Jeremy, OK. All right, good afternoon. Thanks for coming to this last lecture on this topic of celestial holography and celestial amplitudes. So today, the plan is to, now that we have introduced all that we needed to talk about celestial amplitudes, to show you how there are real two dimensional currents that appear in celestial conformal field theory. And we will discuss also some features of celestial operator product expansions. And then I will try to give you some broad overview of this program of celestial of holography, what we have achieved so far, which are the big questions that are still open, and eventually what is the big picture that we want to achieve. So as usual, I will start with a brief recap of the main message of last lecture. So in yesterday, we defined what was a celestial amplitudes. And for this, we represented massless particles, the massless particle across the celestial sphere. Yes. At a point, then a z bar and carrying an energy omega. And eventually, and also a helicity, if it's being particle. So these massless scatterers are represented in celestial holography. Celestial operators denoted by curly O, which carry a conformal dimension delta and a 2D spin. So the 2D spin is identified simply with the bulk for the helicity of the particle. This operator is inserted at some point z and z bar. And delta, importantly, plays the role of the conformal dimension, which can be written as sum of the usual safety language left and right weights, where the spin is the difference of h and h bar. And this mapping, this celestial mapping from a massless carrying to an operator on the sphere was performed via this integral transform, which is a malin transform. So if you have a scattering of n particles, we will perform a malin integral over each leg. So for each particle of energy omega i, we will trade it for this complex, a very complex number, delta. So this is a scattering amplitude written in this convenient boost basis. And in celestial holography, we want to see to which extent we can interpret. And what this interpretation will tell us about the holographic structure of quantum gravity and the boundary of flat space. We can interpret this scattering in this boost basis as a correlation function involving these celestial operators on this putative yet to be discovered, celestial conformal field theory. So here I recall that the whole point of this was to make this SL2C transformation manifest. Good. So now I will turn to the so-called currents in celestial CFT. So these currents will appear when we take specific integer values of the conformal dimension delta. And we'll call them a conformally soft operator. So indeed, in the second lecture, we saw all these nice equivalences between what identity associated to ascentic symmetries and soft theorems. So we saw that all that the soft, I will mostly present, currents for gravity. But I sketch also the story for the soft photon theorem. So the soft graviton theorem can be interpreted as an insertion of a current. Now the question I want to ask is, you see where we had the soft graviton or soft photon were taking the energy omega 2, 0. But now when in celestial holography, we do no longer have energies to talk about. Because we have traded energies for conformal dimensions. So what does it mean in this celestial basis to be soft? Well, what people have came to realize is that there is an analogous, or if you want a dual formulation of softness. But now in the celestial basis, the statement of a particle to be soft. So before we have energetically soft particle, the amount to take on energy to 0, now we will be calling in celestial holography a particle which is soft in a conformal sense as a conformally soft particle. And a conformally soft particle, or a soft operator, will be a celestial object for which the conformal dimension take specific integer values. So these objects will play the role of these two dimensional celestial currents. The which values of delta do we have to look at? Well, today I want to present to you two specific values, when delta is equal to 1 and delta is equal to 0. Because as you will see, these two quantities will give us the two currents that we have insisted on in the beginning of the lecture, which are the so-called supertranslation current and the famous celestial stress tensor. So let's start with the first object, the supertranslation current. The which value of delta do we have to consider? Well, if you remember yesterday, I presented for you these conformal primary wave functions. And if you remember, there was a primary that was becoming a pure differmorphisms for delta equal to 1. And I told you, watch out, when these guys are large case transformation, we should not neglect these terms. And subtlety should arise for these values. And indeed, the supertranslation current in celestial CFT is an object, a celestial object which has conformal dimension equal to 1. So remember, there was the conformal primary wave function that was a pure differ for this value. So naturally, since all this business is related to asymptotic symmetries and large-case transformations, it's natural that something funny happens for this value. And the supertranslation current, more precisely, so if you don't remember what it was, remember that we have these in the usual momentum basis. Let me recall what this was. We had the word identity associated to supertranslation, supertranslation symmetry, where we had a statement that this matrix commutes with the charge associated to this symmetry. I think I've written that in this chart when we had different columns, that this word identity, if you split it into piece, you put the left-hand side of this equality, is equivalent to inserting this P here, S matrix. And inserting this object gives you the soft leading, soft Weinberg, soft graviton theorem. And if you remember, we had this sort of expression.  And now we will write this not in the, so here I'm still in the old basis, the momentum basis. Now we'll write this in the celestial basis. And we'll see that inserting this object makes some funny feature in celestial CFT. So this P inserts a soft graviton. So that was the old basis. Now in the celestial basis, this object will be obtained by taking this so-called conformally soft limit, namely when delta takes some specific values. And actually, it will not be quite exactly the celestial operator when delta goes to 1. So here there is a delta minus 1 factor. This is just a way to put the pole up, whether than down, that coming from omega to 0. Here it will correspond to delta equal to 1. There are different definitions for this, but here I've let the delta equal to 1 in this definition. So you can see that it's not just taking delta equal to 1 of a celestial operator of a spin 2, because I'm talking about gravity operator, but rather I have to take a descendant in the CFT language so derivative respect to z bar of this operator. And so we can count the weights of this object. So if delta equal to 1 and j, the electricity is 2, I can equivalently write in maybe more for a million four meter of h and h bar. So h is half of the sum of these things. h bar is half of this of the difference. So the weight of a delta equal to 1j equal to 0, celestial operator, will be denoted by h comma h bar. So here delta is 1j is 2, so 1 half of 1 plus 2. This equals to 3 half. And 1 half of delta minus j give us minus half. But you see this is not p. These are not the weights of p. p is a descendant of this object. So the weights of p given by the weights of this object, but now since I'm taking one z bar derivative, this will increase h bar by 1. So this is just a descendancy. So the weights of this poor translation current is this equal to 3 half, 1 half. So this is something that we are not familiar with in usual CFT. Typically, we talk about catch moody currents which have 1 comma 0 weight or stress tensor, which are 2 comma 0. So in particular, this p is not holomorphic. So it's a bit an abuse of language to say it's a current. But yet, we will keep calling it like that. And again, now we can rewrite this expression in form of a celestial correlation function. Now in a celestial basis, inserting this super translation current, which I record is in 1 to 1 with expressing Weinberg sub-graviton theorem. Now I'm replacing all my in and out state in terms of celestial operator. So let me use this short hand notation. OK, here we'll carry all the labels of a celestial operator. So if I insert this current into a celestial, now interpreted as a correlation function on the celestial CFT, I will just obtain that. So I'm just doing an n-maline transform. You can check this very easily. This is just by definition of what is a celestial operator as a maline transform of the plane wave. But now you see that before I had this in the momentum basis, I had this omega here in the numerator. Now when I do the maline transform, which is here, because I have delta minus 1 upstairs, if I want to rewrite this in the operator as a correlation function in a celestial basis, you will see that this is just nothing but the correlation function. But now where the conformal dimension of the operator k is shifted by 1. So the shift by 1 is just because we had an omega here. And when you look at the maline transform, if you multiply this by omega, it's the same as shifting delta to 1 in the formula. So again, here we see that the action of the support translation current, very importantly, shifts the conformal dimension of the operator in the celestial CFT, which again is something that we don't usually encounter in what people call vanilla to the CFT. But this is something that we have to deal with in celestial holography. This is just a consequence of super translation symmetry. And we know that this symmetry is there. So we know that whatever is the dual theory of quantum gravity in flat space written in these bases, it will have to obey this infinite amount of relationship implied by super translation symmetry. So you see that already just by recasting what we know from the momentum basis to the celestial basis, we already see some features appearing in the dual theory that is starting to tell us in which way this theory is different or familiar to the one we are used to. Is there any question on this stage on this current? If not, now let's talk about something a little bit less weird, which is the other current, which is very important is this thrust tensor in celestial CFT. So this is really, as I said before, the discovery that we had super rotation symmetry and then that this super rotation symmetry were enhancing the global conformal transformation to the full VR0 group, let people to be very excited and start to dig for more celestial or more CFT2 structure. So how can we get this thrust tensor? So celestial thrust tensor. So this thrust tensor was first obtained in the momentum basis by the work of Andy Strominger and others from really a reverse engineering from the subleading of the graviton theorem. And then they guess what was the form of an object that they needed to insert in this metric so that it's consistent with the symmetry. But in the momentum basis and in terms of the gravitational solution space, you remember this bond expansion I showed you before with the news, the shear. This object is really weird. It's a non-local expression. It's an integral over all non-infinity of some derivative of the new tensor. It was a really funny object that they really didn't understand why how to obtain this in a natural way. And now we can see this object arising more naturally as, again, a conformally soft operator. But there is a subtlety. The subtlety is that this thrust tensor involves a so-called shadow transform in CFT. So I'm not sure that this notion is familiar. So let me define what is a shadow transform first. And this shadow transform actually is ubiquitous in the celestial holography program. It shows up. It leads to many confusions. So it's important to discuss it. It is not just a technical point. It's really actually important in understanding what is the spectrum of celestial CFT. Do we have to include all shadow models or not a lot of debate in the current literature on that? So if you start with the shadow transform of a primary, you have to see primary of weights h and h bar. So again, h and h bar can be traded for delta nj. So what is it? So define xo. Let me write it and then I explain. So the shadow transform will change the conformal dimension. But it will return you till a primary. It also flips the 2D spin. Sorry, Laura, is it primary or quasi-primary? It's a quasi-primary. Quasi-primary, yes. Minus delta. So that's the definition of the shadow. It's an integral over your y and y bar. So there are some normalization here. It depends on the convention. You can find this definition in paper by Osborn, or Osborn and Dolan, for instance. You start with the quasi-primary of these dimensions, delta nj. You do this integral, which depends on which kind of operator you start from. You will have different powers here in z minus y and z bar minus y bar. And this will return to you another primary. But now with different dimension, now its dimension will be 2 minus delta. So if I were to write this in terms of h and h bar, you can see that if you start with something, if you start with the primary of this weight, after a shadow, you will have something with 1 minus h and 1 minus h bar. Now it was realized that the stress tensor of celestial CFT arises as the shadow transform of an operator of weights delta equals 0. So that we'll call a stress tensor. Indeed, if I start with something which has weight 0, the shadow transform will give me something which has dimension 2 minus 0, so 2. So this is the stress tensor in celestial CFT. And there is this delta here. Let me write it up front so that the limit when delta goes to 0 of delta times this expression. And this you can recognize as the, if you take delta equals 0 in this formula, you will see that this exponent becomes 4 and this is equal to 0. If I start with a spin equal to minus 2, it will flip the spin. So it will turn me something that has h equal to and h by equal to 0, which is the dimension we expect from a stress tensor. So it's the shadow of delta equals 0 primary. So again, the realization that this was a stress tensor is really, it's long. It came pieces by pieces by first realizing this stuff, sublatings of graviton theorems, reverse engineer, and then people say, OK, but there is a basis. Actually, this is one motivated to look at this celestial basis, was to look for a basis where the action of this T was diagonalized. So it was built on from all that. And here I'm just giving it to you. So you might think, wow, where does it come from? How could I have guessed it? Well, just looking at the dimension, you can guess. But this shadow transformed me. It's kind of tricky, and it took some times to understand why it has to be like so. So the main property I want to emphasize on is that indeed, remarkably, we have an object that obeys the word identity of a stress tensor in a conventional CFT2. Writing down here for you, you can find in any CFT book. So this is the word identity of a 2D stress tensor, as it should be. OK. So this equation was checked explicitly from amplitude people. So in particular, Thomas Taylor and Stefan Stieberger, they computed a lot of celestial amplitude, starting from the well-known formula that they have basically invented themselves. So they have checked this formula, checked explicitly. So by brute force, you start with the momentum basis amplitude.  So they looked at first Einstein Young-Mills amplitude with gauge bosons and one graviton. So it was really photopolis and Taylor for these. But then people have been looking at many extensions of these sort of facts. So this was really checked by starting with the momentum space Einstein Young-Mills. You do the milline transform. You do a shadow transform. And then you take the limit delta goes to 0. And you land explicitly on this sort of right-hand side. So what it shows is that if you represent gauge bosons by celestial operators, it proved that the celestial CFT operator are indeed full VeraZora primary fields. Because this is nothing but the definition of how a VeraZora primary should, if you want, transform. So this is really in one-to-one. So these sort of check proved that the CFT operators. So before we see, I was talking about mostly the global part. I was telling you about the cell 2C transformation, maybe transformation. But now you see that when we are coupled to gravity, the presence of super rotations encoded by this stress tensor is enhancing the group so that the operators are now primaries under the full local group. Namely, they are VeraZora primaries. And at least we are very happy because this is something we know very well how to deal with in usual CFT 2. So we might hope that because of that, we can exploit the techniques of 2D CFT into this celestial holography program to sort of bootstrap out of the blue, the celestial theory. Is there any question on that? Yes. On the bulk, the computation are done at three-level or they are also loop computation? Right, so this work I was mentioning was done at three-level. The extension to loop corrections is much less understood. There are being some papers on that. Most of the things I'm going to mention are on three-level. But we expect that at loop level, there is each respect exactly the same VeraZora. So OK, so what happens is that the bleeding soft graviton theorem has correction at loop level. But these corrections are one-loop exact. So it's corrected at three-level. It's exact in all order perturbation theory. One loop gets correction, but then there is no more correction. And these corrections are, if you want to recast this soft theorem as the identity of an object, you have to include another contribution to this T here. I'm not entering into this detail, but there is a shifted stress tensor, which is very subtle. It has to do with some vacuum structure of gravity. That's a very good question, but it's under investigation right now. But yes, there is a correction to this T. But it's still a 2.0 thing. But there are some fine details in the account. I didn't quite understand why we care that this T was shadow transform of a delta equals 0 operator. Where do we use it?  If you, let's see, how can I explain it? So yeah, actually there is an extent to which all objects that are all the currents actually, if you want to map them to the gravitational phase space, they're actually naturally related not to the celestial operator, but to the shadow operators, all of them. Turns out that I didn't have to talk about that here, because the shadow of a delta equal 1 mode has also delta equal 1. And there is some degeneracy because of that. But actually, there is a pretty funny aspect that if you want to translate with the gravity phase space, naturally what happens there are somehow all shadow objects. But the reason why we care is because this is somehow what it is. And if you don't have this idea of allowing for shadow transformation, you would not see this object in the first place. And it's pretty non-trivial that there is something of stress tensor in the boundary of that space. I mean, why would it be so? I have not understood this. This is a specific operator, the one from which you construct T, or is that generic operator with delta equal to 0 and j equal to minus 2? This one, is this O? So yes, this is generic in the sense that it represents the insertion of a massless graviton on the sphere with spin equal to 2 or minus 2. I have an analogous T bar, of course, if I take j equal plus 2. So suppose you have more than one operator with delta equal to 0 and j equals to minus 2. How do you know how to generate T, the stress energy tensor? If I have the generacy restain, no? Yeah, so basically, what differentiates these operators will be what differentiate their null momenta, for instance. And their null momenta carries a given energy and a point on the sphere. This is the only thing I know when I'm looking at the sphere. And so what I'm saying is that the stress tensor is related to the operator which insert a conformally soft operator, because delta takes some specific values. But secretly, actually, you can see that it's equivalent to inserting a soft graviton. So basically, taking omega 2 0 is something I'm just stating to you, but taking omega 2 0, you can see that from the structure of the amplitude we'll select some pores when delta takes some integer values. So apart from that, does it answer your question? Sorry, would I say that the uniqueness of gravity, the fact that there is only one spin 2 mass less particle. So if you were assuming that in this matrix, which is a very reasonable, then in this TLSL CFT has to have a unique. There can be any degeneracy of this operator. So it has to be unique. Otherwise, you will have two kind of gravitons that you can't have. Yes, yes, it's a very strong constraint because every time I'm writing this relationship, you have to see this as actually an infinite amount of constraint for the CFT to obey. And this is the whole philosophy of this program, is to try to have a bottom up approach where you derive from what you know constraints on coming from symmetries. And you hope that at some point, this constraint will be so strong, there will be so many of them, that it will, if you want, it will isolate theory that you can identify, or at least you can identify what properties it has to obey. Vice and versa, so I will come to that when I talk about where the program stands. But that's the, so the question was, you can repeat it. The question was, you should both infer property of the CFT by what you know that you should be, like there is just one graviton. So there must be just one operator with this property. Then understanding property of gravities from, let's say, constraints that I know that are on CFT, no? Yes, so the hope and the thing that people are actively trying to do is that now go in the vice and versa direction that you were mentioning. Namely, can we cook up some theory, a celestial theory, from which we can infer new things that we didn't already know from momentum basis and all this very rich literature on amplitude? And this is an extremely complicated thing to do because basically it would amount to basically have a holographic dictionary fully fledged. So this is an outstanding question to have an intrinsic definition of the celestial CFT. OK, very good. Thanks for the questions. Yes, I will come back to these big questions of what we know about celestial CFT so far, what we don't know. So I have a few minutes left before I go into the last section of this summary and outlook. Just the last thing that I would like to tell you is that there is a way to access the operator product expansions or OPEs in celestial CFT. No, it's not 3.2 at all. There are operator product expansions by looking at collinear divergences in the 4D momentum space. So the statement, and you will see why this is the case, collinear divergences of the 4D momentum usual stuff. So again, it's really bottom up. I will start with all these beautiful formula people have been developing and scattering amplitudes and see what it infer we can deduce in the celestial CFT.  That you remember when we parametrized the momentum of the particle like so, not going momenta. If you take the product of two momenta, p1 with p2, just compute that, and you find that it's given by omega 1, omega 2, time z1, 2, z bar 1, 2, or z1, 2. The difference between z1 and z2, and similarly. So in other words, a collinear limit when the two momenta become parallel, selects. Well, if you have a collinear divergence, it will extract the pole when z1 approaches z2. So basically, linear is the same as taking z1 close to p2. So in other words, you have a scattering. You take these two momenta to be linear. These correspond in the celestial sphere to look at z1 approaching z2. Then you have other. So you just take these two point to be close to each other. And then you have other insertions. So this very simple kinematic observation can also be used to now extract OPEs in celestial CFT. So I'm a bit running out of time here. So if you have questions about that, you can ask me in the discussion session. But let me just mention a few results on that. So if you look at graviton, want to derive graviton OPEs, you can do so. Let's take for simplicity 2 positive electricity graviton. So these are represented by two operators with J equal plus 2. They can depend on each, they carry a conformal dimension. So the computation has been done in this paper. I want to look at. So there are basically two ways to derive the OPE that I'm going to write down. The first method is brute force.  And you can compute it. So method one is false, namely, start from an amplitude, look at collinear divergencies. There are formulas for that. Do the melin, do the collinear limit and you will extract this structure. So this has been computed. If you have mixed electricity, it's a bit more tricky. And now people start to know how to deal with things. So this is the OPE you get, where beta is the other beta function, product of gamma. So this is a computation you can do. And the second method, which is a bit more acute, is to just use symmetries. So that's one of the very great things of these infinite dimensional symmetries that they are so powerful that you can, they fully constrain the OPE coefficient. So you can actually don't do any computation, but just if you remember that you have translation invariance and the constrain implied by the leading soft-graviton theorem, you can see that these symmetries, actually a little asterisk you also have to know about the sub-leading of graviton theorem I didn't talk about. But the main message I want to convey is that these symmetries, symmetric constraints, so you don't have to do this computation. They are powerful enough so as to, they actually imply some sort of recursion relations on the OPE coefficients. Before you knew that it was a little beta function. So as to imply an OPE coefficient, actually to uniquely determine the OPE coefficient. So the symmetry you need in this case, you need translation invariance, you need the leading soft-graviton theorem. The sub-leading soft-graviton theorem doesn't impose any constraint because by definition we are working in this SL2C basis. So this is already implemented. And you also need the sub-subleading sub-graviton theorem that I didn't talk about. So you can derive these, you can derive gluon OPEs, you can compute all sort of things. And before I go to the summary, let me mention some comments about which is related to these OPE and these currents. And it's the most more recent literature. But with that you almost, we have almost covered in features of celestial amplitudes. It's this observation that the delta equal one and zero up to, what do you call this shadow transform? It's conformalists of limit, speed to celestial currents. Our recent work have been showing that there seems to be an infinite tower of currents where delta is running at the same time as delta. Currents where delta is running over all negative integers. So, and very recently, Storminger managed to show that if you look at this very complicated structure implied by these infinite towers of current, you can nicely, by a clever field of definition, you can nicely recast this constraint in terms of a single simple algebra. Well, not so simple, but which is the so-called W1 plus infinity algebra, which is an algebra which appears in very different contexts in higher spin, but also in twister space. So now they have been this new interplay between celestial holography and twister techniques. So again, this is a nice combination of different communities working now together. So unfortunately, I don't have much time to talk about that in more details. And I will just, I think, go to the summary of what we have learned in these four lectures and what are the main questions that remain to be answered in celestial holography. So summary and outlook. So what we have seen, and I hope I've tried to convince you that you will remember that the infrastructure of gravity at some particular flat spacetime is very rich and very simple. We have this infinity of symmetries of conservation laws that any consistent quantum gravity theory should obey. They have an infinite amount of symmetries which are constrained in it. And celestial holography, the main point of this program is to make the full use as much as we can use of all these symmetries to constrain the holographic dual of all the gravity, so which again, the main claim on the main thing we are looking after, the theory of quantum gravity in flat spacetime in four dimension, which I recall, describe the real world to some extent. Could be described as a celestial 2D CFT. Leading on the celestial sphere. And there is this promising dictionary which involves this malintransformant equal which makes this SL2C transformation manifest. So we have to summarize and to finish. We have a similar feature that we are used to in usual CFT2. So the whole point is to understand what is the celestial CFT2, does it make sense as a theory? How can we push this and how can we constrain this quantity? And this right hand side here. So let me mention some nice feature that we have seen. So the main resemblance we've seen compared to usual CFT2 is that we have 2D currents. We have catch-moodle currents. We have a stress tensor, which is far from obvious. We have primary operators. We have also the descendants and so on and so forth. But there's a very different. This theory is also very weird to some, to many regards. So let me mention some funny feature we have seen. So first we had this, the fact that this conform my dimension of this primary is a priori any continuous and complex spectrum. We have seen that there is the super translation symmetry give an infinite amount of concern that we are not used to in usual CFT because they shift the weights of the operators by one, as I've written in the formula before. Shift the conformal dimension by one. And there are other things I just throwing at you, but I had some question before regarding that. So it's maybe a time to answer. So for instance, if you look at the four point function of in the celestial basis, you will see that it implies that the cross ratio has to be real and all the point actually forced to be inserted. This is just a constraint from kinematics. They are forced to lie on the equator of the celestial sphere. Just a constraint from kinematics. And this is something that we don't have in usual local unitary 2D CFT. There are also weird thing happening with the conformal block expansion that people are trying to treat the best as possible, but this is also something that is not so fully understood so far. And in the last minute, let me tell you what are the big questions that are left to answer before I take all questions you might have. So of course, this is a very broad topic. I didn't have to cover all aspects of it. And this was biased on my own personal taste. But one question you could have is what is the relationship or is there some way to use what we know from ADS CFT? How could we relate the celestial correlators to the one of ADS? But really celestial holography is very different from ADS CFT. So in ADS CFT, we have this nice time like boundary, the boundary of ADS. But if for flat space, gravity in flat space, we have this annoying null boundary and this is what all the difficulty comes about because in ADS, what we are used to is we are used to put some reflecting or Dirichlet kind of boundary condition where we say that ADS acts like a box. But now in flat space time, as we have seen, we have gravitational wave escaping the boundary. So there is some leak of radiation that we have to account for. So I think we have to deal, if we want to make this relationship precise, we have to understand how to deal with leaks of radiation in holography. So that's our first point. Second point we already discussed, but this amounts to say what is a celestial CFT? Namely, can we come up with intrinsic definition rather than this bottom-up approach that most of the works have been following? So our interesting definition or a list of properties, full list of properties that this theory should obey because so far, it was mostly bottom-up from what we already knew from amplitudes. And we want to top down, we want to start with a CFT and infer something we didn't already know in quantum gravity. And I think that in this regard, we are starting to do so because this nice symmetry structure that is emerging has compelled us to reconsider many aspects of gravity in flat space. We had to look deeper at new kinds of memory effects, new kind of observables. We had to relax very much this Bondi-Mesne-Rosex expansion that I showed in the beginning. And we start to understand how all these things are connected to each other. So I think that we are getting there slowly. And finally, I think one very ambitious goal, which is actually at the core of my own research project, is to have some sense or to eventually use this program not to constrain just a single flat space time, but space time which contain a horizon and which have black hole inside. In the presence of a black hole, this story is way more complicated. But eventually, I think that this paradigm can really learn something new in terms of conservation laws.  So before I finish, I wanted to thank you for coming to the lecture and for your very interesting questions and comments. So thanks.