 Let's start with the afternoon session. OK, so we will now have the third lecture of Laura Donné on celestial amplitudes. Thank you, Francesco. Thanks everybody for coming. Good afternoon. I hope you can hear me in the back, yes? OK, so today we will finally come to talking about celestial amplitudes per se after these two first lectures where we had to make a detour to understand what was the detailed structure of the flat space in the infrared. And we saw these nice symmetries and that they had charges and that these charges were constraining the scattering problem for masses particle in flat space times. And today we will see, so we will go to now the definition of what celestial amplitudes, and we see that this essentially amounts to have a very nice way of organizing scattering elements in flat space time, which will make the SL2C transformation of the Lorentz group very manifest. And these eventually will help us to understand what is the potential holographic structure of quantum gravity in flat space in terms of the so-called celestial conformal field theory. So let me recall briefly what was the last message of last lecture. So yesterday I tried to advocate that there was a nice interpretation of soft photon and we mostly discussed actually some graviton insurgents in the quantum gravity S matrix as a word identity for some two-dimensional currents. And these two-dig currents manifest the rich asymptotic structure at the boundary of flat space. So for gravity we had these currents were given by super translation currents, which I will come back to it when we will talk about celestial CFT. We will come with a nice definition for this current. We also had for gauge theory this U1 catch moody current, which is whose insertion is responsible for the leading soft photon theorem. This guy gives you the leading soft graviton theorem. And I mentioned, and we will come to a concrete construction for this object, but that there is an object which plays a role of the stress tensor in very much like we are used to. It's in usual 2D conformal field theories. So that was the main message of yesterday. And today we will now go to the core of these lectures, which is to talk about celestial amplitudes. So let's, in order to present you this celestial map that maps the particle states to objects in this sphere. Let me just briefly introduce some aspects about Lorentz transformations and the celestial sphere, which I recall is the two dimensional Euclidean sphere at future null infinity, which is identifiably identified with the past null infinity. So we have, let me write down the Lorentz algebra. Just to start slowly, where I use the notation J, I, so I is one, two, or three. J's are my three rotations. J is one, two, three, or three boosts. And these are the commutation relations between these objects. And we also vary I, when we know that there is aomorphism between the Lorentz algebra and SL2C algebra, which is also something you have seen in other lectures here. So the Lorentz generator are here denoted by the usual LMS mode. And this is the order commutation relation, where M and N are also minus 1, 0, 1. And OK, there is a precise way how to obtain this L in function of the rotation and boost generator. Let me not write all these down, because you can find this in most of textbooks. But let me just write one of these relations. Just have an ID. OK. Now we will need, and actually I've already kind of introduced that in the previous lecture, but we'll make a choice for an embedding of how the celestial sphere is embedding into Minkowski, the celestial sphere, into actually the null code. There is no unique way to do that. But there is a nice form of the embedding which I can take. And I've already actually introduced that when we were talking about scattering a massless particle. And this is the choice for, if you remember, the particle has this momentum, p mu, which I've written omega, the energy time, this null vector yesterday. So we have seen this expression already before. And it gives me an embedding of the celestial sphere. And of course, you could make another choice for this embedding. But this one turns out to be very convenient for all these celestial amplitude purposes. So now a very, I mean, the key observation which is behind all this story is very simple. It's just the fact that the Lorentz group on the celestial sphere via Mobius or SL2C transformations on these sphere coordinates W and W bar. This is the usual SL2C transformations similarly on W bar. This is just a symmetry observation. E d minus b is equal to 1. Yes, yes. Yes, yes. So now by using this just simple observation, and I will come to precisely how one can perform this map explicitly. If I write the S-matrix element into a basis which manifests, which expresses in a manifest way the SL2C invariance or covariance that acts on the celestial sphere, then by construction, this object written in this sort of basis will transform covariantly under the action of SL2C. And this is precisely what we are aiming at. We are aiming at eventually interpreting these S-matrix elements in this basis. And again, I will just now define how you can perform this change of basis. So by construction, you will have this SL2C manifest transformation laws. And celestial holography or celestial amplitude is the statement that we will want to interpret these elements as a correlation function. So it's something totally different of operators which represent the massless particle in theory, which is this so-called celestial conformal field theory. So that's the main objective of this transformation that I will present now in more detail talking about conformal primary wave function. Laura, sorry, I have a question. So N is the number of particles involved into the scattering? Yes, thank you. Yes, sorry. So basically, the usual, someone is not muted, I guess, online. So the usual drawing that people do is that if you have a scattering process involving n particle here n equals to 5, you will see or reinterpret this thing as some correlation function, as insertion of operator. So each operator here, operator, which represents massless particle, which here is the celestial sphere at a point, the given point, w, w bar. And the delta will come to it, but it will be the dual variable of the energy. I will precisely come to that. So you will see this scattering process as some insertion of operators. So if you have two incoming and three outgoing, they will all live on the same celestial sphere. So this follows from the fact that, remember, we had some antipodal matching between the past and the future. In principle, you might think there is one celestial sphere at the past and one of the future, but this antipodal matching is precisely made in such a way that these two regions are identified with each other in such a way that a free massless particle will enter and exit the celestial sphere at the same point. So this all comes from this antipodal matching condition I've written. The main message is that there is only one celestial sphere, and we will have insertion of operators which represent a massless particle coming from the past and per se is at a given point the sphere, w, w bar, and we have n of them in principle. So this correlation, can it be interpreted as coming from some specific null coordinate? Because presumably, the scaling dimensions refer to the Fourier transform in the retarded or advanced null coordinates, right? So in some sense, they are smeared all over scry minus or scry plus. So this dimension is not quite Fourier transform of the, neither the energy nor the, there is a Mellon transform associated to that. And I will come to this precise mapping. But I mean, does it correspond to strictly local insertions on some square? No, no, no, no. As you say, it involves an integral over all retarded things. So there's some smearing. Yes, there is some smearing. You're right. Thank you. So OK, let's go to this map. I mean, what is this delta? How is it defined? So conformal primary wave functions abbreviated by CPW. So let's, so all the magic will be done by this integral transform called Mellon transform. So let's consider again, a non-channel plane wave and the incoming or outgoing plus or minus i omega q mu x are the Cartesian coordinates. So here plus or minus is in versus out. So as we have seen, we can parametrize an al-momenta by three quantity, the energy, and the point ww bar at which the particle passes the sphere. And this Mellon transform that will make the magic for us, namely, trading the omega for this delta. And this is how the celestial map is implemented. So Mellon transform, the definition of this transform is as follows. An integral transform over a variable that is here will be the energy on the particle. So this is the Mellon transform of a function f, which depends on omega. It returns a new function, which now depends on this variable here that appears in the integral transform. That's the Mellon transform. There are some assumptions on this function f for the Mellon transform to be well defined. If you're interested, you can go in. You can ask me reference for all this, but I don't want to enter into too much detail. So delta in principle can be generally complex in this expression and take this form. And I will come back to that. So in principle, delta is an arbitrary complex. There is a well-defined inverse transform. So you can invert this Mellon integral, modulo some assumptions. But there is a way to invert this map. So notice that there is an inverse Mellon under some assumption. It's about the analyticity of this f tilde on some complex strip. We will not need all these details, but just to tell you that you can go back and forth under some assumptions. So what I will call a conformal primary wave function is simply a Mellon transform of a plane wave packet. So now I'm starting with a scalar conformal primary wave function. So this is the definition of what I mean. So we'll have a bunch of labels, so we have to be careful about all these labels because we don't want to neither confuse you nor be too sketchy. So a conformal primary wave function, which can be incoming or outgoing depending on whether the plane wave is incoming or outgoing, is defined as the Mellon transform of the wave packet and it will carry. Now it will no longer be labeled by energy omega but rather now by this quantity, this complex delta. So I'm just writing this Mellon transform for a plane wave and this epsilon here gives some regulator which is there to ensure that the integral converges, which is positive. So that's the definition. Let me introduce some stupid extra label that is not necessary here, but when we'll be talking about spinning wave functions we will have to introduce another label which is spin and here since I'm talking about scalar conformal primaries this level will just be done and just be equal to zero. But in general we can define more general spinning primaries. So this integral can be computed. Actually you can, in Mathematica it just gives you what's the answer. And the result is given by this where you have this appearance of this regulator here in the dominator. So it's plus or minus i to the delta gamma function of delta divided by q at x to the power delta. Why are we interested into this? Well because now the nice thing about these functions as their name suggests is that phi transforms now as a bulk, as a scalar under the bulk Lorentz transformations. So you see that this object depends both on the bulk coordinate x and also on point ww bar on the sphere. It's just a scalar under the Lorentz but under the action of the Lorentz which is induced by this embedding. So on this SL2C action acting on ww bar it transforms as you will see a 2D conformal or quasi conformal primary because now I'm just looking at the global part of the conformal group with the given weights. Like the weights that we are used to write down into the CFC h bar under SL2C. So I will write down the transformation rule so that this sentence takes some flesh. So if I do a Lorentz transformation on the coordinate which is accompanied by a Mobius transformation on the angles ww bar and then I could have just written down w prime, w bar prime. The transformation rule is that you can check that this is true and there are two ways of expressing the weights either in terms of h and h bar either in terms of the conformal dimension in the spin and they are related via this equation. The conformal dimension as usual is the sum of h and h bar while the spin, the 2D spin is the difference where it's actually zero but when we will generalize this to spin in particles it will no longer be zero. So equivalently this is actually equal to if you are more familiar with this way of writing the conformal transformations here is an unequivalent way of writing that. So you should recognize this as the transformation rule of a primary of weights h and h bar. So what we are just doing is we are making this integral transform on the plane waves and the reason why we are doing that is because now we have made the SL2C action manifest. So in summary usually we write in momentum basis where particles are labeled by their energy. Here in this boost let's call it celestial basis particle will be represented by this complex number delta and we go from one to the other via this valence transform and if we have a spinning particle in the bulk it will carry a, so that's the energy it will carry a helicity L and this will be simply identified with the 2D spin it just will be equated. So this is 2D spin conformal dimension. So now there is a common I would like to make which turn out to be rather important, before I write that maybe I just mentioned that this phi delta another way to state this is just that they are complex now highest weights with respect to the Lorentz group. So this is just another way to write down what the weights are. So since this is scalar 0 I will write this relationship where this L or the Lorentz generator I have written before. So basically now the action of the boost is diagonalized while in the usual momentum basis we have a nice, the plane wave transform manifestly nicely under translations but the SL2C transformation is obscured now it's the other way around. We will have in these boost basis they said the transformational analysis to see will be very nice but now the price to pay is that the transformations under translations will be more obscured and I will get back to that later on. Is there any question on that? I have two questions. First of all this action of SL2C on W can be just derived from the definition of the complex coordinate on the sphere? Up to, well there is no unique way that this will induce a unique way to write this down so this will depend on this embedding so it's actually equivalent to choose a little group choice for a little group but if you give, basically if you give me a given embedding then yes you can derive this you can derive this in a unique way. Okay, but okay this two coordinates W is essentially, does not contain more information than X, no? It's just a way to parameterize the limit for the distance going to infinity. Yes, so the Lorentz group will the transformation on the bulk coordinates will induce a transformation on the angles at the sphere at infinity so they go hand in hand. Okay, so I'm not understanding why when you write the fields you write both X and W like if they are independent coordinates. Well, so in principle it's like a plane wave and in principle it depends on the bulk point and which is defined but also has a momenta which is pointing towards a point in the sphere so if you want here also this plane wave is also an object which depends both on X and on this Q and now this Q I'm choosing an embedding where this Q is parameterized by WW bar so it's just the same stuff but now in these bases somehow. We are not used to that because we are not used to write the null momenta I guess like so, but this is nothing. Okay, so W is the large distance limit of the dual variable of X. Yeah, exactly. So this object is defined at any bulk point X but now I will want... So it's a bit what we did yesterday when we pushed this wave, this function to the boundary by taking an enlarge R limit and then I mentioned this saddle point approximation computation that I didn't show but basically when you will push this to the boundary it will make, you know, the angles let's say Z here which are included, you know, in boundary coordinate you have U, R, Z, Z bar. It will make Z and Z bar to coincide with W and W bar when you take the enlarge R. Yes, I know it might be confusing this double thing but it's nothing but again if you look at the plane wave just inherited from this relation. Now let me make a comment. The statement that plane waves form a delta function normalizable basis which is statement where these things denote the Klein Gordon in your product. I can write down quickly the definition I'm using for the Klein Gordon to... So we are familiar with the fact that the plane waves form a delta function normalizable basis which press like so. So this statement translates into the celestial language in the following way. This statement will impose some constraint on the value of delta. So more precisely what you can do is just you take this expression here you do two melin transform, one for the first plane wave one for the second and you will find the Klein Gordon in our product of two primary wave functions. Let me just write W and not every time W and W bar there is one evaluated that depends on W on W prime so it can be in and out. So if you do a melin transform the two melin here you can compute this so the left hand side is equal to that by definition of what is conformal primary but at some point you will hit some integral and you will see that this integral doesn't converge unless you restrict delta and delta prime. See here sky depends a priori can carry another conformal dimension delta prime so it will constrain delta and delta prime to take this form where lambda is and lambda prime are real and you've encountered similar stuff in Matthias lecture before so these are the continuous here it's called the principle series of irreducible representation of SO 1.3 So this is imposed by the convergence of the melin integral and this is something that was worked out by Sabrina, Pasteur's key and Schuhang-Schau maybe I can give you some reference by the way for these conformal primary wave functions so I guess Dirac already looked at this but more recently they were investigated by the Bohr and Solodukin and also Pasteur's key Schau and Stromminger 2017 here's the reference for that and you can find the reference for the Bohr-Solodukin paper inside this paper so basically this statement about the delta function normalizability of the plane waves is translated into the melin basis as some restriction on the conformal dimension to lie on the principle series so I will come back to that because there are a lot of subtleties about this spectrum if you want but I think this is something that is worth mentioning because actually these modes these celestial primary wave functions which have these values of delta correspond actually to normalizable states which have a usual radiative fall off when they go to the boundary of spacetime but in general you might wonder what happens if I go off the principle series so if I take the real part of delta to be 2 for instance and this is also a very interesting thing to look at as we will see the stress tensor actually lies outside this principle series and we will come back to that when I will talk about tomorrow about 2D currents any question or comments? I guess this is just not a question but just a microphone left open otherwise I didn't understand the question so let me say a few things few comments about now spinning conformal primary wave functions because at the end of the day we will want to discuss catering of photons, gluon and gravitons so spinning so this was I've written here the scalar k's conformal primary wave functions so spin 1 first j equals plus or minus 1 so now the confusion is arising because there will be some plus or minus coming I mean plus or minus elasticity and there are some plus or minus which meant outgoing incoming I will actually now I will suppress the in and out labels because otherwise we will have too many indices to carry on so a spin 1 primary wave function will carry now well it will have some tensor indices and let's see let me write it here so j equals plus 1 so I'm looking at elasticity plus here so it has spin j equals plus 1 so it is defined roughly speaking as a scalar conformal primary times these tetrad m u and I will explain that but let me just write down the expression for the negative elasticity just to be complete so now j is equal to minus 1 we'll have m bar now times the scalar conformal primary so this carries the elasticity piece and you might think that I could have obtained a spinning primary just by multiplying the scalar by the polarization tensor but this is not quite true this m is not the polarization tensor it is the polarization tensor of plus elasticity this is a notation that I used yesterday corrected or shifted by a quantity that depends on the bulk coordinate x this q mu times q dot x and similarly m bar gives you the object for the opposite elasticity so again here plus and minus they don't denote incoming or outgoing but the elasticity these things are required in order to have a nice transformation under SL2C if you just take this polarization tensor times the scalar primary you will see that this guy doesn't transform nicely under SL2C you have actually to take this tetrad thing and so now with this spinning one primary so transforms under Lorentz and the Mobius now it will be also transforming as a 2D primary of weights h and h bar but now it's also transforming as a vector under the bulk transformation so this is the same transformation as for the scalar except that now it's transformed as it should under the Lorentz action notice that there is some intrinsic gauge fixing into this definition I don't want to mention too much on that but basically these guys are naturally in Lorentz gauge but just because of the equation satisfy for the scalar primaries and similarly and then I will stop bombing you with definitions we have a spin to spin to primary which is plus or minus 2 and this will define for us later on the currents that play a role in in SLSLCFT so for plus helicity this is nothing but now you need to take two of these amps here and similarly for j1 minus 2 where you take the bars and now these guys will transform as a tensor under the Lorentz group and it satisfies the so this thing satisfies the linearized equation of motions so yeah basically if it's in the so called the Donder gauge this will just be a box of h minus equal to zero a bit more complicated if you relax the trace and the Lorentz condition on it but yeah so just to finish so it's an important it's an important thing that the fact that to have this nice transformation you have to add this piece will make this object quite not equivalent to just the polarization tensor time the scalar one but they will actually be gauge equivalent to that but we we have heard in the first lecture that the gauge all that has to be with gauge we have to be very careful because there are some gauge transformation or diffeomorphisms that do not die at the boundary and we have to keep track of all that and actually it will be it crucial to to keep these gauge transformations because some of them can be non-zero at the boundary so just to tell you that the relationship between simply the polarization tensor time the scalar primary these guys is gauge equivalent or diffeomorphic equivalent to it so I'm not writing the explicit expression for XI, XI mu you can find it in the in the Pastersky paper the exact form is not very important but you can already see that there is a weird thing happening when delta equal to 1 so for this value of the conformal dimension you can see that the naive spin to conformal primary is no longer well defined it just reduces to a pure diffeomorphisms and this is actually a tight and in one to one correspondence with the presence of super translation and of this super translation current that I have presented yesterday and I will go back to it so here I just want to say watch out when delta equal to one and also when delta equal to minus one but this I don't want quite to discuss this when delta equal to one this guy is a pure diffeomorphism or sometime called a goldstone mode associated to the breaking of symmetry so far it's not clear why this has to do with super translations but I wanted to mention this already at this stage and similarly for the A mu so the A mu is gauge equivalent to epsilon mu times the scalar guy and the factor is delta minus one over delta so you can see again that delta equal to zero and delta equal to one seemed to be important value of the conformal dimension and it did this guy will define for us some goldstone mode associated to the breaking of large gauge transformation at the boundary so now this alpha gauge can be large namely it doesn't die at the boundary transformation again watch out when delta equals to one and this will actually define for us what people have been calling a conformal soft operator in celestial CFT for these specific values of delta any questions for now we have all the all the relevant quantities and transformations to talk about celestial amplitude to define celestial amplitude the last 10 minutes are you saying that all A mu's are gauge equivalent to that particular form with delta minus one over delta so this is an exact statement this this is just so what I mean is that this A mu so this is just an equality that follows if you take you remember you the expression for phi delta was roughly speaking one over q the text to the delta times some i and some gamma of delta so if you take this and multiply it by that you will be able to write it in this form so the first term will just give you this piece and the other guy I'm sorry that I'm not sure I have the expression for alpha here oh yes I have it so that you can take if you want that you agree with this but the expression for alpha is just epsilon times x divided by delta times minus q dot x to the delta so this is just an equation you can check easily what I mean is that this A mu is not exactly equal to epsilon time in the scalar one but it's gauge equivalent to it they just differ by gauge transformations but usually we don't care about gauge transformation we just say yeah I mean we drop this but as we have seen in the first lecture there are some gauge transformation that are non-zero at the boundary and they act non-trivial on the states so we better keep them and indeed we'll see that this will define for us when delta equal to 1 these 2D currents and the point is you need that extra term for it to transform nicely in the SL2C so it's because this guy doesn't transform in a nice way but this object does do you also get like a funny 0 at delta equals to 2 in some other thing that you had the super translations not to somewhere so the special value for us will be 0 and 2 so delta equal to 2 as you might guess we have to do with the stress tensor for the gravitation conformal primary and the delta equal, did you ask about delta equal to 0 or delta equal to I think it is the super rotation which is called the Vida-Sauder thing and then I will mention some this shadow transform which exchanges delta with the 2 minus delta so the delta equal to and delta equal to 0 are related we have a shadow transform ok that they I will define what is a shadow transform tomorrow so in the last 7 minutes we can finally define what is a celestial amplitude so basically we have already encountered that so a celestial amplitude 3.3 so if I write a usual S matrix element these are the usual momentum basis so we talk about state somehow and now we will talk about wave functions we will talk about amplitudes and particles now we have a precise way before I was a bit sketchy but now so we have a precise way to define that so as I have said over and over mass is particle is parameterized by an energy a point that the bar pushes the sphere and this is the radiosity so this is the usual way we express amplitudes in these momentum basis so for mass less actually I am focusing on mass less case maybe I will say a word on massive case the celestial amplitude so I will call it curly C now involving n particle will be labeled now by two other quantum numbers this conformal dimension and the spin and it is just defined as taking n times a maline integral one for each leg of the usual momentum basis amplitude so we are here the identification between the L and the J is just really in the identification between the electricity and the spin as I have mentioned before the energy for this delta and this is definition of a celestial amplitude for the mass less case this map is simply defined with a maline transform but for a mass less case it is a bit more tricky but there is also a way to define to make an integral transform which is such that the scattering of massive particle will always transform nicely under SL2C so far by this inverse so this is just to go a little bit beyond what what I have presented so far where I focus only on mass less particles the map involves now no longer the maline transform another which is built from the tool from CFT embedding formalism and involves the bulk to boundary propagator from ADS a gator so this is just a common if you are interested into the massive case which I have to say is much less understood from a celestial holographic perspective than the mass less case so I am just writing down the reference for that if you are interested in looking at this how you can obtain a celestial amplitudes for massive particles so the important point and the upshot of all this is that for both cases here so this is not a maline what I meant is not a maline transform but something else but for both cases by constructions by construction celestial amplitudes transform co-variantly under SL2C this meaning again that if I act with SL2C by design I will have an n for endpoint function and time these sort of factors where the h is the weights of each associated to each scatterer so hi is again it's just delta one half delta i bar delta i minus j okay so what you can do now that you have this recipe to build celestial amplitudes you can take any formula for your favorite amplitude and the first one that were computed in this celestial basis were the three point maximally elicited violating gluon amplitude and I will not write down this because I'm running out of time but just so that you have an idea of what it look like you can really explicitly compute check that the amplitude you obtain say example three point mhv mhv gluon amplitude you can do the exercise this is the reference for the paper you do three mhv and then you get so you start from the amplitude you know from the formula and then you obtain the celestial amplitude for let's say two minus elicity and one plus gluon so there are some 30 you have to go in the split signature because and take z and z bar real and independent but basically you will find expressions that you recognize as a three point function in a 2D CFT there are some subtleties that I don't have time to discuss but you can ask me there is some delta function in the z bar coordinates this is inherent to the slow point functions you can ask me more on that but you recognize this as the usual form of a three point function with the given weights this is standard expression with definite weights where this lamb does so remember that delta I so here they are also lying on the principle series and delta I is one plus I lambda I and this is just I'm just and then I'm done writing this the specific form it takes so you can actually check for a bunch of amplitude that performing this integral makes all this CFT looking like transformation to be very manifest so this has also been checked for massive particle and for other sorts of particle but I will just stop here so thanks very much for listening