Intro to Celestial Holography - Part 3 - Mina Himwich

[00:00:15]Okay. Um so today we're going to just go through more um celestial opes and get to um talking about the algebra of soft graviton or or um you know soft symmetry operators from both the celestial op perspective and then back from the momentum space perspective. uh so last time we ended on talking about uh colinear limits in the bulk as uh corresponding to the op limit on the boundary. So as two moment so we're talking about massless momenta here. So if we have two massless momenta um when they become colinear in the bulk that corresponds to their distance on the celestial sphere going to zero. Um and I think we went over this a little bit fast at the end yesterday. So if we have a um amplitude with these two massless particles P1 and P2 the leading term in the colinear limit as P1P2 goes to zero will correspond to this just this three-point vertex where the two momentum become colinear meet at some vertex and then we have a propagator of the sum of the momentum and if we um expand this out in omega and Z and Z bar the three-point vertex is going to involve uh factors of omega with the dimension of the vertex minus 3. So dv is the dimension of the bulk vertex. The propagator has a factor of omega squar.

[00:01:43]So omega's here the sum of omega 1 plus omega 2. We're going to be looking at the leading singularity in z. So leading holorphic singularity and then there'll be some power of z bar to the p.

[00:01:54]And um if we convert this to the melon basis, what we see is that these factors of omega uh are going to change under the melon integral are going to change the dimension of the operator that's appearing on the right hand side of this op. So uh what the yeah the melon transformed version of the statement is this op that I wrote here. So we're looking at again the lead this is just means the leading term as Z12 goes to zero and the dimension of the operator on the right hand side is fixed by just the melon integral to be delta 1 plus delta 2 plus dv minus 5 from these factors of dv minus 5 here and then uh asking that this thing transforms both that left hand right left and right hand side transform the same way under sl2c fixes what P has to be and what the spin of the third operator has to be because the spin and the delta of these operators are related to um just H plus H bar. We can look at how each each side transforms under holorphic or anti-holorphic uh half of Laurens transformations and those tell you that P has to be EV minus 4 and uh the third spin has to be S1 plus S2 minus P minus one. So this and this coefficient this OP coefficient um corresponds directly to just like the the melon transform of the bulk colinear splitting function which is what we have here and you can just compute the colinear splitting function in whatever theory you're interested in and then you'll find this OP coefficient but from the we also are interested in finding the the OP coefficient from a boundary perspective um because we want to be able to do you know everything in the bulk that we want that should have an anal log on the boundary and on the boundary um we can show that this OP coefficient is fixed by pointy symmetry action on the boundary in the same way that the splitting function is fixed by poner symmetry in the bulk. So I'm going to go through that and then we'll then we can move on to um looking at specific opes of these soft operators. So because each operator each each of these O's come from a melon transform of a bulk um like single particle igen state and so since these each of each each igen state dep corresponds to a family of these deltas the OP coefficients will actually also depend on delta and so the OP coefficients generally can depend on the dimension and the spin of the operators or equivalently um we're going to be writing that in terms of the har weight and a spin. So a general onsets for this OP takes the form of an OP coefficient that depends on H bar.

[00:04:48]Uh so OP Yeah.

[00:05:10]So we're going to label our operators now by H and H bar.

[00:05:22]Um I'll try to write bigger than yesterday also. So let me know if it's not visible. So we have our leading.

[00:05:28]This is we're looking at the one over Z pole. And now I'm going to take a sum over the leading term in Zbar and also all of the uh anti-holorphic descendants. So there'll be a sum over derivatives in Zbar.

[00:05:44]And each of these terms, the ZBar descendants will have its own coefficient.

[00:06:10]So this OP coefficient uh The leading term is the m equals z term we wrote over there. But and I've also included this expansion over the just the sum of the right holorphic anti-holorphic descendants and um this op coefficient I've written as a function of h bar. It's also a function of spin but I've left the spin dependence implicit and we'll see how it comes in uh at the end. So the action of we're going to try to constrain this thing from the action of 2D plunker symmetry or sorry 4D punk symmetry on the 2D celestial sphere. And so uh these symmetries are just the global part of the symmetries that came from the leading and subleing soft theorems.

[00:07:00]So they should look familiar. Um so the lorren symmetry as we saw takes the usual form here I'll be using the right half of lorren.

[00:07:26]So just LB bar acting on an operator takes the usual form from 2D CFT.

[00:07:56]And the global part is just the n= minus one 0 and one. And then pankare we saw or sorry translations we saw because p is proportional to omega there's going to be this shift in dimension from the action of translations.

[00:08:16]Uh so translation generators can be labeled by M and N where M and N are uh plus and minus 1/2 and the action uh of these global translations just gives you back different components of the momentum which here we're parameterizing with Z and Z And then because of this proportionality with omega, there's going to be a shift in the dimension on the right hand side.

[00:09:01]Okay. So we're going to use the right half of So we're going to use the LB bars and also the PS with M is equal to um just pick out these modes uh to constrain the OPE because these modes are the ones that um only involve guess there should be a minus minus one half here. These are the modes that only involve um zbar. So the action of this generator and the lb bar generators are only going to mix z bars in this uh op. So we'll be constraining uh just the the coefficients of this. We don't mix the z uh primary with its descendants. Yes.

[00:10:08][clears throat] >> Yeah.

[00:10:12]>> The translation has a lower.

[00:10:15]>> Yeah.

[00:10:20]>> Um it will change the it will change like this powers of Z and Z bar that appear on the right hand side. So it will be changing the spin in this with if you cons consider the overall uh action of this there'll be a Z and Z bar factor here. Yeah. Yes.

[00:10:37]>> I missed maybe something in your explanation. I don't understand why this makes sense because you treat Z and Z.

[00:10:45]>> Yes. Because we're here we're only looking at the leading the coefficient of the leading one over Z pole. So we're trying to only constrain the one over like we this is a very chyal construction from the beginning. in that we're looking at the holorphic li we've taken Z and Z bar to be independent and looking at the leading term as as one over as Z12 goes to zero but Z bar one2 doesn't go to zero so yeah this is a very like chyal construction um uh this is just well if you if you think about this from the colinear limit there's It's just the one over Z12 comes from the propagator. So this is like the leading term that you'll get. Um yeah we can yeah we can talk more about that also afterward. Yes.

[00:11:40]>> CF perspective.

[00:11:45]>> Yeah. So the O's are global conformal primaries here. Yeah. And yeah I haven't talked about the verauro structure at all. This is just we're just using the global conformal group. Yeah.

[00:11:57]Okay. Good. So, I'm not going to go through this whole derivation, but the onsets you can show is it's already consistent with the action of LBR minus one and zero.

[00:12:12]uh but we are we can use the action of L1 bar for example on the left and right hand sides to get a constraint on the OP coefficients and also with this uh these P modes. So um if we're looking at the action of L1 bar for example we can take uh L1 bar on the right hand side and this thing is then going to be equal to the action of L1 bar on the first operator.

[00:13:06]Uh so we do this then we take its op with a second operator.

[00:13:14]Uh this should be equivalent to taking it on the right hand side where here we just act with L1 bar on the operator.

[00:14:00]Okay.

[00:14:01]And so from evaluating this and demanding that they're equal, you find a constraint on the OP coefficients.

[00:14:10]Uh, and doing this for Okay, good. So there are three constraints from L1 bar and then P minus a half plus and minus a half and uh imposing these three constraints leads to recursion relations.

[00:15:14]for the OP coefficient which are uniquely solved by a beta function. So we find that these OP coefficients involve a beta function in the anti-holorphic weights.

[00:15:45]And then there's also going to be a spin dependent coefficient which depends on S1, S2 and P or equivalently on S1, S2 and S3 because S3 is trained again to be P + 1 minus S1 minus S2 and the spin dependent co the spin dependent coefficient directly comes from the coupling constant of the bulk interaction. So um and we'll just see an example of this in a second but this is the general constraints that you find on the OP um from from cloner and you would also find this same thing if you directly mel if you direct directly just melon transform the splitting function from momentum space so um yeah this is a nice use of 2D symmetry to derive the 2D So now yeah just we'll do an example um for gravitons.

[00:17:19]this board a little more.

[00:17:31]Okay. So now we want to specialize. We're going to study graviton ops. So we want to specialize to the case of S1 is two. So this first the spin of the first particle is graviton. Um uh so let's consider yeah our first particle being a gravitton uh in 4D minimal coup the 4D minimal coupling corresponds to the fact that the spin of the other two particles S2 and S3 are the same. So And then we're going to also be taking both of these things to also be a graviton. So um this will be two. And when S2 and S3 are the same, this implies that P is one, which is consistent with the fact that P is DV minus 4.

[00:18:40]So the bulk vertex dimension for this three graviton coupling is five.

[00:18:46]Okay.

[00:18:50]So again our graviton operators we're going to write as this g delta plus we'll consider a positive helicity graviton and this thing has weights h bar is delta + 2 over two and delta minus 2 over two.

[00:19:12]We'll also be considering this conformally soft graviton which is the limit of this um as delta goes to integers that we saw yesterday.

[00:19:28]So this conformally soft graviton we defined as HK or I guess this is a new definition of the conformally soft graviton as hk and here we'll be taking this graviton to uh negative integers well mostly negative integers Okay.

[00:20:01]So yeah, here we're taking this limit as K goes to one for the leading soft graviton, zero for the subleading one, minus one, etc. This is actually like negative of the of the uh like indexing of the pole and omega structure that we saw yesterday. So sorry that that's confusing but this is the notation that was in the that we introduced in the in the literature. So I wanted to stay consistent with that.

[00:20:30]Um so right these HK operators now have weight have a right weight in particular. Well both of their weights start getting negative but the right weight starts becoming negative as we take K further and further down here to pick out the further subleading terms in the soft expansion. And um the right mode expansion of this operator H uh has You can show that just for any operator that has a negative weight uh on the right given by this uh k minus 2 over two. You can you can do a mode expansion of this H k into modes hkn which are now going to be we take modes on the right and keep them as a function of z and they're going to be a closed set of modes under the action of lb bar 1 0 and minus one. So the action of lbar one zero and minus one will keep us within the set of modes.

[00:21:42]Um so there's this closed finite antiholic mode expansion we can do for Her SL2R And um we'll see how this is that closed expansion is relevant in a second. So when we just when we specialize our op and with with the coefficient that we found uh to the general uh yeah we specialize our general expression to this graviton and we find that the op takes this form.

[00:22:49]So again, there's going to be a bulk coupling constant.

[00:22:56]We have our pole in one over Z, which is uh there.

[00:23:02]Um then we're going to have our OP coefficient, which is a beta function.

[00:23:12]And now I've written it back in terms of delta.

[00:23:31]Okay.

[00:23:34]And now we can take the conformally soft limit of both of these things. And basically when you do this, the beta function is going to have poles as these deltas go to negative integers. And these zeros that multiply these operators are going to pick out some there's going to be some cancellation between poles and the beta function and zeros that we introduce from the conformally soft limit uh to give a finite result for this OP coefficient.

[00:24:00]And so uh doing that we find an op that now involves a finite sum over descendants which is related to this finite antelomorphic mode expansion that we that um these operators admit because of their weight. So this becomes now a sum just to 1 minus k.

[00:24:39]There's some combinatorial coefficient.

[00:24:47]This is a binomial coefficient.

[00:25:07]Okay.

[00:25:10]And so we find this uh op. And now these we can do a mode expansion of both of these again of these h operators on the right and define a commutator.

[00:25:33]So we'll define a commutator of holorphic things in a familiar way from 2D cft as an integral.

[00:25:58]uh of this uh product of of holorphic currents. So we're going to each of these are going to correspond to um modes of h. So we're that's going to give us commutator between these uh HL modes.

[00:26:36]So there'll be some uh big we just do this we find some big uh combinatorial factor here and it and it looks like this mode computator.

[00:26:51]Okay, I guess I put these at C1 Z2.

[00:26:59]Okay, so I didn't write the expression here because it turns out there's a really nice redefinition you can do uh to redefine um just write it here the modes of H in terms of the modes of a different operator W. So we can define the modes of this operator W as given by 1 over kappa times a combinatorial uh coefficient.

[00:27:42]And when we do this we find that this algebra uh becomes the w 1 plus infinity algebra which I'll write Okay. So just so we reorient ourselves this so the WPs now have weights uh 3 minus p.

[00:28:57]So now p which is uh right which corresponds to this h with a - 2p + 4.

[00:29:13]So when we had these negative l's there going to become positive ps. So now we have operators that have positive right weights where P starts with three halves which is corresponds to um L is one or the leading term. Then there's P is 2 which corresponds to L equals Z or the subleading term. And um these things will have a closed set of modes between 1 minus P and P minus one which is just the closed set of modes a rewriting of the closed set of modes up there.

[00:29:51]There's also a central term of this algebra with p equals 1 uh which we don't have a soft theorem interpretation of but uh it's still there and it's interesting we can talk more about it um in the discussion and in terms of the mode in terms of the k modes we can rewrite this weight as k + 2 over 2 And then uh this is 4 - k / 2 which is equal to 1 - k - 2 / 2 which was the weight of that uh hk operator the kus 2 over2 and so this actually has the weight of like one minus h bar of this h operator. which is actually which is the same weight shifting EG wonder light transform which is like a half of the shadow transform u that I introduced yesterday. So um I know this is like a lot of stuff coming at you. So just uh just it's just to notice that we've changed the weights in the same way as like the shadow transform changed the weights of the subleading soft operator to be from h to h bar it goes to one minus h1 - h bar.

[00:31:16]Here we're doing a a shift just of the right weight of this operator from h bar to 1 - h bar. Okay. Um so we have this algebra. It's also possible. So in addition to taking just um in in addition to like leaving these as a function of Z, we could also take the modes in Z which would and so we could introduce a new label for modes in Z as well. And that gives you something called a loop algebra. Here I'll just focus on um this mode and Zbar modeing in Zbar.

[00:31:57]Uh good.

[00:32:05]So I'll start over here again.

[00:32:16]>> Yes, sorry. So this is a 2D light transform also. Um yeah so the W's the HK operators have weight uh right weight k minus 2 over two uh and these w operators have right weight 1 minus kus 2 over2 so the w's are like the light transform of the h's uh and you can actually and you can construct them with al like from as an operator as a 2D light transform as well Here we're just doing a mode kind of mode reshuffleling to make these into the modes of a light transformed operator.

[00:32:59]Uh okay.

[00:33:04]So Oh yeah.

[00:33:33]So it's possible to compute the act. So as well as finding this is this is the algebra between two soft conformally soft graviton operators on the celestial sphere. Um so in addition to finding the algebra between the soft gravitons themselves, we can also find the action of the soft graviton on massive particles or or sorry massless particles on hard particles by considering uh this commutator WP with just a regular hard operator.

[00:34:08]And you can compute this You can compute this action in the same way as we computed the commutator from the op for the two graviton operators.

[00:34:38]You just you can just look at their op from the general form.

[00:34:42]uh take the conformally soft limit, do the mode redefinition and you can find this action and you when you do this you can show that the action in successive commutators uh respects the W algebra. In particular, we find that if we act in one order and then the other order, they give you back what we expect from the commutator.

[00:35:43]And so also in addition to this form of the OP that we wrote down being coariant under plunk array transformations you can also show that it's coariant under this W symmetry action and the proof of this type of statement and and yeah so the proof of this W algebra and a lot of the um things related to W algebra can be done uh by induction where we start with.

[00:36:11]So yeah to prove this we we can just use the base cases of P is three halves 2 and five halves because we can see from the form of the algebra uh that I wrote over there that the P equals um two modes will take you from yeah are just these are just like the right half of SL2R. They take you from a WQ to this W with the same Q, but then a shifted uh right mode. and the W5 halves shift you in the P uh index.

[00:37:22]So using the action of just well two and five halves really you can get down um to all other values of p.

[00:37:34]So you can show that this action holds doing an inductive proof in um the p and p and m labels.

[00:37:46]Good.

[00:37:48]Okay. So I think for the rest of the time I wanted to I know the OPE formalism invol like involves a lot of new notation and the melon basis and stuff. So I wanted to go back and talk about how we think about the W algebra in momentum space and um that will also allow us to think about the W action on massive particles which are not captured in this OP formalism which really relies on the massless colinear limit. Okay.

[00:38:13]Yes. Any any questions? I feel like I see confused faces. Yeah. So a lot of the analytic structure here like having single valued um like analytic refactors your ops and the poles of the data functions being in the right place. This is sensitive to this nice conspiracy that's on the complex side of the board.

[00:38:31]>> Yeah.

[00:38:33]>> The dimension the bulk dimension being what chips your operator makes ensures that all the powers that appear by OP are like integers and everything is nice. But I think the the argument that I usually This is like a tree level, right? But if we go to one loop, we can the effective vertex have the dominance dimension. Is there a reason do we think that this breaks if we go to one in the or is this just like statements about level as a whole theory or what happens?

[00:39:05]>> Um yeah, I think generally there will be other structure if you start going to loops. Um, right now we can just take this as a tree level statement.

[00:39:14]>> There's like some consistently complicated theory that we're studying.

[00:39:18]>> Yeah, I guess we're we're just studying this the tree level statements right now. Um, I think so some things uh we know like the soft like the soft operators like for example the soft graviton uh theorem tells us that that thing is not corrected by loops. So some things in some cases we know that there is like uh more general it holds more generally. Um >> yeah.

[00:39:49]>> Yeah. Uh okay good.

[00:39:56]So right we're going to now yeah we'll go back to momentum space now and hopefully uh we can yeah connect with some of this and make it a little clearer.

[00:40:47]Okay. So again, momentum space is useful to both extend this symmetry action to mass to acting on massive particles and also um it's going to connect to light ray operators 4D light ray operators uh which will be the subject of some of the next lectures. So yesterday we saw that we can we're going to be considering just a tree level soft expansion.

[00:41:09]So we have our amplitude where we have now um yeah just external particles P1 through PN and the sum starting with the WER pole uh of an expansion of the amplitude in the graviton's energy. Okay.

[00:41:43]Okay. So it turns out that at all L uh we can find the following form of the uh AL things here.

[00:42:07]as a soft factor acting on just the original amplitude as in the case of the leading and soft leading and subleing uh soft theorems plus additional stuff.

[00:42:29]So we have our first piece that's just going to be a soft factor for each particle that depends on the gravitton's direction multiplying the original amplitude.

[00:42:40]And then there's going to be some non-universal pieces that are there starting at L uh greater than or equal to one.

[00:42:58]Okay. So I'll write on a new board the form of this SK.

[00:43:25]So the soft this this SKL generalizes uh which depends on Qhat or on Z and Z bar it generalizes the leading and subleading uh universal soft factors to higher powers of angle momentum generator which we saw appeared in the subleading sock factor.

[00:43:43]So it's going to be So it takes this form And uh these nicely also exponentiate.

[00:44:25]But so at the leading order we get we just L is minus one and we just get back to the leading soft theorem leading soft factor. At L is zero we get back to the subleading soft factor. And in that case this these B terms are zero. But once we get to L equals 1 which is the subsleing soft vector um it there can be corrections to this uh to like the splitting of the amplitude into just the soft vector times the rest of the amplitude. So we'll be focusing on this uh this SK piece and you can show that this gives you a coariant splitting between what we call this SK piece of the amplitude and then the remainder term which something that you know we it's it's consistent to just separate this piece out by itself. And this thing has weights again of minus L + 2 over two minus L minus 2 over two.

[00:45:24]Sorry again that L and K from before are like inverse of each other but there we go. Okay. So um actually maybe I'll go over here actually. Yeah, it's fine. So we can so in the case of a leading soft theorem, we saw that if we took two derivatives of the soft factor, it became a delta function and so that gave us a local action on the hard particles.

[00:46:18]You can similarly do the same thing for these uh soft factors at higher L and you can just show.

[00:46:25]So anything with this weight if we take Zbar derivatives if we take L + 3 Z bar derivatives of it.

[00:46:38]Uh so yeah this is an SL2C primary with these weights. You can show that if there's an SL2C primary with these weights and we take Zbar uh L+3 Z bar derivatives of it, this gives us an SL2C primary descendant or really SL2R primary descendant which has weightus L + 2 over 2 and - L sorry L + 4 over And this L+ 42 we're going to call P.

[00:47:17]So this thing again has the same weight as a light transform. It's 1 - L -2 over2. So taking these derivatives of the primary descendant. We're sort of doing the same thing. Now we're in back in, you know, momentum space 31 signature as this changing of the weights that we did on the H bar modes uh in the 2D perspective.

[00:47:40]So we can again consider modes of this light transformed or primary descendant operator.

[00:48:06]We can define the action of a w generator on momentum space uh particles as these modes of the primary descendant of the local primary descendant of the soft factor.

[00:48:23]So here again okay yeah this soft factor I've also written for massless particles only. I'll talk about the mass of generalization uh in a second.

[00:48:37]So now we can integrate over the 2D plane to find these modes in Zbar.

[00:49:01]And this yeah delta is going to be some differential operator in Z and omega of the hard in like zk and omega k for the acting on the hard particle. And you can again show that this thing satisfies the action of the w algebra acting on hard particles.

[00:49:20]Um good. So you can do the exact same thing for massive particles where now um we need to define a different soft factor for massive particles.

[00:49:51]um I won't get into it here but we find we can just construct a different soft factor s prime and for massive particles um instead of yes basically the difference is that instead of taking derivatives of the soft factor and finding that it localizes to a delta function in the plane um we we can take derivatives of the soft factor and find that it becomes a bulk to boundary propagator on time like infinity like we saw in the case of massive particles yesterday. So there's a nice form you can find a separation of this you can find some soft soft separation of the soft expansion like this such that there's a soft factor that transforms coariantly and that has this nice derivative structure where um our dzbar Primary descendant takes the form of some bulk to boundary propagator which is this one over qhat p to the 2p and then it'll also involve this maybe I should write this a little bigger Huh?

[00:51:43]there's some normalization that I won't put and then it involves this just so right remember that this one this thing is like the multi boundary propagator where the dimension delta is 2p and then we have uh also So an angular momentum generator but now with the opposite holicity which comes from change you know taking the primary descendant.

[00:52:23]Okay. And so you can using this primary descendant for the mass of particles, you can again construct this delta PM action by taking modes in Zbar and show that it satisfies the W algebra as well.

[00:52:40]And and okay. Yeah.

[00:53:15]So yeah, so one I think nicer nice way to think about these primary descendants of soft factors and also to get sort of reoriented with the soft expansion is um that for mass massless particles you can think about these primary descendants as actually the action of light ray operators at null infinity or null integrals of the stress tensor acting on hard particles at null infinity. So for example So the W3 house generator corresponds to our usual the two derivatives of the leading saw factor. So this is leading u and its action on hard particles is given by the action also of limit as r goes to infinity of just the leading component the leading component of this uu component of the stress tensor as r goes to infinity.

[00:55:02]Okay. [clears throat] So again the global modes of this thing which are the modes between 1 minus p and p minus one. So here P is three halves and the global modes are minus a half and plus a half.

[00:55:17]Those are the two there's a that's a kyal half of translations. It's two translations and you get it by integrating this thing over functions f like with zbar and zz bar.

[00:55:32]Okay. So or one and Z bar.

[00:55:39]There's a similar picture at all subleading orders. So let's do the subleading one which is W2.

[00:55:53]Here we're going to be taking three derivatives of SK0.

[00:56:02]And this thing will be like uh the integral of this other combination of stress tensor components that now involve one of the transverse directions uh acting on your particles at infinity.

[00:56:32]Okay.

[00:56:35]And these if you ex if you um like expand the Lorren generators in these plati coordinates, these will look like um a local version of the Lorren transformation generators at null infinity. And just like and like we like we saw before the subleading soft theorem corresponds to uh the super rotation symmetry the global part of which is lorren. So the global modes of the m equals 2 thing are 1 minus 1 and zero which is our kyal half of the lorren symmetry.

[00:57:10]Um, good.

[00:57:14]And then a sub subleing. I'll just write one more This is some uh more general transformation of these hard particles at PK and it now involves four a set of closed four closed modes. So this pattern will continue at all subleading orders.

[00:58:23]and we'll find uh that they all involve further integrals of the stress tensor and infinity. Yes.

[00:58:42]>> So at the leading order you can you can get both the chyal and the non-chyro but at subleading order um you have to use both holicities and this is related to like the shadow of of the uh leading the leading thing like and this shadow being uh uh related to each other. Yeah.

[00:59:07]Good.

[00:59:09]So we can label all of these So all these yeah all these primary descendants of a soft vector can be labeled um again by in the same way as the w operators they transform in the same way and I'm going to introduce a not another notation of this thing as uh this operator WP bar where here the bar just oh sorry actually all these should have bars we're looking at the Z bar Okay. So we can introduce this notation of big W for the primary descendants of the soft factor. uh with right weight P.

[01:00:30]So WP has weights same as W 3 minus P and P and uh right again by construction we know that they obey the W algebra acting on massless particles at null infinity.

[01:00:54]So um one question you can ask is so right at null infinity all of our particles are sort of effectively free and we're looking at things at tree level and we wanted to ask how I mean this question is about how this algebra is corrected at loop level and what happens to it as we add more interactions. Um one partial answer to that question comes from a sort of completely different direction of considering these operators. So these these light ray operators the W bar P which you can construct just out of integrals of the stress tensor you can construct these W bar operators also in a CFT without gravity. All these stress tensors are just um matter stress tensors and you can construct them using the matter stress tensor of a CFT. And so um one sort of recent result is that you can show that if you're in a 4D CFT with no gravity necessarily and we take T to be the CFT stress tensor, you can do an independent there's an independent derivation of the fact that these W bar operators satisfy the W algebra.

[01:02:15]And in particular their commutator uh takes this form.

[01:02:28]Uh so it's a commutator of these local operators in the transverse directions Z and Z bar prime.

[01:02:39]This is a prime.

[01:02:57]So um the operator yeah the commutator of these operators takes the form where if you integrate it against the modes.

[01:03:09]So if we define modes WP M as the integral of It's a mode integral of these W bar operators.

[01:03:28]Um the structure of these derivatives immediately gives you the little w algebra when you do these integrals over the modes and the dot dot dot terms are all terms that um vanish upon this integration but they can be also computed explicitly.

[01:03:45]So this is a another instance where we we have found a W algebra in a generic interacting 4D CFT and sort of the connection with this action of the um soft factors at null infinity is because we're effectively considering this free action at null infinity and free CFTs are a CTF. Yes.

[01:04:10]Yes. You can also get the loop index. So here there's no Z structure here. So you can just also add powers of Z in this integral and you'll get the loop. Yeah.

[01:04:23]Okay. So right the last comment that I wanted to make about this um which is sort of a connection of how these soft factors yeah these soft factors also appear in the CTF context.

[01:04:55]So I know we haven't gotten to we haven't we are sort of partway through the CTF uh lectures but this is this will be hopefully um helpful for connecting with those lectures uh during the rest of the of the school and also it would be great to discuss more um so in the CTF context you can also Consider the threepoint function of these w operators uh with two scalers.

[01:05:20]So if we take two scalar operators O or we can yeah same scaler at two points sandwiched between one of these W operators or sorry we sandwich this W operator between two of them.

[01:05:45]This can be directly computed from just the uh tu new stress tensor. This so we we we can find this three-point function. It has a a general it's fixed by conformal symmetry as we'll see in the in the CTF lectures and then you can just take the appropriate um integrals of it to find the threepoint function of the w operators.

[01:06:19]And when you do this, um, you find a result that looks exactly like this descendants of the soft factor.

[01:06:36]Oh, we also for transform the same momentum space. So good.

[01:07:04]and um yeah just again to reorient ourselves the p= 31 is the leading soft theorem which would correspond to the anc operator which we know just gives us um the momentum back so Good.

[01:07:26]But and again here we have we're use we're we find that the primary descending that corresponds to this massive or offshell uh soft factor because these operators are not on on the mass shell in CTF on the massless um momentum support in CTF.

[01:07:42]Okay. So that's what I wanted to get through. I know it was really fast and there's a lot of new notation. So, I'm happy to discuss it uh during the rest of the school. Um, any questions for now? I also will try to type out my notes and send them around so people can have something to reference.

[01:08:21]uh so the central extension right is this p equals 1 term which you can just see from the algebra as central um there's no soft theorem interpretation of that okay so remember that yeah the p equals in this map of indices like the p equals 3/ corresponds to the leading soft theorem which is like a pole in one over omega so if there was some soft theorem interpretation of that central term it would have to be like a one over omega^ squar thing which is just there's no soft the like we don't know of an interpretation of that um from like momentum space perspective it turns out to play a role in defformation of this algebra um if you consider it in like ads um which I did have not had time to cover here but I'm happy to talk about more um but there's no direct uh soft theorem interpretation of it from what I've presented this way Yeah.

[01:09:14]Um >> yes talking about fix >> great yeah um so there's the same you can do this story totally an analogy where instead of having a yeah a graviton you have a photon you can take the conformally the Photons have a series of soft theorems as well. You can can take the soft limit or conformally soft limits that correspond to these different terms in a soft expansion and you'll find that there's an algebra that you get which instead of a w algebra is this different algebra called an s algebra which is basically like you know where the structure constants are just fab for your color color group. And also this these S generators you can show when you have mixed photons and gravitons they form an adjun representation of the W algebra and you can go through the whole construction of these stress tensor action on null infinity with currents instead of stress tensors in the in the cage theory case.

[01:10:22]>> Yeah.

[01:10:25]If you consider a series with both so >> uh no it'll just be gravitons. Yeah.

[01:10:41]>> Yeah. A graviton photon will contain a photon.

[01:10:48]>> Yeah.

[01:10:49]>> Well, it's because gravitons aren't electrically charged. So there would be a Yeah.

[01:10:59]>> Yeah.

[01:11:09]>> of which OP sorry the 40 OP the threepoint function is just fixed by kinematic is or yeah the kinematic structure of the three-point function is fixed. Um, so we're just using this o to threepoint function and then taking integrals of it. So there's no it's just a um I don't think there's a radius of conversion issue with it. Yeah.

[01:11:37]>> Yep.

[01:11:40]question.

[01:11:42]How do you take this operator terms of the terms of for example dress >> so I think uh that will be covered that's some there's some recent work on that by Sabrian Sabrina and Ian and Brett and Suy have been working thinking about these detector operators and dress states and maybe that'll be mentioned further in the conference we should definitely discuss Yeah.