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SPEAKER: So welcome, everyone. This is the second of the three beta lectures this week. Before we get started, I want to remind you that the last lecture is tomorrow over in Kennedy Hall at 7:30. So our speaker continuing beta series is Andy Strominger and the title of today's seminar is the Holographic Principle in Flat Space.
ANDREW STROMINGER: OK. So yesterday I gave a physics department colloquium, tomorrow will be a public lecture, and today I'm going to be mercilessly-- I want to talk about some recent results, which I'm excited about. And so it will require some actual mathematical equations, which we didn't see much of before.
But let me start by just giving less than a cartoon of what's become a rather large and active field within which this result emerged. And it's a result-- just to say in the beginning, it's a result which I think makes some substantial progress on this question of what are the symmetries of nature, both in how we characterize them and what the symmetry group-- though we certainly haven't solved the problem.
So the starting point that has enabled a lot of-- that has provided some really powerful mathematical tools is to we start by considering the S matrix. So here's past and all infinity, here's future and all infinity. And we have some particles coming in, they interact, and then we get some particles going out.
So the idea is to map this to a correlation function on the circle where for each incoming particle you mark a point on the circle-- sorry, on the sphere-- and similarly for each outgoing particle. So this 3 to 2 S matrix would be a five point correlation function where we also have to label the points in the correlation function according to whether they belong to the in region or the out region.
Now, why would we want to do this? Well, one thing that's really easy to understand is that here, the Lorentz group is S031, which is SL2C. And SL2C-- of course, we're just defining this to start out with is a rewriting of what's over here.
And that SL2C acts as the conformal group on this two dimensional sphere, and therefore this correlation function will look a lot like conformulation functions in a two dimensional conformal field theory. And even more significantly in the last five years, [? Kichauso ?] and I proved that using a subleading soft theorem that, in fact, in quantum gravity, which is our real target, there is an even bigger symmetry.
And you get not only SL2C, but you get the whole infinite dimensional enhancement to the full infinite dimensional conformal group. So there's going to be a lot of similarities in the study of this object, to the studies of two dimensional conformal field theory. And just to say one other thing, what point-- let's take a holomorphic coordinate Z on the celestial sphere.
And what is z equals to? So we'll label these by z1, z2. So zk is equal to p1k plus p2k plus ip2k over p3k minus i p0k. And you can show that if we do a Lorentz transformation on the p's, then z goes to az plus b over cz plus d, where ad minus bc is equal to 1. And these are arbitrary complex numbers, so that's SL2C. So SL2C, Lorentz transformations act very simply on the celestial sphere.
Now there are different points of view that one can take on this. One is that we're just rewriting scattering amplitudes in a different basis. In fact, Dirac proposed doing this when he was-- I think it was part because of the outcry over the outrageousness of the Dirac delta function, but he didn't get very far with it. But we're getting more mileage out of it. Now you didn't get much use out of it, so you just think of this as let's rewrite this, the S matrix, in a different basis and see what we can learn, see what properties are evident in the new basis.
A more far reaching goal is to understand flat space holography. And so the natural gas that you would make here when you have to duel theories-- usually you would have a quantum gravity theory in the bulk, the bulk of this case being four dimensional Minkowski space, and then you would have a dual theory without gravity on some smaller space, which is in some sense a boundary.
And the guess here-- and the symmetries are the first guide to how this can work. And so the guess or the hope would be that in quantum gravity that the dual, the flat space holography, the dual is some kind of conformal field theory that lives on this celestial sphere. And I say some kind because as we'll see, these conformal field theories we don't understand very well, the locality properties and positivity properties and so on. So they're conformal field theories in the sense that they transform under the conformal group, but in other regards they differ.
But I'm really, here, going to be thinking in a bottom up approach. And I am going to just use this rewriting, and this is-- so there's an outstanding problem which I posed at colloquium yesterday is, what are the fundamental symmetries of nature? And this seems to be there are many formalisms to introduce that in. There's the walled conserved charges and there's S matrix relations and there's all kinds of ways that you might try to answer that question, but this seems to be a really streamlined way to do it.
Now just one more word about this and then I'll move on. So when we do this, over here we always work in momentum basis eigenstates, and that is not a natural thing to do from the point of view of the celestial sphere. In fact, translations do something very odd over here. They change the conformal weight of an operator. It's better to work in some kind of conformal primary basis, the kind of thing that we're used to in conformal field theory.
And the way you get conformal primaries-- so you have conformal primaries of suppose we have an endpoint amplitude. We call this the celestial amplitude, z1 to zn. And of course, we can have spin indices and so on, which I'm neglecting here. It's just equal to a Mellin transform of the momentum space amplitudes p1 through pn.
Now to evaluate this, we have to solve for the p's in terms of their z's. It's a couple steps, not too hard. And obviously they're omegas. And then do this integral and we get something which depends only on delta and z's. So we traded. Here these p's are each three numbers, because they all have a mass shell constraint. So we traded the three p's for the z's, which are the point on the celestial sphere where the particle exits. And it's conformal weight, not its energy. Yes?
AUDIENCE: So by going to null infinity, are you assuming that this theory only has massless particles?
ANDREW STROMINGER: Great question. So yes, when you go to null infinity-- well, yes and no. I am here considering only the case of massless particles just because it's simpler. But it also works, as it must, for massive particles. And I'm not making any assumptions here, you know? I'm just rewriting things in a new basis and I can certainly rewrite massive particles in a conformal basis.
And instead of a simple Mellin transform-- do that instead of using a simple Mellin transform, very easy to get a massless particle into good formal primary basis. Instead of a Mellin transform, you have an integral over the momentum space hyperboloid, sometimes called ADS, of a bulk to boundary propagator. And that gives you a massive particle in a conformal primary eigenstate.
And interestingly, massive particles and conformal primary eigenstates can make it to null infinity. Sounds very counterintuitive, I know, but it's true. It's only finite energy that prevents them usually. But if you take a massive thing and accelerate it forever, it will go to null infinity and you can write down solutions.
We have written down solutions to the massive wave equation and you compute the Klein-Gordon current and it crosses null infinity. They're not finite energy states, neither are these. We're not working in energy organizations.
AUDIENCE: And then you could still use them to account for [INAUDIBLE] as matrix elements for [? finite ?] energy states.
ANDREW STROMINGER: You could by Mellin transform-- by convoluting back. It's a repackaging of all-- but it's a nice thing that in this repackaging of the theory, everything can be thought of as living on this celestial sphere. OK. So now I'm going to-- yes?
AUDIENCE: You seem to have lost a degree of freedom going from [INAUDIBLE] momentum to 1p and 1 delta. Is the other degree of freedom a z bar or something?
ANDREW STROMINGER: z is complex. z and z bar. It's the coordinate on the sphere. OK, great. So now I'm going to skip like 20 papers and I'm just going to write down a formula and explain it and ask you to accept it and work from there. So we now think about the theory, the four dimensional Minkowski theory, in terms of operators on a sphere. And so therefore we can talk about operator product expansions.
What happens when we take-- of course, that's the bread and butter of conformal field theory, understanding operator product expansions. And we can talk about operator product expansions and of course, those operator product expansions have already been discussed in the quantum field theory literature. It's the collinear expansion, which people study jet physics and so on are-- well, everybody's interested in it, I guess. And so let me write down the formula for the operator product expansion-- and I'll explain what everything means-- of two positive helicity gluons.
And I guess I am going to neglect the label, which in the end doesn't matter that tells me whether it's an in or out gluon. In our paper we keep track of all those labels, but it's not conceptually important. OK, delta 2. z1, z bar 1, z2, z bar 2. Goes like 1 over z1 2 minus ifabc Euler beta function of delta 1 minus 1, delta 2 minus 1 times o plus c delta 1 plus delta 2 minus 1.
And I remind you that the Euler beta function, which comes in all over the place, gamma of x, gamma of y, or gamma of x plus y. OK. So this formula says that if you take two conformal primary gluons-- and actually, conformal primary, let me use the word boost eigenstate. They're eigenstates so that's the same thing. They're actually highest weight eigenstates, they're not just any old eigenstates.
Well, we call them boost eigenstates. So if I take two gluons and boost eigenstates with color A and color B and with positive helicity I let them get near each other, I get this. I get a third gluon. And this weight here is just dictated by the conformal invariance, which is Lorentz invariance. And then there's this coefficient. So this coefficient was first computed by Steve Berger and Taylor by direct transformation of the known results for collinear expansion.
But later it was interestingly realized that this could be computed by no work and that consistency with actually translation and variance which shifts the weights of these operators completely fixes it. So it's, again, showing you how certain things are more transparent. And when you work in this language. OK.
Now let me just say, because these things have spin one, they have a left and a right conformal way-- well, everything has a left and a right conformal way-- And it has h and h bar equal delta 1 plus 1 over 2 delta 1 minus 1 over 2. OK. So this is a formula which I didn't derive, I just tried to tell you what it was and define all the things. There are a lot of papers on it deriving it in different ways, but before I go on, let me just ask if there are any questions about this formula. Yeah, Anna?
AUDIENCE: Is the right outside supposed to be the limited of left hand side when the z's collapse on top of each other?
ANDREW STROMINGER: Yeah, right. So there are corrections here. Good. Yeah, I should have said that . There are corrections here which are suppressed by powers of z and z bar. So this is the leading term in the operative product expansion. One can also write down the whole op block and get the whole thing. There's some exact expression where you put an equals there and you get an op block with. Yeah, Tom?
AUDIENCE: So I know you said it doesn't matter if it's in or out, but why is that? How do you understand that? Is it like there's two copies of a CFT, or why did it collapse to one? It seems like you would get a very different operator, the ope, depending on whether you took them together like this or on top of each other.
ANDREW STROMINGER: Well, why doesn't it matter whether they're in or out? Yeah. I would think of it this way. This expression-- it's a little backward reasoning. I'm not sure from the first, but a little backward reasoning. You notice this thing has poles and these poles are going to be related to the soft theorems. And so this whole expression is controlled by the soft theorem.
This weird fact engaged there that the soft theorem-- I mean, it's well known from, say, MHV amplitudes that they're completely controlled by their soft limit. And in soft theorems if you have some hard charged particles coming in, you get the same answer if you have a soft photon if you attach there or attach there. There are some subleading corrections.
They're not identical for in going and out. If you look at the subleading corrections to these expressions, they're different. They're not the same operator, the in and the out. But the leading poles, which is what I'm interested in now, are the same for in or out. I guess I don't have a great answer to your question, but it's true.
AUDIENCE: I mean, this is for the 2D conformal field theory. I apologize, I came in late so you may have addressed this already. But from what I inferred from your abstract is that you were going to use the subgraphics on theorems to find some holographic mapping onto a 2D conformal field theory on the celestial sphere, and I was just wondering-- help me as you start going through this. That's kind of jarring because you're two dimensions down in your holography rather than the usual just one and so I'm wondering what's specific to this case that I should be paying attention to in advance that will explain that to me.
ANDREW STROMINGER: Well, it's not true that you're two dimensions down. So for example, in ADS5 cross S5, you start at 10 and you go down to 4. So we don't always lose a dimension. And that's because some dimensions are compactified. And the relic of the compactified dimension here is the delta. The delta, which is, at this point, continuous, is, if you like, can be thought of as being related to the expansion of the modes in the extra dimension in the-- you can think of it like this.
AUDIENCE: [INAUDIBLE]
ANDREW STROMINGER: Yeah. So you can think about it like this. So we have anti-de Sitter space. So this is Minkowski space and we can slice it into anti-de Sitter space top and bottom. We can do a hyperbolic slicing. And then if we just concentrate on one of these anti-de Sitter slices-- it would be Euclidean radius 3-- we would expect a conformal field theory on the two dimensional boundary with a Virasoro reaction. Then what do we do about this extra dimension? Well, we kind of Fourier transform in that dimension and that is what leads to this delta index. This delta index rebuilds that extra dimension.
OK. So this is the formula and now one of the salient features of this formula is that it has poles when delta 1 goes to 1. And these poles-- in fact, it has poles for delta 1 equals k equals 1, 0, minus 1, dot, dot, dot. You get poles and these poles are very interesting. Let's try to understand what these poles are. So let's first define an operator, r1 of-- let's first talk about the pole when delta 1 goes to 1.
The poles are all different in an interesting way, and these poles are going to be related in a very simple and direct way to the infinite tower of soft theorems. The first one is going to be the standard soft gluon theorem at tree level soft gluon theorem, and then the next one is going to be the subleading one. And then the new ones are ones that people have sort of mumbled about but not said anything very precise about, but we'll now be able to get a very precise formalism and moreover get the algebra of the whole tower.
So we define an operator r1a of z. Maybe I want to put the 1-- I put it there now. r1a of z is defined as the limit as delta goes to 1 of delta minus 1, o plus a delta of z and z bar. So first of all, why isn't this operator just 0? We're multiplying by 0, the o's were supposed to be nice operators.
We're multiplying by 0, why would we be interested in such a thing? Well, the point is that those o's have poles in their operator product expansions. And therefore when we multiply by precisely this amount, this operator that sits at the pole will have finite op coefficients. And moreover, notice that its weight is 1 comma 0, so we expect it to be a current. And indeed, that is one of the beautiful things that happens when you go to the celestial sphere.
The soft theorems are generated by currents, holomorphic currents like this. In general, they're Kac-Moody currents. And we're going to see in this talk that, in fact, there are a whole tower of w infinity currents and all the consequences of soft theorems are equivalent to the consequences that we're used to of a Kac-Moody symmetry on a two dimensional sphere. Yes?
AUDIENCE: The delta minus 1 should get preferences, right?
ANDREW STROMINGER: Yes.
AUDIENCE: And in the op extension that you wrote above, it only depends on the z, not z bar. Is that intentional?
ANDREW STROMINGER: It's just true that that's the leading term. We're doing the leading pole as z1 goes to z1 and z2. Very importantly, we treat z and z bar independently and that is something which has become ubiquitous, so ubiquitous, it's usually not even stated in the so-called amplitudes program, and it amounts to going to 2 comma 2 signature.
So effectively, we're going to be working in 2 comma 2 signature and we're used to working in a Euclidean signature than 2 comma 2 signature, but there are very definite rules. I think Mason and Skinner wrote a very precise paper about this 20 years ago about how you take amplitudes and 2 comma 2 and analytically continue them back to 3 comma 1. And in fact, in many ways I think it's nicer than going from Euclidean space because in 2 comma 2 you can stay on shell.
AUDIENCE: That might be the answer to the question I was asking earlier that maybe they look different when you come back to 3 comma 1.
ANDREW STROMINGER: OK, there is a very beautiful answer which I can't resist giving since you asked it. When you go to 2 comma 2, the boundary-- so there's only one [? scry. ?] There's only one null infinity, there aren't two. And it's a torus, not a sphere. And there are two patches. It's a Lorentzian torus and there are two patches.
And when you analytically continue back to 3 comma 1, one of them goes to the past celestial sphere and the other one goes to the future celestial sphere. So it's actually a much nicer way to-- there's a lot of calculational techniques, advantages that had been exploited in the amplitudes community by treating spinner. You know, they go to the spinner helicity formulas. And I guess this goes back to Penrose.
AUDIENCE: [INAUDIBLE]
ANDREW STROMINGER: Well, when you analytically continue-- just when you analytically continue and conformally compactify, that's what happens. I can't say it in two seconds, but that's what happens. So the operators on the torus corresponding to in and out states live at different points. OK.
So now if we take the op of this operator, ra1 of z, with any-- let me call this a soft-- I should really say conformally soft, but it's too much of a mouthful. o plus b of delta of z and z bar, we get minus i z1, z bar 1, 2, 2 bar. We get minus ifabc, z1, 2, o plus c of z2, z bar 2, which you will recognize is the standard. So this is any operator on the sphere.
Let me call it a hard operator in a conformal basis. And you'll recognize this as a standard ope relation for a non-abelian Kac-Moody symmetry in a two dimensional conformal field theory. And so if we start taking correlation functions, this is a current. Looking at correlation functions we can look at contours and we can integrate these currents around these contours and we can get identities by deforming the contours around the two sides of the sphere.
And that is an elegant reformulation of all the identities on scattering amplitudes that we get when we study space time scattering amplitudes. OK. But now there are other poles. What happens when-- what do we do?
AUDIENCE: Why did it become singular when delta went to 1? Like if you didn't put that 1 minus delta factor in there you would've gotten something more singular.
ANDREW STROMINGER: Yes?
AUDIENCE: Why does it [INAUDIBLE]?
ANDREW STROMINGER: Well, it's the conformal. If I had been working in energy space, I would have gotten a pole in w. That's the famous soft pole in the soft theorems. If you insert a soft photon in a S matrix element, you get a pole as its energy goes to 0. That pole becomes a pole in delta when you go to the conformal basis. OK, now what about delta 1 goes to 0?
Well, when delta 1 goes to 0 we get what we want to define is the limit. When delta goes to 0 there's not one current, but two currents, and we write it as delta. Sorry. We take the limit as delta goes to 0 of delta oa plus delta, and we expand that as the square root of z bar times ra minus a half z plus 1 over square root of z bar ra plus a half-- wait what, did I do here?
Of z. Of z. ra plus a half of z. And so we get a doublet of currents here, and these are closed under the action of SL2R bar. And notice now that h and h bar become a half minus a half. So the h bar weight here is becoming negative. And that is the kind of thing that one usually wouldn't consider in ordinary conformal field theory, negative conformal weights.
But when you have negative conformal weights, you have nice, finite dimensional representations and a negative conformal weight minus a half, the SL2R bar. You have an operator which is in a doublet. So you get a doublet of subleading soft gluon currents, and this was known. That there were two I think it was already known by Francis Lowe in some sense in the '50s.
Mina Himwich and I made it very clear that there were two separate currents with ward identities, but we didn't understand what their relationship was to another. And now we've understood it all in a more systematic way that these are just in SL2R bar doublets. And if you look now, these are a of z's, both of them have ope's with the o's, which we could easily write down, and those ope's are equivalent to the subleading soft gluon current. Or if we were doing qed, the subleading soft photon current.
So more generally, there's a whole tower. Now the fact that there's a whole infinite-- so we see that we can get poles at any negative end. We can get an infinite poles at any negative end. And the fact that there are infinite tower of soft theorems-- I think it's kind of in Penrose and Newman's early stuff that they didn't quite understand what to do with it. But more explicitly, it was in recent things by Hamada and Shiu, and Guevara, and Lin and Jiang.
People have been writing it, but when you write it in energy space it's very confusing. And in fact, in energy space there are-- nobody knows how to define the soft S matrix in the usual translation basis because the double soft limits in energy space, when you have more than one thing, it doesn't compute. So we don't know what-- here in the formerly soft limits, there is an unambiguous S matrix, including the boundaries where the weights go to integer values.
So now we consider a whole tower generalizing these things as the limit epsilon goes to 0 of o plus comma a of k plus epsilon z, z bar, which we write as the sum n equals k minus 1 over 2 to n equals 1 minus k over 2. Remember that k is-- so n is always 0 or positive because k is getting negative. Of rak comma n of z, z bar to the n plus k over 2. All right.
So these are all SL2R multiplets just keeping the z dependents straight. These are all SL2R multiplets of higher and higher, but finite size, as k grows. They go like 2k plus 1, I guess. Sorry. Minus 2k plus-- something like that. OK. So these are all soft currents and they will all have finite ope's and so we get an infinite tower of these things. And these are what I was talking about in my talk yesterday.
This whole tower generate their symmetries. They're part of the symmetries of nature that I was talking about yesterday. They have observable memory, like there's predictions that come from them. Now one thing. Let's see. OK, let me-- but what's even better is I've got all these things and now I can compute their ope. And I can find the algebra of this group of all the symmetries.
AUDIENCE: Excuse me, I have a question.
ANDREW STROMINGER: Yeah?
AUDIENCE: What about the goals for delta 2? Is there something special with delta 1? I mean, if I remember correctly, the formal of function b was like a product of delta functions over--
ANDREW STROMINGER: Well, I mean, I was writing o plus a delta 1, o plus b delta 2, 1 over z1, 2, beta of delta 1 minus 1, delta 2 minus 2, 1. So it's symmetric in one and two, right? o delta 1 plus delta 2 minus 1. And what I'm doing right now is I'm making both of them go to negative integers. And so then we're going to see the algebra of them.
And what you find is that the commutator rkam, rlbm is equal to minus if-- and this is a slightly long formula, but I promise it's the last long one I'll write-- 1 minus k over 2 plus 1 minus l over 2 plus n plus m. I'm just doing this so you'll be appreciative when I write a much simpler formula later. Factorial 1 minus l over 2 minus m-- actually plus m factorial, and then times the whole thing with n goes to minus n, m goes to minus m times rck plus l minus 1c n plus m.
And you can check that this obeys the Jacobi identity. And this commutator is defined by a commutator b where z is equal to the contour integral 1 over 2 pi i, a of z, b of w, dz. So this is a two dimensional commutator. It's not a four dimensional commutator. I never looked at things that earlier or later times. If you want, you can think of this as a commutator in the dual one dimensional Hilbert space on a circle of the two dimensional field theory.
So to evaluate this explicitly, you would take S matrix element and you would evaluate two points like this and then you would consider what happens if you move one point around the other point. You're not putting some operators later in time and some earlier in time. And this result was obtained in collaboration with Alfredo Guevara, Mina Himwich, and Monica Pate. OK. I'm going to move on to gravity, which I'm going to be much faster with, but yeah.
AUDIENCE: So you started with what I thought were local operators, but maybe they weren't local operators because then their ope is a non-local operator. Where does this operator R live? Is it integrated over [? scry? ?]
ANDREW STROMINGER: Wait, why do you think they're non-local?
AUDIENCE: Well they're acting like conserved charges, they're not acting like conserved currents. They're not functions-- the R's are not functions of z, are they?
ANDREW STROMINGER: Yes.
AUDIENCE: Oh. Oh, they are functions of z.
ANDREW STROMINGER: Sorry.
AUDIENCE: This looks like the algebra of charges, but it's-- those are at the same z?
ANDREW STROMINGER: Yeah. Oh. Sorry. Sorry, sorry, sorry, sorry, sorry, sorry. It's rr of z. So what you do is we have two holomorphic operators. It's this thing, right? Well, here's the definition.
AUDIENCE: OK. I see. So they're currents.
ANDREW STROMINGER: They're currents.
AUDIENCE: OK. Yeah.
ANDREW STROMINGER: OK, so let me now turn to gravity. So we worked out, just using the symmetry, all the leading singular terms in the ope for gravity, gravitons, and gluons of any helicity just using the symmetries. Anna and Monica and [INAUDIBLE], and I did this for their leading ope's. Yes?
AUDIENCE: Sorry. [INAUDIBLE]. The unitarity of the scattering matrix, did you mention what constraints that puts on the [? spins? ?]
ANDREW STROMINGER: I mean, we're rewriting a unitary theory. So the four dimensional S matrix is-- if it's unitary in one set of variables, it's unitary in another set of variables. It's a very interesting question. There's lots of interesting questions like that, what unitarity of the four dimensional S matrix means for the two dimensional theory. I don't know the answer.
Actually, there was a paper by Ho Huang, Huang, and Yu-tin Huang, some amplitudes people who were addressing this in part and understanding the unitary constraints, translating the unitary constraints on the EFT. He'd drawn into constraints on the things in the celestial amplitudes, but I think there are things to say there. There isn't some really, at this point, some sharp, simple-- there's lots of other interesting questions like, what is crossing symmetry and 4D mean in 2D?
And even more interesting is, what is crossing symmetry in 2D mean for the 4D theory? Because the 2D crossing symmetry looks very strange from the point of view of the 4D theory. It looks like a very, very powerful constraint. So a lot of questions like that that haven't been answered. But I mean, there isn't any question here about what the theory is. So they must have answers somehow.
But how exactly? What are the right tools to get at them. ? So in gravity, well, there's a similar formula. We have gravitons. So we have delta 1, delta 1, z1, z bar 1 with the second graviton, again, plus helicity. And let me say, because I'm not sure I mentioned it before. I am, in this talk, only considering plus helicities. It's not that plus helicities are more interesting than mixed helicities with plus and minus.
In fact, the mixed helicity case with plus and minus is probably more interesting. It's also much harder and we haven't figured out how to make sense of it. But the operator product expansion in the plus helicity sector closes, so we're free to consider this as a separate subset. But we should be fully aware that we're far from the end of the story even when we fully understand this separate subset.
I think it might be a little bit like when you try to make a consistent conformal field theory you start with the conformal blocks or the Virasoro blocks of primaries of different weights and they're easy and simple to describe and have lots of properties. And then when you try to glue them in together into left movers and right movers in with a modular invariant partition function, that's the really hard problem in conformal field theory. So maybe what we're doing here is solving the easy problem.
This goes like, again, the ubiquitous Euler beta function. The reason these beta functions appear all over the place is that they have the ope there to ensure translation invariance. These have to do something special when you shift the weights by integers and beta functions have nice properties, recursion relations under integer shifts that make it all work out.
Now there's a z bar 1, 2 over z1, 2. This is the leading singularity in z1, 2, g plus delta 1 plus delta 2. That's the leading ope. And now again we do an expansion where we write limit as epsilon goes to 0 of epsilon times g plus epsilon plus k where now k equals 2, 1. Before we started at 1, but now we're starting at 2.
And this is equal to some sum from n equals k minus 2 over 2 to 2 minus k over k, hnk of z over z bar to the n plus k minus 2 over 2. And these h's now are currents. They're in multiplets, higher and higher and bigger multiplets, and they're exactly the multiplets-- they're the exact analogs of the R's I was talking about before. It's just all the numbers are a little bit different at gravity.
Now instead of writing out the commutators of these h's, I'm going to redefine something new. wnp is equal to gamma of p minus n, gamma of p plus n, hn minus 2p plus 4. So I get some complicated commutators for the h's, but now I'm going to make this field redefinition. In this field redefinition in terms of the z bar dependence is non-local, because I'm mixing up the modes.
And it's actually related to something that's very much studied and conformal field theory called light transform. And one finds that the w's, after doing this, that the operator algebra is wnp. wnq is equal to m, q minus 1 minus n, p minus 1, wm plus n, p plus q minus 2 where all these w's depend on z's.
So this is a Kac-Moody algebra. The Kac-Moody algebra of the group is called-- this group is called w1 plus infinity. And w1 plus infinity is a famous group and in mathematical physics that we'll talk about where it's appeared, but let me mention that m is also restricted in its range because n was restricted in its range here.
And so this is actually called the Kac-Moody group is the wedge algebra of w1 plus infinity. Now exactly this group has appeared many times in mathematical physics. Just to mention a few, the first time that I noted appeared is Penrose used this in his construction of the general self-dual field configuration and twister space or in 2 comma 2 signature.
And basically they were all generated, all the different solutions were generated by the wn's. Klebanov and Polyakov found the generators of this as the discrete states of c equals 1 string theory. A deformation of this algebra, there's an interesting quantum deformation of this algebra, which I think should have some role to play in things but I don't know what, describes the algebra of a free field in two dimensions or free fermion in two dimensions.
There's something called w gravity that Chris Pope, and [? Sean, ?] and others worked out. It appeared in the matrix theory in the construction of the large end limit. Not BFSS, but before that the [? Hoppa ?] and the super gravity people started out a large end limit of su infinity and they got w infinity out of that. Without the z's, this is the algebra of area preserving diffeomorphisms. So it's a very important infinite dimensional higher spin algebra in two dimensions. It's probably the only one with such a simple closed expression.
So four good things have happened all at once here that made me, anyway, happy about these developments. The first thing is that we knew that we had-- we've known for a long time that there had to be more soft theorems because we knew that Lowe's subleading soft theorem doesn't commute with itself and gives you more symmetries. And so there had to be some tower that were higher and higher in something, but there was no way to organize.
It got very messy. Nobody could guess what the next thing is, there was no way to organize them. Finally, we organize them in terms of these h's, but they had these insane complicated looking commutators that obeyed the Jacobi identities, but still, why should it be so complicated? Then I did this field redefinition and I found the whole thing reduced to probably the most studied, nicest infinite dimensional algebra of this type.
So that seemed like a good thing. The other interesting thing is that somehow everybody is bothered by the negative weights. Somehow in this field redefinition if we classify it according to wn representation, all the weights are positive. So we lost this negative weight business. Now, this is not all the operators in the theory. Maybe the negative weights will come back. But somehow the negative weights have disappeared.
Secondly, there's been a number of papers out of the twister community at Oxford. They knew this formula, they think. We've been spending months trying to reconcile our notations, but they knew this formula. Maybe even Penrose knew this formula. They knew about light transforms and they knew that it was a good idea to work in this funny basis and they certainly knew that there was a w infinity hanging around.
And moreover, some of the amplitudes that one computes in the celestial amplitude formula-- we don't know what basis of operators to use, but Sharma recently shown in the basis of operators that twister theorists would naturally use, all the singularities go away like the delta function on the equator. Whoever has followed this program knows that there are various kind of singularities that we don't quite know how to deal with.
So that seems a good thing. But perhaps the most exciting thing as all is that this w infinity is a big and powerful group. In the past it's always arised in integral models. The twister people loved it, but I don't ever think they had any idea that it would persist into the real world. And we didn't use anything here except soft theorems that apply to the real world.
So we're learning that this powerful infinite dimensional group, which was, we thought, relegated to the domain of really fun, integral models in two dimensions. In fact, it's the symmetry group of any theory-- it's part of the symmetry group. I've only done the positive helicity part. It's part of the symmetry group in the limit. There's two limits that we've taken here. We've ignored the cosmological constant and also, I'm going to say something about quantum corrections in a moment.
It exists as a subset of the symmetry group of nature. There should be experiments associated with it, memory type experiments, and it exists as a symmetry of the subgroup of the symmetry of nature. And we've also ignored here the effect of quantum corrections. Now let me just say a few more things and then I'll come back. So it's in the classical limit.
So one is that I can now do a similar transformation on the gauge fields. sm gamma is equal to gamma q minus m gamma q plus m times my r's, ra, q. ra, what is it? It's ra 3 minus 2q comma m. And then the sm's, instead of that horrible algebra I wrote on the previous board, just obey sma, snb equals minus ifabc, scm plus n.
And then there's weights here, q, p, and then this becomes q plus p minus 1 I think. Yeah, minus 1. So they form a very, very simple algebra. And moreover, the w's, all the soft gluon currents are in one irreducible representation of the w algebra. So that is wmq. Let me just get all the numbers exactly right. wmp, wmp-- where'd it go? sqan is equal to mq minus 1 minus n, p minus 1, sp plus q minus 2 comma a, m plus n.
So the gluon soft operators or soft photon operators are all in an irreducible representation of this w1 plus infinity. Now important corrections or important question is, what kind of corrections are there? Just some, and there are several papers on this. It's very hard to correct a soft theorem. And some of them, the leading soft theorems, are known to have no corrections to all orders in the loop expansion. They can't be correctioned.
The subleading ones are a little more vulnerable. They can sort of have anomalies or there's a few-- they each have a short list of higher dimension operators that can correct them, which are nicely discussed in a paper by Henriette Elvang. Other people discuss it too, but I like to look at that one. There's a short list of higher dimension operators. We have to think about those corrections and we were also very interested in quantum corrections.
What do quantum corrections do? So what Elvang looked at is sort of in a Wilsonian effective theory, which operators will correct this algebra? But in Wilsonian effective theory, you also have to integrate over loops of massive particles and those could somehow correct those algebras. And we checked it in the one place where we could, and that is the Yang-Mills theory in gravity have all plus amplitudes.
So MHV amplitudes are sort of mostly pluses and two minuses, but at the one loop level you get all plus amplitudes and those all plus amplitudes, even in gravity or ultraviolet finite, they take a very specific and very complicated form, much more complicated than MHV amplitudes. We haven't put the paper out, but we've shown-- and it actually surprised me-- that the w algebra is completely intact.
It's not deformed, it doesn't have an anomaly. It's exact, even at the level of the all plus corrections. So kind of expecting it was going to get deformed, but haven't seen where that comes from yet. So we don't quite know what's going to happen next, but we think it should be interesting. So maybe I'll stop there.
SPEAKER 1: Questions?
AUDIENCE: Yeah, I have two questions. The first question is so you are saying there is no central extension if you consider [INAUDIBLE] correction?
ANDREW STROMINGER: I hesitate to say that because I wanted there to-- I was anticipating a central extension at the loop level, but there's more currents in this story. These are just the soft currents. They also have partners which are currents which you could think of as the Goldstone currents. Right? So these are symmetries, like in a two dimensional conformal field theory even without a level.
Think about a two dimensional conformal for the u1 current algebra no level, but charged fields. And the charged fields are you could Bosonize the charge and you can make a current out of it. So these theories tend to have off diagonal levels. They have a matrix of levels with other currents.
So we've understood something about how that happens in the u1 case, not so much in gravity or the non-abelian case. Yeah. And these things are the-- it's the things that source the soft currents, so the Wilson lines. And some work has been done representing these things that's how you compute their correlators. We understand how to compute-- there's a soft algebra that succinctly characterizes a lot of discussion about the factorization of the S matrix into a hard and a soft part.
And at least in my mind, it's more elegantly done when you work in a conformal basis in your factorize into the conformally soft and the conformally hard part. And then you find a soft current algebra that generates the soft S matrix, and that includes not only these soft modes, but Goldstone modes. The Goldstone currents, the things that make Wilson lines.
The correlations of the soft Goldstone modes are the correlations of the Wilson lines which have, of course, been studied quite a bit in quantum field theory and we know about them. And then the levels in these things become the cusp anomalous dimensions, which can be thought of as one loop thing. So in some sense, there's levels arising at one loop, but in a different object. And it's in the matrix of currents and we haven't studied yet the whole matrix at this level. We're actually starting to do that.
AUDIENCE: OK, I see. Another quick question is, how can I see this is as related to super translations for h's?
ANDREW STROMINGER: How can you see that it's?
AUDIENCE: Related to the super translations [INAUDIBLE].
ANDREW STROMINGER: Yeah. So the current-- let me just talk about the h's. So let's see. h with delta equals 1 plus of z. It's actually a doublet because it has weight 3/2 minus 1/2. So we should write it as h.
Well, let me just tell it to you this way. We can write that as h plus or minus a half. Those are the two indices, the doublet in the SL2R representation of z And then that's in a multiplet-- well, yeah. h minus a half is equal to d bar, the limit as delta goes to 1 of d bar, which takes a descendant.
Takes you from l1 to l minus 1, d bar of g of 1 of z, z bar. And this is the thing. This is exactly the thing that I was calling 5 or 10 years ago the super translation current. And this is the thing whose ward identities I proved were equivalent to Weinberg's soft graviton theorem and generated-- yeah. This is the field that I showed was equivalent both to the leading soft theorem and one of the first papers I wrote on the subject.
So all these fields are related in a very, very concrete way that the formulas are not very long. One thing that has been a simplification here is that in the earlier work, one was talking just about soft currents and analyzing things according to powers of the energy omega. And it's almost equivalent, but it's conceptually much clearer to talk about conformity soft currents. Yeah. Yeah?
AUDIENCE: So in ordinary holography, we have a strong [INAUDIBLE], the bulk gravitational coupling is 1 over Yang-Mills. Is there something like that here?
ANDREW STROMINGER: No, I wouldn't-- yeah. I think gravity in the bulk at any end is dual, or any g Yang-Mills is dual or any g string is duel to Yang-Mill's theory on the boundary at any n and any g Yang-Mills. Well, they're related in some way. But you don't have to be in the large end limit for the duality to hold. So Yang-Mills theory has two-- if you like, Yang-Mill's theory has two classical limits.
One is where g Yang-Mills becomes strong-- small, and quantum corrections are suppressed. And one is when n becomes large and there's factorization of correlation functions of operators and there's a completely different classical limit. And we would identify that with the classical limit of gravity in the bulk. Now we might not be so lucky in constructing a duality to have such a simple situation.
For example, theory well, in m theory in 11 dimensions where there's no parameter. There's no dimensionless parameter. There can't really be two descriptions. I suppose we can think about expanding it scattering amplitudes and momentum over the Planck mass. So we're speaking in very general terms here about any theory of quantum gravity and any theory with a gauged sector.
And so there isn't there isn't another parameter that we've put into the discussion, but I think if one were to construct an actual useful top down example of this starting from string theory or something, starting out with saying here we construct the celestial CFT, here's an intrinsic definition, we would likely need such a parameter to do it.
SPEAKER: OK, we should probably wrap up. Let's thank Andy again.
[APPLAUSE]
Professor Andrew Strominger, Gwill E. York Professor of Physics, Harvard University, presents "The Holographic Principle in Flat Space" part of the Bethe Lecture series.
Gravitational soft theorems imply that scattering amplitudes in any 4D quantum theory of gravity have a symmetry which acts like the 2D conformal group on the celestial sphere at lightlike infinity. This motivates efforts to holographically recast 4D quantum gravity as a 2D conformal field theory on the celestial sphere which I will describe.
