Intro to Celestial Holography - Part 2 - Mina Himwich

[00:00:16]Okay. Um, so I guess we'll uh continue.

[00:00:20]Yeah. I guess I just wanted to make one comment about um the spontaneous symmetry breaking picture from the last lecture. So really the presence of any hard scattering process Um we see from the conservation of these charges that there's going to have to be some associated any any hard scattering process will have some associated vacuum and this um yeah if we have things that that vacuum that's sort of created by these hard particles is what's shifting under the insertion of a soft of a soft graviton. So um the vacuum is spontaneously broken but if we had no scattering of course there wouldn't be any spontaneous symmetry breaking.

[00:00:57]Uh okay good. So um right I wrote down the vector field for uh that with this yZ um that is a conformal killing vector on the sphere that's one of the asmtoic symmetries you find upon asking for what are the asmtoic symmetries of an asmtotically flat metric and we'll see that they're equivalent to the subleading soft theorem. Um but we're going to I I think I'm going to wait to introduce some more language about celestial symmetries um from the 2DCFT perspective and then we can see the action of the verosaurus symmetry in that context. So I'm going to switch gears in this lecture to talking about uh celestial amplitudes which are amplitudes that transform in a highest weight in highest weight representations of Laurens SL2C.

[00:02:00]Okay. So again this comes from our basic observation that the Lawrence group is the same as the 2D global confirmal group or yeah 4D lens group and uh these can be parameterized by matrices uh ABCD with a unit determinant.

[00:02:28]Okay. So one way to uh sort of quickly see how the Lawrence group acts as global compromal transformations on massless particles for example. So imagine we have like a massless particle that's traveling in the Z direction. Uh then we can do a boost along this direction and that will just multiply. So we can have our massless particle.

[00:02:55]If we boost it, it will just shift it to like some multiple of the momentum.

[00:03:07]And um this is acts like a 2D dilation on the sphere. And we also have uh so 2D dilations are like these boosts along the particles direction and 2D rotations are just little group rotations around the mass like a massless particles momentum.

[00:03:30]So this is also familiar from spinner holicity variables.

[00:03:36]Um if you've seen them in spinner holicity formalism we can parameterize the momentum as uh lambda alpha lambda alpha dot and these spinners lens transformations act as sl2c on these spinner indices. So for example we can write lambda alpha as a square root of omega um parameterized by z. So we have this energy scale here and point Z we can think again these Z and Z bar are on the sphere which we can also think about as projective space um and we have our alpha dot here I'm looking at an outgoing particle but there would also be a sign if we had incoming versus outgoing and our momentum so the lambda's transform under this SL2C transformation just by this matrix.

[00:04:44]And you can recap, you can rewrite this as a phase and a re rescaled energy and point omega prime and Z prime where this omega prime transforms as CD plus CZ plus D^ 2* omega and Z prime is this usual moious this transformation in 2D and then there's also a phase uh which will cancel out uh when we look at our transformation.

[00:05:36]So overall the momentum uh this momentum will transform as so this thing is going to be omega q hat I'll call it p hat omega p hat this will transform to omega prime uh p of zprime Z bar prime which can also be repackaged into just the usual lorren transformation.

[00:06:16]Okay.

[00:06:20]So Lawrence yeah. So Lawrence transformations act as these mobious transformations on the celestial sphere.

[00:06:28]And the program of celestial amplitudes is to you know as in 2DCFT we want to consider operators that transform and highest weight representations of SL2C. And so we're going to have things that are labeled by um a left and right SL2C weight. So we're going to have uh delta which is H plus H bar and S which is H minus H bar. And so this is going to be like 2D dilation the 2D dilation value which is like boosts along the null momentum.

[00:07:20]And then this is going to be like the 2D rotation which is like the helicity in 4D.

[00:07:38]Okay. Um so to define celestial amplitudes uh uh there are two steps. One is to solve the bulk wave equation subject to highest weight conditions.

[00:08:06]And we'll get into what that is in a second.

[00:08:22]And the result of this is that we're going to find wave functions that are labeled by H and H bar uh and also the bulk point X and boundary points Z and Z bar that carry the same labels as operators in a 2D CFT.

[00:08:43]Um so these H and H bar are the confirmal weights and Z and Z bar are position uh in 2D. And then the second step is that we can use these in LSZ or do a basis transformation of the amplitude itself if you don't want to use LSC.

[00:09:12]So uh the result of this is that we're going to find uh an amplitude that carries the same labels as a correlation function uh of global confrontal primaries in 2D coot.

[00:09:39]So we're going to define um a celestial amplitude suggestively written as a correlation function of operators as So this is a definition of just our regular timeordered um correlation functions but using these wave functions instead.

[00:10:20]So here I this I label is going to include the H and H bar labels and then other uh any quantum numbers that this has.

[00:10:45]Um and again you can also just instead do this basis transformation that we'll talk about in a minute at the level of amplitudes.

[00:10:56]So by construction these correlation functions transform under uh z goes to this transform point a z plus b over cz plus d as as a correlation function of 2D global conformal primaries. So in particular um it will get mapped under this transformation to just a product of the conformal factors.

[00:11:46]or sorry not in control practice it'll just get multip transformed with its usual uh weight of a correlation function under SL2C transformations.

[00:12:03]Okay, great.

[00:12:05]So now we're going to go over the the two steps that are involved in this um procedure for constructing celestial amplitudes. The first is to solve the bulk wave equation subject to highest weight conditions. So I guess I can write over here. So for step one, we're going to find highest weight solutions.

[00:12:42]of our wave equation.

[00:12:53]Okay, so here we're going to find that fi is labeled by x and z. So our highest weight conditions are going to be that L minus one 5X is equal to DZ L0 is equal to H + Z and L1 is equal to 2 D H + C ^2 DZ and these are the familiar action of the uh uh global modes of the stress tensor in 2D CFT. So we can find so these L's are constructed from particular combinations of the Laurens generators and um so there's an analog with the Z bars also. So if we have our low end generators mu just this anti-ymmetric combination then we can combine we can combine them in particular ways and I'll just write these here for completeness uh and then L plus one I should write bigger. Sorry.

[00:14:36]Okay. And then LBar M is LM star.

[00:14:46]All right. So um when we do this we find that there are two families of solutions um which I guess I can write over here.

[00:15:07]So the first type of solution is called the melon transform of a plane wave or is given by the melon transform of a plane wave.

[00:15:28]And it it's given by integrating uh our regular wave function momentum space wave function over energies weighted by some power of omega. So here we have e to the i p which is a function of omega z and z bar time x.

[00:16:01]And so this is a scalar one. If it there also there also would be factors here.

[00:16:07]If there's um for higher spin there'd be like polarizations that involve like dz of of uh p hat and stuff like that. Um and there's also epsilons that I'm not writing down. But this wave function again is going to transform with the thing that I just I guess the yeah the delta is h plus h bar and s is the just bulkity and so if we remember that we saw that omega transforms with this weight 1/2 1/2 so omega prime is cz plus d ^ 2 * omega So this from this transformation you'll immediately see that O delta um well okay if we have a scalar then you can immediately see that O delta will transform a CZ plus D to the delta to delta.

[00:17:14]Okay. So, right again. So, yeah, usual CFT operators are diagonal under rotations and dilations of the point that they're located at. Um, momentum states of course under already diagonal under rotations around the little group um rotations but not under boosts along their momentum. So, we're instead considering this boost state which has a definite scaling under boost.

[00:17:41]Um, okay. So the second family of solutions is called the shadow transform.

[00:17:50]And maybe I could put it over here. So yeah, like it's useful to keep this one up here.

[00:18:17]Okay, so the second family of solutions are given by the shadow transform of this and the shadow transform uh is a coariant transformation you can define in any number of dimensions but in 2D it looks like I'll just write it for the shadow transform of our wave function.

[00:18:52]So it's going to be an integral of the first family of solutions um over the complex plane such that uh it takes a wave function or some object that transforms with weights 1 - h and 1 - h bar to one that transforms with weight h bar.

[00:19:22]Um and so okay couple comments.

[00:19:32]So the existence of this two families of solutions is because the bulk wave the bulk wave equation uh is invariant under inversions and these bulk conversions will map between these two families of solutions.

[00:20:05]Um so another comment is in this first family of solutions the point uh Z and Z bar here has a clear uh interpretation as just the uh 4D moment just labeling the 4D momentum of this particle. Um but in this shadow solution this point Z and Z bar is not some easily interpretable point. It's just some point on the sphere and then we have a non-local integral.

[00:20:34]um over over this other wave function. So there's no clear 2D interpretation of this point. Even though it's labeled by a point, there's no like clear interpretation of what this is. It's just some uh non-local integral transform. And I'm also going to note that uh you can rewrite this as the lens the integral over uh lens just the usual momentum of 1 over p.q Q where P here is a massless momentum uh where here I'm again using that delta I'm assuming it's a scaler and taking delta as h plus h bar okay so just write scalar Yeah.

[00:21:52]version.

[00:21:59]Um >> yeah, I guess when you do an inversion, you're kind of like mapping infinity and the point. So when you're like doing a dilation, it's sort of in the opposite direction. And then there's a conformal factor from doing the inversion. Yeah.

[00:22:30]Um but maybe yeah I'll get we can talk about it in the break too. Uh good right. So in this form it makes it sort of more more clear what the generalization would be to massive particles. Um so these two solutions I've written down so far for massless for massless particles. Um but for massive particles you can also write down a solution.

[00:23:01]See >> that just takes the same form but instead of integrating over null momenta we integrate over massive momenta. And uh just write that down here. So massive momenta like we saw in the penrose diagram instead of traveling to null infinity they travel to time like infinity and they're labeled not by just an energy scale and a direction but they're labeled by a mass and then a three-dimensional or a a direction vector that has three parameters instead of two. And those can be thought of as the points um parameterizing a uklidian ads 3 which uh which also parameters parameterizes time like infinity. So >> you're considering >> just yeah these solutions are this is for massless >> for so >> oh we haven't done that yet I'm gonna generalize this one Yeah, for acid particles and here yeah I used I used the fact that h and h bar are uh the same to repackage that it's just a dotproduct of these two vectors.

[00:24:39]Yeah. Okay. So for massive okay good. So mass and momenta again uh we can parameterize them by a direction vector that's labeled by y w and w bar where these yw and w bar are in a uklitian ads3 with this metric.

[00:25:24]Okay. And so um the solution in this case will be an exact analogy with sort of that form.

[00:25:33]It's uh I guess I can hopefully it's clear that this we're talking about the massive ones now. It's given by an integral over uklitian ads3 uh times this one over p.q to the delta where now p is massive and that thing is known as a burebound propagator.

[00:26:12]So again here P is the mass of P and this again is just one over P to the delta.

[00:26:33]Okay.

[00:26:36]So um this is by construction covariant and you'll again get something that transforms with or this scaler with a weight delta and again there's no importantly there's no simple interpretation of this boundary point we're integrate. So this massive momentum is going to this point WW bar. Yes.

[00:27:06]to >> Yeah, >> but >> we'll get to that in a second. Yeah.

[00:27:14]Yeah. Good. That's a great question.

[00:27:16]Yeah. So, um for massive particles, right, there's only this one type of solution. There's this one family of solutions.

[00:27:25]Well, okay, let me let me um finish my point about So, yeah, there's no there's no simple interpretation of this boundary point. we have to integrate over the coariant thing to do here is to integrate over all of ADS3 uh and then label it by a point this point on the sphere and so if you want to have a massive particle in this usual description in celestial holography it's going to be just a non-local operator um and there's been we've computed some explicit celestial amplitudes with massive particles and they have some you know they have branch cuts and stuff on the celestial sphere in Z so not usual nice CTF operators But you know we think we want to consider massive particles and this is one thing that we're still trying to figure out. So good. So in terms of the yeah so the massive wave functions now um the massive wave equation is no longer invariant under inversions. Um and there's only the single family of massive solutions that are allowed. Um but inversions do act non-trivially within the solution space. Um mapping these ones with h and h bar to ones with one minus h and one minus h bar. But we're going to we only really need to consider uh one set.

[00:28:43]And right about this point about the two families. So when you take um actually maybe I can just do it here. So if I try to take the um boundary limit of this bulk to boundary propagator uh then I'll actually find so this thing is going to look like y over y^2 plus the 2D distance zus w ^2 to the delta. Hopefully this is clear that these things are supposed to be equal. Okay. So if I take the if I take the y goes to zero limit of this then there's two terms. One where there's a delta function in z minus w and then one where there's a pole.

[00:29:52]And there's coefficients here that I'm not writing. Um, so and these are these correspond to the massless melon basis and the massless shadow basis.

[00:30:13]So the massive one is when you take them the limit to the boundary involves these two solutions, the melon and the shadow.

[00:30:20]Okay. And then another comment is that so you can ask put it over here. You can ask um what set of deltas will allow you to go back will allow you to like take this set of um wave functions and then transform back to a complete set of solutions to the wave equation in momentum space. And the analysis was done um by Perski and Sha. And I probably should write down.

[00:30:58]So the reference for all of this section on celestial amplitudes is uh okay.

[00:31:10]And now I'm worried because this one has five numbers, too. So maybe I'm just like Hopefully it's okay. This is good. So if we take delta to be 1 + i lambda where lambda is real. These deltas are known as the principal continuous series and they form a basis for delta normalizable solutions to the wave equation.

[00:32:22]And you can transform back by doing this inverse melon transform over delta.

[00:32:38]Guess I called this I guess slightly different. Okay.

[00:32:47]Is this a schematic?

[00:32:49]Okay.

[00:32:53]Okay. Good.

[00:33:00]So, right. So, we think that these deltas because yeah because they form a a basis for solutions of the wave equation. We want to think about potentially these deltas as in our spectrum of the celestial CTF.

[00:33:13]Yes.

[00:33:17]>> Yeah. Yeah.

[00:33:20]Or Yeah. dus one. Um and right so we um but we also know from we also know that there's going to be other operators that are not on the principal series. um because those operators correspond directly to the soft insertions um that generates asmtoic symmetries and I'll we'll go through how that works now. Um so the next section is going to be about how asmtoic symmetries look in celestial amplitudes start with because I can use this.

[00:34:08]>> Yeah.

[00:34:13]massive case.

[00:34:14]>> Um for the so for the massive case this is where this is what the this uklidian ads3 is what's parameterizing our massive momenta.

[00:34:28]>> So and it's just a point that they asmtote to at future time like infinity.

[00:34:31]Yeah.

[00:34:34]Okay. Good. So over here okay so now we're going to do asmtoic symmetries.

[00:35:06]in celestial amplitudes.

[00:35:18]So we're going to now consider the soft expansion. So yeah, like in the case of leading soft theorem, we could consider expanding an amplitude with an outgoing graviton in the energy of that in the the energy of that graviton as it goes to zero. So we're going to do a soft expansion of our amplitude and this is going to be at tree level.

[00:35:52]We can just write it as a sum over tailor expansion in the graviton's energy with these terms a l that are labeled by the graviton's direction and then the other momenta.

[00:36:08]Okay, so again yeah cap is a gravitational coupling.

[00:36:13]I think we've already introduced a limitation. So um if we want to look at okay actually maybe I should have some little more general discussion of these melon melon transforms. Um so that over here.

[00:36:40]Yeah.

[00:36:45]So yeah, so just this melon transform yeah just as as an integral transform.

[00:36:58]If we can think about some function of omega and if this function falls off as omega to the minus a as omega goes to zero and omega to the minus b as omega goes to infinity then the melon transform is defined So in the delta plane um as in this strip between delta equals a and delta equals b and we know from the universal soft factor that this term is this a is going to be one for gravity.

[00:37:48]So, and then it's possible there's going to also be further poles uh on the negative axis which we'll see in a second will correspond to further subleading terms in the soft expansion.

[00:38:10]uh and then these poles on the positive real axis will come from you know operators in in your EFT or something how the amplitude falls off of high energies and in particular if there's exponential suppression of the amplitude all these poles will be erased. So here we'll be just we'll be focusing on these poles on the negative real axis.

[00:38:37]Okay. So and you can roughly see that if we take this thing and then um do an integral from zero to some cutoff of uh of our amplitude.

[00:38:59]Then it's going to be like the integral of um the sum and should use a different board. Sorry. Um and then this maybe I can put it up down here.

[00:39:40]So this thing is going to then become just the sum over 1 / delta plus l a with some factors of lambda.

[00:39:54]So if we're just worried about if we're just thinking about um the soft expansion of this thing around zero um these different powers in the expansion are picked out by different uh residues as delta goes to different values different integer values. So another way to write this is okay if I rip me to one of the older ones.

[00:40:29]Okay, this one as we can pick out the elf term in that expansion by taking the limit as epsilon goes to zero.

[00:40:58]of our melon transform to a negative uh dimension plus epsilon of full amplitude.

[00:41:13]And from this expression which is maybe now covered we can immediately see that this is just like the melon transform with something of weight minus delta. Yeah.

[00:41:32]>> Oh sorry this is also a uh sum.

[00:41:40]Yeah. So we want to pick out each if we take the limit as delta goes to minus L of delta plus L times this thing we're going to pick out the residue of this of the term that as delta goes to minus L.

[00:41:55]That's what we're doing over and so we're here we're just doing that but I'm switching to notation where we have epsilon.

[00:42:00]Um okay. So this tells us that the sub L leading term uh is already in a state of definite 2D conformal weight uh with dimension minus L. So it has this weight. HH bar is minus L + 2 over 2 - L minus 2 over two. So usually yeah HH bar is like delta plus S over 2 delta minus S over two and here S is two for the graviton.

[00:42:45]Okay. So this means that for example the leading soft graviton uh you can isolate it by taking this uh you know minus L to be one and when we interpret the bleeding soft graviton uh as an operator on the celestial sphere we're going to interpret it as an operator with inter dimension. Yes.

[00:43:21]Yeah.

[00:43:23]>> Yeah.

[00:43:30]>> Yes. So there'll be logs at loop level here. I'm just sticking to tree level for simplicity.

[00:43:35]>> Yeah. Do these generate new >> they'll generate higher higher order higher order poles >> um >> like or >> um yeah they can be so they can be some of them can be resummed also we can discuss maybe that during the break uh but yes they'll generally be higher order polls integers yes Um I think right now we're just we're imposing momentum.

[00:44:27]>> Yes. So right at subleading orders it's going to get more complicated because to right when you're taking the momentum of something to zero then looking at the subleading terms you're going to have to start shifting the other particles momentum to keep momentum conservation uh true. So yeah there's going to be some I'm going to present a soft expansion that actually um and we'll talk about momentum conservation at that point. Yeah >> yeah so far there's a delta and I think you can always We'll discuss the delta later. Yeah, great. Um, good. So, in the case of the leading soft theorem, we don't need to worry about the delta.

[00:45:10]And I think it's useful to now write uh the leading soft theorem in this new notation of celestial amplitudes.

[00:45:27]Oh, actually want to save that one.

[00:45:29]Sorry.

[00:45:35]Okay. So, Good. Right. So remember our leading soft theorem is that we take the limit as omega goes to zero.

[00:46:13]Now I'm not going to strip off. Now I'm not going to take the residue of the pole.

[00:46:25]So we'll have this leading term where again yeah Q hat's the gravitton PK the external particles and we have our original amplitude.

[00:46:46]So um right this thing just separating it out it's 1 over omega now times I guess I have my coupling out here omega k and then z bar minus z bar k over z minus z bar for massless particles sorry z - Okay. And so we want to isolate the 1 / omega term in the soft expansion. And to do that, we're going to take the limit as delta goes to one of delta min. We're going to take a certain limit can formally okay these limits. Um where did I write it?

[00:47:42]Anyway, the limits as uh I guess it's down here sorry these limits of uh the dimension to these integers are known as conformally soft limits instead of soft limits. So we're going to take a particular conformally soft limit to isolate the one over omega pole and then this the factor of omega here in the melon transform of the massive particle pk it's going to actually cause delta k to be shifted to delta k plus one that's just because there's going to be an extra factor of omega in the melon transform for that particle.

[00:48:19]Uh so just writing this out explicitly I'll get the top Okay, good.

[00:49:06]So, just we're just going to translate that notation to the notation of celestial amplitudes.

[00:49:27]So the easiest thing to start with is just transforming our endpoint amplitude. That thing is going to look like a correlation function of operators that are labeled by H and H bar from the melon transform. And because we want to isolate the one over omega term in this expansion, we're going to take the limit as delta goes to one of delta minus one acting on our graviton operator. So the graviton we're going to represent as this g delta which again has weights delta plus 2 over two delta minus 2 over2. This limit is going to correspond to the limit um that picks out the residue of the one over omega piece in the expansion.

[00:50:13]And then there's going to also be a positive a plus here for positive elicity. And then the other massless particles well I guess yeah here there are massless are going to be labeled by their points and uh h and h bar weights under the melon transform. So the left hand side is going to look like this.

[00:50:39]And then the right hand side now we'll have our factor of cap over two sum over k of z bar minus z bar k over z minus z k but and we're going to have our original particles but now the kith particle is going to have its HK and HK bar shifted up each by a half because of that extra factor of omega.

[00:51:19]Okay. So the action of this leading um soft or conformally soft graviton operator is actually you know it looks like a usual if you if you took a Z bar derives for of it for example you would find that has this usual one over Z minus ZK pole but there's also this shift of the dimension of the operators which is obviously something that's not familiar doesn't happen in usual 2DC cf so the leading soft graviton theorem which again is the global part of it equivalent to translation symmetry has this really strange action on the celestial sphere where it shifts operator weights. Um but because it's the symmetry of our 4D amplitude it's going to give us powerful it'll we can also use it to constrain the 2D CFT and we'll see that uh shortly but right so from this expression of the soft leading soft theorem in the celestial basis we can pick we can interpret this g delta or this with the limit as delta goes to minus one this operator uh is interpreted as a weight three halves minus one/2 operator and it has an op we can interpret this uh this saw factor as a factor and an op. So if we take the limit as delta goes to one delta minus one g delta with some operator its op will look like this factor of Z bar over Z times the original operator with shifted weights.

[00:53:13]Okay.

[00:53:15]So right again you could take a Z Z bar derivative of this and it will look like a current in 2D but there's this additional doesn't have the standard weight of a usual current and that weight actually comes from the fact that the the weight of the operator the hard operator is being shifted.

[00:53:33]Okay.

[00:53:35]All right. So now we're going to do the subleading soft graviton theorem in the same context or in the same notation.

[00:53:43]And we can do that here.

[00:53:46]Okay. Actually, I can reuse part of this expression. So, so we we've expanded our we've written our general general expansion here. The leading term is this L is minus one term. The subleading term is the L equals0 term. And it turns out that the L equals Z term can also be written as the sum of a soft factor acting on the original amplitude. So um >> Uh we haven't done that limit yet. Yeah.

[00:54:41]Um you can also if you do look at the co so we'll talk about that shortly that um when the limit of these two points comes together that limit is the colinear limit for massless particles. And so this term is actually also the leading term in the colinear limit between the graviton and the hard particle. Um but right right now we're talking about this as the soft limit.

[00:55:02]uh but we'll see it come up more generally in colonial limits shortly. Yeah. So good the subleading soft factor.

[00:55:21]So at at subleading order, our amplitude, which is a function of the graviton's direction and all of our external particles, can also be written as the sum over a saw factor acting on each of the particles, which looks like the leading term uh multiplied by something involving the angular momentum.

[00:55:51]generators here. I'm going to just write it for scalers, but there'd also be a piece um that involves the spin of spinning spinning particles. But here, I'll just for simplicity, we're going to focus on scalers.

[00:56:03]And it's just this thing acting on the original amplitude.

[00:56:11]So here I've defined this is going to be our S0.

[00:56:19]It also depends on qhat. epsilon remember is given by this derivative of qhat.

[00:56:25]And this f m new is a symmetric or anti-ymmetric combination of epsilon and qhat.

[00:56:37]And l is our usual angular momentum generator.

[00:56:56]too messy.

[00:57:05]Okay, good.

[00:57:08]And right like finding this term in the sum finding this term is is much more complicated than a leading one because you have to take into account subleading terms and you know as was mentioned in the question uh you need to be careful with a momentum conserving delta function but it turns out that's this soft this subleing soft theorem um is also universal it was proposed by kachazo and strumminger in this paper.

[00:57:46]Oh yeah, I also meant to say somewhere.

[00:57:55]Yeah. So, more details about like celestial amplitudes and uh different different behaviors found in this paper by Monica Anna, Nema and um Ellis on and Andy or sorry, no just Nema and Andy.

[00:58:12]And this is the uh archive number.

[00:58:19]So the subleading soft factor we can we can write it out like we did with the leading soft factor in momentum space. So this thing uh is going to look like when you expand it out it's some big uh well not it's not that big some it's a differential operator acting on the amplitude.

[00:58:52]So again we have our 1 / z minus zk and now we have this uh differential operator and under the melon transform omega d omega k uh becomes like minus delta and Okay, I'm going to erase this. Is that okay?

[00:59:26]Okay, it's hearing Noah.

[00:59:30]So now we're going to be interested instead of taking limit of delta goes to minus delta goes to one, we're going to find take the limit as delta goes to zero to isolate this subleading term.

[01:00:00]So in the celestial basis uh the subleing soft theorem is going to look like the limit as delta goes to zero of rg delta Um, which again I'm going to assume we're having we're looking at scalers here.

[01:00:40]Uh, so that delta is 2 h bar.

[01:01:08]Um so again this looks like uh the action of a current with some now more complicated structure in Zbar and Yeah.

[01:01:25]Delta. The delta's here. Yes.

[01:01:28]>> Oh, yes. Good. Thank you.

[01:01:31]Great. Yeah. Uh and in so but instead of considering this limit, we can also think about uh taking the shadow of the subleading soft graviton.

[01:01:48]and the shadow of the subleing soft graviton um ends up being what looks like the nice Verauro stress tensor symmetry generator. So I'm going to instead consider this uh okay I guess I'll this leading one is also from and we can take its shadow which I put I left the shadow up there so hopefully Um so now h is two is delta plus 2 over two um and h bar is delta minus 2 over2. So we take this Now we're taking instead this transform of the amplitude and upon doing this so this is going to be defined as Um uh this this this um shadow of our subling soft oper operator is defined this way and when you do this you find that its action becomes the same as a 2D stress tensor. in particular.

[01:04:16]Well, it's a antiholmoric 2D stress tensor.

[01:04:32]So, we think about this thing as being like uh the action of some t bar and then we can also integrate it against uh an anti-holorphic conformal killing vector y of zbar to recover the anti-holorphic part of the super rotations that we wrote down earlier. So of course with the global the global part of which so if it's just one zbar or z bar squared those correspond to a kyal half of the global lawrence transformations or the global sl2c um a more general y bar will give you the super rotation transformation which corresponds to the more the higher like lb barbar modes in the veraural algebra.

[01:05:30]So and you can also you can also go through the same construction with the negative helicity graviton which you can then shadow transform to get the positive helic the z or positive helicity stress tensor.

[01:05:43]Okay.

[01:05:45]And then this uh construction of this 2D stress tensor was done in this paper.

[01:05:59]Okay. So another comment is so the leading soft theorem the shadow transform I guess didn't I guess I um okay so yeah it turns out in the case of leading soft theorem the shadow of the leading leading soft theorem and it shadow are just are related to each other so we don't need to consider the shadow transform in the leading case, but in the subleing soft theorem, it's really the shadow operator um that we want to consider as being part of the celestial CTF as the thing that generates the verosaurus the verosaurus symmetry or the local confirmal symmetry. Um, another comment is that the existence of this thing that looks like local confirmal symmetry really relied on the fact that we're using massless particles because we have this we had to you know we did this shadow transform of something that involves this pole in 1 / z minus ck. So you can go through the whole construction of the soft theorem and the asmtotic symmetry for massive particles as well.

[01:07:09]But when you do this transform, it's not going to look like the action of a stress tensor on a massive on on the massive particle because the massive particle again is labeled by this point.

[01:07:20]Um, okay. Well, the analog of the massive particle is labeled by a point that has come from a smearing over uklidian ads 3. So there's no like local action of the stress tensor on massive particles, but you can but there still is an action that that has a virus symmetry. It's just not this nice local realization of it. It's a realization on ADS3.

[01:07:43]Okay. So, check the schedule.

[01:07:50]Great. Uh, we have a little bit of time.

[01:07:53]So, the next thing I wanted to get into, which I think we're not going to definitely finish today, but we'll go well into the next lecture, uh, is talking about more about opes in general as collinear limits. Um so we've seen an example of how the like this colinear pole as Z and Z bar approach each other as two masses particles become colinear in the bulk the points Z will uh approach each other on the celestial sphere. This colinear limit looks like a term in the OPE again. Um and yes so there's a more general story about colinear limits and OP coefficients. Yes.

[01:08:36]Why?

[01:08:39]Why is the shadow the one that shows?

[01:08:54]>> Um, Okay. So I think I'll have to like look at look um we'll have to discuss them more in more detail like as after the lecture but I think in if you go to higher dimensions and do this construction it becomes clear there that the shadow is really what you want to consider um and but we yeah I have to go over that after the lecture with you.

[01:09:29]But yeah um but there's some story that the shadow I think is the more natural object from the higher dimensional perspective and then when we come down to 2D this is what we get. Um, we can also talk. Yeah, even though it looks like a non even though it looks like a non-local transform here, it still is what is giving us something that looks like a nice local operator. Um, I do have massless particles and there's some more comments I have about that after lecture too. Yeah. Uh, good. So I was going to just start going into some colinear limits and ops and then we can continue that tomorrow.

[01:10:41]Okay. So, right. So like we said before, massless momenta have an inner product that's proportional to their uh 2D distance on the celestial sphere.

[01:11:15]And uh we can think about taking the colinear limit of these momenta uh which corresponds to taking z12 and usually z bar12 to zero if they're complex conjugates. But now we're going to think about z and z bar as being independent which means we really have gone to bulk 22 signature.

[01:11:37]Um but uh yeah so this is a um sort of common thing to do in in defformationations of amplitudes and we are we are really going to want to try to interpret Z and Z bar differently in in the celestial CTF which we'll see um in a little bit and I'll make some comments about the symmetry interpretation with the Z and Z bar uh independent versus complex conjugates when we get there. So the leading colinear singularity in an amplitude comes from when a propagator goes on shell.

[01:12:22]So we'll have some vertex three point vertex And uh so yeah we take our anoint n plus one point amplitude involving uh n plus1 particles and we're going to take p1 and p2 colinear then in this limit as p1 p2 goes to zero we have this threepoint vertex X divided by the propagator and then an endpoint amplitude with now another particle that's coming in with the sum of the momenta or this one fewer particle with the sum of the momenta and if we write this in terms of omega which is omega 1 plus omega 2.

[01:13:36]Then the way it scales with omega will be as 1 over omega^ 2 from the propagator. There'd be a factor of omega to the dv minus 3 from the bulk vertex.

[01:13:50]So dv is the dimension of the bulk vertex.

[01:13:59]And then we're going to look at the leading singularity as z12 goes to zero.

[01:14:03]So there'll be some 1 / Z12 singularity here and then there'll be some power of Z bar12 which we'll call P.

[01:14:15]Okay.

[01:14:16]So I should not have written it on this board. Okay. But um I think we're technically out of time.

[01:14:32]Um, I'm just going to write down one more equation and then we can uh continue it tomorrow.

[01:14:48]Okay. So now when we take this equation and transform it to the celestial basis, we interpret this singularity in 1 / z as um the leading singularity in the op two operators. So we now have our delta O delta 1 um S1 second operator O delta 2 S2 um and this thing is going to be some power of Z bar 1 2 Z bar 1 2 the P over Z12 and now Our operator on the right hand side is going to involve delta 1 plus delta 2 plus dv minus 5 which comes from the fact that there's an overall power of omega to the dv uh minus 5 here.

[01:16:01]So yeah took this thing it has power of dd minus 5 that means that in the melon transform of this operator there's going to be a shift by dd minus 5 and it has spin s3 let's add z2 and basically coariance of this thing under the lorren group fixes that the spin of this third operator has to be s1 + s2 - p - 1 and that p has to be equal to dv - 4.

[01:16:45]Okay.

[01:16:47]So this coefficient um it can be determined either by just looking at the splitting function.

[01:16:55]Sorry. Yeah. This this thing is called the colinear splitting function. You can either compute the colinear splitting function from momentum space or you can actually also determine the colinear splitting function uh using 2D symmetries on uh in the boundary CFT.

[01:17:12]And so of course it makes sense that we can do it from either perspective. Um but it's nice to have a intrinsically 2D way of seeing what these OPA coefficients should be. And so in particular trans they're they're fixed by poner asymmet the op coefficients in the form of collidarial splitting functions in momentum space are fixed by poner asymmetry and uh the action of ponkery symmetry on this OP on the celestial sphere also will fix what these what these op coefficients are and because single particle states uh momentum igen states correspond to this to a family of operators O delta all of these OPA coefficients are going to actually be non-trivial functions of delta and we'll see exactly how that works uh I guess next time. So I think the plan for next time is to yeah go over how we fix the OP coefficients using point array symmetry on a celestial sphere. Then we're going to go over looking at the OP of two gravitons and then to two conformally soft gravitons. So the algebra of how these different symmetry generators act on the celestial sphere that will give rise to the W algebra and then we'll talk about the W algebra from a more position space perspective at the end. Yes.

[01:18:31]>> But this is still only for massless particles. Yeah. Okay. So you are continuing like there is always this disting distinction between massless and >> Yeah. So here >> this colinear split story is all from massless particles. I'm going to go through yeah so and we're going to go through like the OP of two gravitons which is you know massless.

[01:18:54]You can also act about ask about how the gravitons act on hard hard particles which would be like an op of a graviton with something else. That only works for massless hard particles for the massive hard particles. Um I'll show you the generalization tomorrow or yeah tomorrow.

[01:19:10]Yes, >> you mentioned that the subleading theorem is also universal and I'm assuming that means the sub sub and so on haven't been figured out yet.

[01:19:20]>> Uh the sub I mean universal in the sense that it doesn't depend on um the external matter and also there's a one loop exact correction uh so it doesn't get corrected by um you know higher dimension operators or more loop corrections and stuff like that. uh the subleading soft expansion is just something you can do in principle there's so tomorrow we'll talk about a a particularly nice way of writing the subleading terms but um in principle those that nice way of writing them can just be corrected and as theory dependent yeah >> yes >> um so in the beginning you wrote that the boosts can be represented by moius transforms uh but that's only true for 3 + 1 dimensions, right?

[01:20:11]>> Yeah.

[01:20:12]>> Um and so what part of this discussion sort of gets changed in higher dimensions? What relies on that?

[01:20:19]>> I guess so far we've only talked about representations of these operators. Um so you can still label operators by their dimension and spin in higher dimensional CTFs. there's just not going to be this like local looking Lorent action on them. Um but they'll still transform under the global conformal group in higher dimensions which are the Mobius transformation. Yeah. Yes.

[01:20:48]>> Is it the case that if I were doing this with the subleing colinear singularity in order to fix these sort of coefficients I wouldn't look for coariance under meth. look for it under like like >> um >> so okay there are a few things one is that the subleaning terms there's not necessarily like a nice subleing colinear term like this that can be non-analytic uh behavior which I think Lance is going to talk about further in the workshop but also yeah I think in the true CTF like you'd want to use the real you want to use verosauro symmetry to constrain subleading terms in the P.

[01:21:30]Um, so far basically people have just been using the global global SL2C. Um, yeah.

[01:21:49]Okay. No more questions. We'll start again tomorrow and we'll discuss it in time.

[01:21:57]Yeah.