representative there's just one particular reason yeah that yeah that that's right so I've been a little bit loose on my language this is just a modulo so it's a representative class but there's a very natural candidate which will arise and it'll be specified in terms of a UN represent I want to say that story quickly and then get to actually looking at a two particle example or something like that so you can understand it as quickly as I can so anyways the whole point of okay so I have essentially need to look at the decomposition of these spaces and if I say that lambda AI let me represent a little bit more abstractly this is the space for the index a which runs from one to two and then w is a space for i which runs from one one to n and this is the sort of magic formula in this talk which is that this decomposes into a sum on partitions of n little n of now these are sure functors which are symmetrizations of v left cross the same symmetrization pattern applied to w all right this has a very simple physical I mean not physical operational meaning what I mean here is I've taken n copies of lambda that's what the symmetric product means and if I apply a symmetrization procedure to the A indices it's going to apply the same symmetrization procedure to the I indices so that's why these are going to be in the same representation and then this sum is restricted such that then this is a partition it means that the number the sum is restricted by the sense that I can't anti-symmetrize I can only anti- symmetrize so much so since a runs from one to two I can only do it like rank two anti-symmetric so this is restricted to the number of rows in row is a partition is only two so at the end of the day I will have in this whole object things which look like something a symmetrization pattern on v left for lambda AI which tells you the Lorentz spin on the A's some other symmetrization pattern on v right for lambda tilde and then so these are irreducible as Lorentz representations but here as the UN objects basically the I indices it's not irreducible and this is where pulling out the primary comes and so what you essentially have is a.ij1 so the only way to reduce this object in terms of u1 representations is so this has some definite symmetry pattern is you could start contracting the indices between here and if I contract any indices I'm going to pull out a factor of p so the claim is that the object which combines these two tableau into a UN representation without contracting indices is primary and with that there's a very simple rule for it which is that if this represents the tableau symmetrizing I don't know two states here and actually why don't we just do one because it will be very easy to see so what I mean here is lambda AI lambda tilde a.j can be written as lambda lambda tilde minus 1 over n delta ij pAA dot and then plus the 1 over n delta ij pAA dot so this is equal to the adjoint representation plus a singlet the singlet involved pulling out a factor of p and that's my this has no factor of p in it it's the natural candidate for modding out by p this is primary and it's very easy to see just by picking even some explicit weights inside of the representation it will be holomorphic there will be a component of it which you can just see is trivially annihilated by k so what's fascinating about this is that it means that they sort of so you take kind of the object glued together in the maximal sort of way out of this and that's primary but that thing determines exactly the spin representation in vice versa and so they're controlling one another so this so the conformal group or we could say it is sl2c for this primary piece and this un together are sort of dual in the sense that their decomposition in here determines one another more precisely you could look at this and say that the chasm years of one group are determined by the chasm years of the other group which you can do by explicit calculation with the form of the generators here so the upshot is that the representations as un representations um actually let me maybe not give all that detail because i'm going to end with that we got to see an example or else this makes no sense so in this two particle ring so why don't we pick the stress tensor this was asked about before so the stress tensor t mu nu is in spinner indices it's going to be symmetric combinations of these two okay so the tableau that is ending up bringing in was one which symmetrized the left handed indices the a indices and then there's one which symmetrizes the right handed indices stress tensor of what theory that's you just ask all the questions i want to get to so so the thing is that what this is at the level here is that i started with an object which contains every holicity representation i took a tensor product of it so it's just building the entire Hilbert space of an arbitrary you have to select your theory in here and that's it's important to figure out what that means here so you're going to need to be able to figure out how to interpret what comes out of here in terms of particles and we're going to see something which is this explicitly contains only the vector fermion and scalar stress tensor which is not an accident so as we have this u2 as u2 representations i'm going to take this tensor product and put them together in the one which forms a primary and so this as some this is i don't know how i'm going to explain this very shortly is that if i take the dual of the i'm just going to raise the indices on lambda tilde i and that gives me the dual of it and so now everything will have an upper index and i can view this as this and i want to take the one which causes no contractions and so this thing is the claim is primary it's relatively easy to check that's annihilated by k you just pick a single state in there and so i have an overall object which has these indices on it completely symmetric and ijkl and it's a so and these run over one and two so in particular there's five states inside of this and if i pick ijkl is equal to one one one one one one two etc so picking in for example all ones on that it's automatically symmetric this thing is lambda one lambda one lambda tilde upper one which corresponds to lambda tilde lower two and so if you come back to this thing up here it involves the first particle the one with one on there has two lambdas so it's the left-handed field strength so there are say scar three theory if i have a direct sum of two three theory i want to compute primary by hand then you realize that there are primaries so momentum enters in primaries i'm saying so it's not not obvious that momentum we understand that that's not true not true you can have a combination so once you want to compute a primary genetic primary in say even in the direct sum of two free field scalars then you get clear you get trigger parameters like phi one square phi two square and so on but then there are primaries which have been a combination where derivative do enter now which are some combination will be annihilated by k and the other would be the December so the question is do you do you can you see this primary here i have the thing that you would not be able to find this sort of so maybe a primary which is annihilated by not annihilated by k yeah they're not in a neutral way but here he has for can you consider in your scheme theories where there are two types of scalar particles yeah yeah but they're not so do you do i do that i mean this is a very simple statement yeah so there are primaries that have momentum into when you go when you build the primary out of you know elementary fields derivatives can enter let me tell you as simple as that yeah right so do they appear here somehow yes you're going to contract the lenses with the lambda tildes and there's those are going to be the derivatives right when one time the tildes come together are you thinking that even appears in this sort of sense that here's an explicit form of momentum a total derivative inside this primary you have a contractual you will you will declare that it is at the same so i misunderstood so well i would say here that it's like actually removing the total momentum component so i i think i need to see an explicit example of what you're talking about this no you just means like very simply like in this in the stress tensor you're going to have defi defi that's yeah oh yeah so so what okay so this is the stress tensor and i maybe we should say offline because it will take a few things to say the indices out in here but you actually get to like get away with not using that specifically because the derivative goes over to p mu which goes over to lambda lambda tilda and so like there's a way of this if i wrote out the stress tensor here for the scape this one turns out to be the scalar it may not look immediately obvious to you that it's what you and it's only because what because let's take a theory which contains so slava this one here is this one's very easy to look at so i have two things here it's f left with one particle and f right for the second particle so your identity contains two types of f or two types of scalars then i expect to see several fields yeah so good there are maybe i potentially oversold things but that's a very simple thing to include by just adding another index to this and playing that game okay so i don't have a beautiful story for arbitrary all right so there's some modification yeah there's some modification to it so but what you do need to know is how you interpret what the state is made of and that's all labeled by it to u n representation so that that's just the number of basically lambdas minus lambda tildes for each i tells you the sort of particle type which shows up in there and if you play this game here this would be like a right-handed uh fermion for two left-handed derivative acting on right side the derivative acting left right on one left-handed this one is more or less specifically because of what was just asked a second ago but i'll write it in the more familiar way where everyone knows that this is sort of what's this one's interpreting and so at first it also looks like you might need to apply some sort of symmetrization procedure to if these are supposed to be identical particles so like this itself is actually automatically symmetric under one two exchange and so but all of that happens when you add different types of flavor is that u two goes to u n where n is the number of flavor of scale as you have right yeah this doesn't have to be u two here yeah sure um there was one of the things that's special about the sort of this explicit example which you all know is that the stress tensor is conserved so instead of having nine components in four dimensions it has just five components from this conservation equation it's no accident that the u two representation is dimension five here and it's also no accident that you only the stress tensor as determined in this sort of procedure the one which has that symmetry only contains vectors fermions and scalars as those are the only local stress tensors you can have for free theories in four dimensions so i find that sort of remarkable that it's being controlled by this u two as well okay in the interest of time i need to say just uh that in d equals three it's basically the same story just take lambda tell them and delete them in all my previous equations call the u n it's actually now an on because you can write a momentum p a b equals lambda a lambda b in three dimensions the same same sort of story here but now if i have an i index on them these are real so it's on if i if you look at this whole story all these modes which show up in either the on representation or u n representation they all correspond to exactly the harmonic modes on the u n minus u n minus two or o m o n minus two and this at the n equals one level just as before we started with the u one little group this was a z two little group so you're really only getting out scalars and fermions which again are the only free cfts that you have in three dimensions perfectly consistent and so the on again like the un is a direct generalization of that taking n particles in the little group and putting them into the full group that they sort of in terms of computations of things i've been a little sloppy throughout this by disforcing it algebraically and calling these states but this is a Hilbert space this needs to be a unitary representation there's an inner product which you need to use to compute anything that this audience wants to do and so i won't say too much but let's say in 3d again you had these sorts of things here dot dot dot this disappears but in particular i want to focus on also this e to the ip dot x and then some polynomial in p maybe p i that from our previous talk this just transforms into many a product you know what i'm saying okay transforms very nicely into i'm saying 3d this sort of quadratic thing again in the numerator or the exponential which means you have Gaussian things which you're computing and so everything turns out to be sort of wick contractions again and it's very familiar and i think that there's sort of combinatoric structures there which become more obvious at least to me the simplex was confusing so to sort of summarize and say is that there's a very nice way in my opinion of formulating how the primary fields and various space time dimensions go in 40 we saw the role of this s you to the conformal group cross un and 3d it's the conformal group cross on and then in 2d i didn't say anything about this but it's again the conformal group sl2r and it's also on and from the previous talk it just amounts to turning the momenta in the spinners which i just mean to say turn this into some quadratic thing we're here maybe i'll call it u just replace those variables and you'll see this entire pattern show up in everything that you and 2d so instead of working with p minus call that u squared the when viewing this as harmonic oscillators it i didn't get into a story about that which i think is nice and complementary because there's actually different ways of computing in here that go between the spaces that's worth exploring in terms of speeding up computations but it also makes it clear what you would want to do if i want to do supersymmetry which is just add in appropriate creation and annihilation operators which have anti-commuting statistics just as we had sort of lambda d which have commuting statistics and you can probably pull out the representation story that way it may or may not hopefully it's sort of obvious to everyone in this audience who's worked a little bit with cfts but the ope data is exactly what i'm computing here for the free field which is that so for free field theory the ope is literally just doing the tensor products of conformal representations and decomposing them so you should be able to pull out the cft three point coefficients out of this procedure it'd be interesting to explore if that can be done in a sort of fast way which has potential other uses including maybe uh into the sort of input assumptions into the conformal bootstrap so i think i'm over time thank you okay it's the first exposure so it's clear that something deeper is going on but just to understand so suppose that i'm at equal four and i know that my even theory just has only scalar fields or you know or equal to the only scalar field and i i don't want these vectors fermions flying around i just want can i modify your procedure in a way that i only get operators made out of scalar fields on the nodes or should i do this global computation and then in the end throw out the the biggest part of the field because they contain some fermions which i did not have in the first place um so i presented a very global computation it after it's at the end of the day you're just going to write down some tableau and then what you want for your state if you have a specific thing in mind it's very easy you just fill the tableau with the sort of correct helicity numbers you can view this as a computationally you can do this automatically by looking for using the un generators in terms of so the un generators themselves are lambda dj minus lambda sum on the a indices the torus generators of this if i call this eij determine it's the helicity of each individual particle so you can just look for the ones which eii minus the helicity you're interested in acting on just the block pulls out the sort of state that you want that's a very brute force way i haven't given a lot of thought to implementing that in a fast way and it's absolutely something that would need to be thought about in d equals two can i understand the vertex operators in this framework uh that's a good question i don't think so it's very hard since this is all again locally based but yeah they're not hiding in there somewhere i i haven't given it any thought i thought about this when an hour ago when this was being asked a lot that i have to say i didn't come up with anything clever my guess would be no can i ask you a question on myself i'm sorry it was enough yeah so i was a bit confused about this statement about the opeco efficient so normally you think that you can freak theories these opeco efficient a encode something kinematical so normally they're very complicated products of gamma functions and in general they're not known for three arbitrary primaries and freak theory you don't have a simple way of computing them so somehow all this complexity should be reproduced by by your formulas right yeah that that's i would say there isn't a ton of free lunch necessarily but you have to go through the sort of usual exercise of how do you compute clepsch gordon coefficients which is you take the chasm you're written in terms of the other one again it's like something i've done maybe with a few of these but haven't exploited so no you i think it's promising on some of it but there is some algebra i'm hiding as well that's right um no not that i've thought of it be like interesting again i think that there's lots of very easy routes to go on here um yeah basically i've just given you the generators of the global conformal group so it's a building up on the slava's question so there has been just to be concrete in the in genetic dimension if we say free scare you know that they or we have a unitary bounce we give you a highest in currents which are the only specific no list of tower operators which people have been able to compute to the fact that they have conserved currents the precise form in terms of the double trace by five so in terms of beginning by our dreams and we will excellent question so team you knew is the first one in that entire tower and then everything in this ring except for the holomorphic polynomials so this if you compute d here it's again just the number of lambdas lambda tilde plus the overall constant is two and so if it's not holomorphic this goes over to j one plus j two plus two which is the unitarity bound so you have all these short multiplets which are these currents and you're able to compute in terms of the five the elementary consequence the specific form I mean you you extract it from these yeah it yeah that if I if I can use that also for there's something there which is that you're essentially looking at these quadratic terms which can turn the conserved currents if this thing here has taught me anything I wish I got to say it more is the quadratic nature of these operators and so I myself am curious as to if pi is the canonical conjugate variable to pi if these have any sort of meaning and so there's an overall symplectic symmetry which these are actually hiding in and this is sort of an infinite dimensional version of it which if it's integrable it would actually integrate to this sort of thing so I've been a little bit curious if there's an interpretation of things here of that and I should say that I think I forgot to mention this is aspects of the story at least in terms of this like duality go in the math literature by how duality or reductive dual pairs and this is called an oscillator representation and or seagull shale while representation who actually introduced it in field theory originally in this form but I don't know much about it so yes there's something I noticed that was very special about this two particle space you do get all the conserved currents and get them explicitly and it one starts to wonder if you can actually just redo this whole procedure now in an infinite dimensional way and kind of look at it and get something from there