 Okay, so let's start with the last lecture by Laura. And so please, the floor is yours. Thank you very much. Hope you can hear me. Okay. All right. Good afternoon. Thanks for coming to this last lecture on this topic of celestial holography and celestial amplitudes. Okay, the plan is to now that we have introduced all that we needed to talk about celestial amplitudes to show you how they are real to dimensional currents that appear in celestial conformal field theory. And we will discuss also some features of celestial operator product expansions. I'm going to try to give you some a broad overview of this program of celestial of holography what what we have achieved so far, which are the big questions that are still open. And eventually what what is the big picture that we want to achieve. So, as usual, I will start with the brief recap of the main message of last lecture. And yesterday, we define what was a celestial amplitudes. And for this, we represented massless particles. Massless particle cross the celestial sphere. Yes. At a point and a dead bar and carrying an energy Omega. And eventually, and also a electricity if it's being particle. So this massless scatters are represented in celestial holography celestial operators, denoted by curly Oh which carry a conformal dimension delta and a 2D spin. The 2D spin is identified simply with the bulk for the electricity of the particle. The center is inserted at some points at the bar. And delta importantly plays the role of the conformal dimension, which can be written as thumb. The usual safety language left and right weights, where the spin is the difference of each and each bar. The mapping the celestial mapping from a mass is 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 million integral over each leg. 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 a, this convenient boost basis, and in celestial holography we want to see to which extent we can interpret. This interpretation will tell us about the holographic structure of quantum gravity and the boundary of 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 conform celestial conformal 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 safety. So these currents will appear. When we take specific integer values of the conformal dimension delta. We will call them console conformally soft operator. So indeed, in the, in the second lecture, we saw all these nice equivalences between what identity associated to to asymptotic symmetries and sub theorems. We saw that all that the soft. I will mostly present current for gravity. But we, we, I sketch also the story for the support on theorem. So the subject on theorem can be interpreted as an insertion of, of a current. Now the question I want to ask is, you see where we had the soft gravity tone or so photon we're taking the energy omega zero. But now, when in celestial holography, we do no longer have energies to talk about, because we have traded energies for conform my dimension. So what does it mean in the celestial basis to be soft. Well, what people have came to realize is that there is an analogous, if you want to do all formulation of softness. And now in the celestial basis. The statement of a particle to be soft. But before we have energy energetically soft particle, the mountain to take on energy to zero. Now we will be calling in celestial holography, a particle which is soft in the conformal sense as a conformally soft. particle. And the conformal is soft particle or soft operator will be an celestial object which for which the conformal dimension take specific integer values. So, these objects will play the role of this two dimensional celestial currents. Values of delta do we have to look at. Well today I want to present you two specific values when delta is equal to one and delta is equal to zero. Because as you will see, these two quantities will give give us these the two currents that we have insisted on in the beginning of the lecture, which are the so called super translation current. The famous celestial stress center. But let's start with the first object the support translation current. The which value of that that we have to consider. Well if you remember, yesterday, I presented for you this conformal primary way functions. And if you remember, there was primary that was becoming a pure different morphisms for for delta equal to one. And I told you watch out. When these guys are large case transformation, we should not neglect this, these terms and subtlety should arise for these values. And I said the super translation current in celestial 50 is an object a celestial object which has conform my dimension equal to one. So remember, there was the conformal primary way function that was pure detail for this value. So naturally, since all this business is related to asymptotic symmetries and large gauge transformations. So that's something funny happens for this value and the super translation current more precisely. So, you know, 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. The world identity associated to support translation for translation symmetry, where we had statement that this matrix commutes with the charge associated to the symmetry. I think I've written that in this chart when we had different columns. And this world identity if you split it into piece. You put the left on this left hand side of this equality is equivalent to inserting these. Here, as matrix and inserting this object gives you the soft leading soft Weinberg's of gravity on Karen. And if you remember we had this sort of expression. And I told you this looks like you won't catch money current but the form with this energy Omega. And now we will write this not in that 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, this object makes some funny has some funny feature in celestial safety. And this P inserts a sub graviton. So that's what the old basis now in the celestial basis this object would 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 one so here there is a delta minus one factor this is just a way to to put the pole up, whether than down this that that coming from Omega to zero here which will correspond to that they want to one. There are different definitions for this but here I've led to the delta equal to one in this definition. So here that is not just taking delta equal to one of a celestial operator of a spin to because I'm talking about gravity operator but rather I have to take a descendant and safety language. So derivative respect to a Z bar of this operator. And so we can count the weights of this object. If delta equal one and J. The elicit is two. I can equivalently right and maybe more for the million four meter of age and each bar. So age is half of the sum of these things. Half of this of the difference. So the weight of the delta equal one j was zero. Celestial operator. Will be denoted H, H bar. So here that is one j two so one half of one plus two this equals to three half. And one half of delta minus J. Give us minus half. But you see this is not P these are not the weights of P is a descendant of this object. The weights of P given by the weights of this object but now since I'm taking one Z bar derivative is will increase H bar by one. It's just a descendancy. So the weights of this poor translation current is this equal to three half one half. So this is something that we are not familiar with in series in usual safety. Typically we talk about catch moody currents which have one comma zero weight or stress tensor which are two comma zero. So in particular this piece 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 these expression in form of a celestial correlation function. So in now in a celestial basis. I'm inserting this support translation current, which I recall is in one to one with expressing Weinberg. So gravity on theorem. Now I'm replacing all my in and out state in terms of celestial operators or let me use this short hand notation. Okay. Here will stand for will carry all the labels. So I'm going to 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 and Malin transform and check this very easily this is just by definition of what is a celestial operator as a Malin transform. And now you see that before I had this in the momentum basis I had this Omega here in the numeric numerator. Now when I do the malin transform, which is here, because I have delta minus one upstairs. I want to rewrite this and the operator as a quite 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 one this so the shift by one is just because we had an Omega here. And when you look at the malin transform if you multiply this by Omega is the same as shifting delta to one in the formula. So again here we see that the action of the super translation current, very importantly, if the conformal dimension of the operator in the celestial CFT, which again is something that we don't usually that 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 the symmetry is there. We know that whatever is the dual theory of quantum gravity in flat space written in this basis, it will have to obey this infinite amount of 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 in the dual theory that that is starting to tell us in which way this theory is different or familiar to the one where we are used to. Is there a question, are there any questions on this stage on the, on this current. Okay, if not, now let's talk about something a little bit less weird, which is the other current which is very important is this trust answer 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 of your zero. Group, let people to be very excited and start to dig for more celestial or more safety to structure. So how can we get these stress tensor. So let's tell us the answer. So this, this stress tensor was first obtained in the momentum basis by the work of Andy Stronginger and others from really a reverse engineering from the sub leading soft gravity on theorem. And I guess what was the form and 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 the 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 the some derivative of the news tensor. It's it was a really funny object that they really didn't understand why how to obtain this natural way. And, and now we can see this object arising more naturally as again a conformally soft operator, but there is a subtlety. The result is that this stress tensor involved the so called shadow transform in in safety. So, I'm not sure that this notion is familiar so let me define what is the shadow transform first. And this shadow transform actually is ubiquitous in the celestial holography program it shows up. It is too many confusions. So, it's important to discuss it is not just a technical point is really actually important in understanding what is the spectrum of celestial CFT. Do we do have to include all shadow models are not a lot of debate in the current literature on that. So if you start with the shadow transform of a primary. A primary of weights. h and h bar. So, again h and bar can be can be traded for Delta and J. So, what is it so we find like so. Let me write it and then explain. So the shadow transform will change the conformal dimension. And it will return you a primary. It also flips the to the spin. Sorry Laura is a primary or was a primary is a quasi primary. So that's the definition of the shadow it's an integral over your why and why bar. So there are some normalization here depends on the depends on the convention, you can find this definition in paper by Osborne or Osborne and Dolan. You start with the quasi primary of weight of these dimension delta and J. You do this integral which depends on which kind of operator you start from, you will have different powers here in the Z minus y and red bar minus y bar. So I return to you another primary, but now with different dimension. Now is dimension will be two minus delta. So if I were to write this in terms of h and h bar, you can see that you start with something. If you start with the primary of these weights. If you start with the shadow, you will have something with one minus h and one minus h bar. Now, it was realized that the stress tensor celestial CFT arises as the shadow transformer of an operator of weights delta equal zero. So I will call a stress tensor. Indeed, if I start with something which has weight zero, the shadow transform will give me something which has dimension two minus zero. So to. So this is the stress tensor in celestial CFT. And there is this delta here. I write it up front so that the limit when delta goes to zero of delta times this expression. You can recognize as the, if you take that I was zero in this formula, you will see that this power become this x one become four and this is equal to zero if I start with a spin equal to minus two it will flip the spin. So this is certainly something that has h equal to an h by zero, which is the dimension we expect from a stress tensor. So it's the shadow of that they were your primary. So again, the realization that this was a stress tensor is really it is long. So it became pieces by pieces by for first this realizing this stuff, submitting some gravity on theorems reverse engineer and then people say okay but there is a basis. Actually this one motivated to look at this celestial basis was to look for a basis where the action of the C was was a diagonalized. was built, built on from all that, and here I'm just giving to you so you might think why what does it, where does it come from. How could I have guessed it why just looking at the dimension you can guess that the shadow transform is kind of tricky and took some time to understand why it has to be like so. But so the, 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 in a conventional CFT to writing down here for you. And you can find in any CFT book. Okay, so this is the word identity of the 2D stress tensor as it should. Okay, so these equation was checked explicitly from from amplitude people. So in, in particular, Thomas Taylor and Stefan Stieberger they computed a lot of celestial amplitude, starting from the well known formula that from that they have basically invented themselves. So, they have checked this formula, checked explicitly. So by brute force so you start with the momentum basis amplitude. And then you do a bunch of many transform. So they looked at first Einstein Young Mills amplitude. So with N gauge bosons and one graviton. So he was pretty photocopulus and Taylor for these but then people have been looking at any extension of these sort of facts. So, this was really checked by starting with the momentum space, you know, Einstein Young Mills, you do them in transform, you do a shadow transform and then you take the limit that that goes to zero, 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 safety operator I indeed full your as our primary fields because this is nothing but the definition of how your as our primary should if you want to transform. So this is really in one to one. So, sort of check proof that you see if tea operators. Before you see I was talking about most global part I was telling you about this to see transformation, maybe transformation. But now you see that when we are coupled to gravity, the presence of super rotations in encoded by distressed answer is enhancing the group. So that the operators are now primaries under the full local group, namely, they are. There are zero primaries. At least we are very happy because this is something we know very well how to deal with in usual safety too so we might hope that because of that we can exploit the techniques of to the CFT into this celestial photography program to sort of bootstrap out of out of the 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 also look computation. Right so the this 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 most of the thing I'm going to mention are on three level. But we expect that look level. There is it respect exactly the same. So, okay, so what happens is that the bleeding soft Graviton theorem has correction at loop level. But this correction are one loop exact. So you just corrected it's corrected at a tree level it's exactly no order perturbation theory, one loop gets correction, but then there is no more correction. And this correction are. If you want to re recast this of theorem as well entity of an object, you have to include another contribution to this tea here. I'm not introduced entering into this detail but there is there is a shifted stress sensor which is very subtle it has to do with some background structure of gravity. And that's a very good question is that it's under investigation. Right now, but, but yes there is a correction to this tea, but it's still a 2.0 thing but there are some fine, fine details. Okay, thank you. I didn't quite understand why we care that this tea was shadow transfer over that equals your operator. What do we use it. Because that's the only way you can, you can get it. Somehow, if you, if you, let's see how can I explain. So, yeah. And then there is an extended to which all objects that are all the currents, actually, if you want to map them to the, the gravitational phase space, they're actually naturally related not to the celestial operator but to the shadow operators, all of them. I didn't know that I didn't have to talk about that here, because the shadow of a delta equal one mode as also delta equal one. But actually, 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. And actually, what happens there, so somehow all shadow objects. But this is why we care. The reason why we care is because this is somehow what it is. And if you don't look at if you don't have this idea of allowing for a shadow transformation, you will not see this object in the first place. It's very non trivial you know that there is something of stress tensor and the boundary of that space I mean why, why would it be so. I have not understood this is a specific operator, the one from which you construct T, or is that generic operator with delta equal to zero and j equal to minus two. Yes. Oh, so yes this this is a generic in the sense that it represents the insertion of a mass less gravity on on the sphere with spinning world to to here to two or minus two. I have an alagos T bar of course if I take j one plus two. I suppose you have more than one operator with the with delta equal to zero and j equals to minus two. How do you know how to to generate the stress energy tensor. Um, you mean if I have the generous you just say no. Yeah, so all, I mean all. Yeah, so basically, what different differentiated these operators will will be what you differentiate their null momenta, for instance, and they're not momentum carries a given energy. And a point on the sphere this is the only thing I know when I'm looking at a series here. And, and so what I'm saying is that the stress tensor is related to the operator which insert a conformity soft operator because delta takes some specific value, but secretly actually you can see that it's, it's equivalent to inserting a soft, soft to grab it on. So, basically go taking omega to zero something. I'm just saying I'm just stating to you but taking a big omega to zero you can see that from the structure of the amplitude will select some pores when delta takes some integer values. So apart from that. Can you answer your question. Sorry, would I say that the uniqueness of gravity decided there is only one spin two must less particle so if you were just assuming that in this matrix which is a very reasonable benefit in this. The chelsea safety has to have a unique there can be any degeneracy of this operator. So it has to be unique. Otherwise you will have a two kind of gravity turns you can tell. Yes, yes, it's a very strong constraint because. So this you can every time I'm writing this, this, this relationship, you have to see this as actually an infinite amount of constraints for the safety to obey. 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 on coming from symmetry. And you hope that at some point this country will be so strong there will be so many of them that it will. And if you want to, if you, you will isolate theory that you can identify, or at least you can identify what properties it has. Vice versa, so I will come to that when I talk about where the program stands it, but that's the, so the question was. You should both infer property of the CFT by what you know that you should be like there is just one gravity on so that there must be just one over with this property, then understanding property of gravity is from let's say constraints that I know that there are on CFT Yes, so the hope. And the thing that people are actively trying to do is that now go in the vice versa direction that you are mentioning, namely, can we cook up some theory a celestial theory from which we can infer new, new 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 was a month to basically have a holographic dictionary fully fledged. So this is an outstanding question to have an intrinsic definition of the. Okay. Very good. Thanks for the questions. Yes, I will, I will come back to. I will come back to, to these big questions of what we know about celestial CFT so far what we don't know. So, I have few minutes left before I go into the last section of the summary and outlook. And 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. The operator product expansions by looking at collinear divergences in the for the momentum space. So the statement and you would you will see why this is the case. Collinear divergences of the 4d momentum. Usual stuff so again it's really bottom up we'll start with all this beautiful formula. We have been developing scattering amplitudes and see what it in for you can deduce in the celestial CFT. So these things extract for us. And there is a very stupid reason for that. That you remember when we parameterize the momentum of the particle like so, I'm going momentum should take the product of two momenta P1 with P2 just compute that. And then you find that is given by omega one omega two time. And one two z bar one two or Z one two or difference between Z one and two. And similarly, so in other words, a collinear limit when these two the two momenta become parallel. Well, if you have a collinear diverge divergence, it will extract the pool when Z one approach is Z two. So basically, you know, you want is the same as taking that one close to P2. In other words, you have a scattering. You take these two momenta to be in here. These correspond in the celestial sphere to look at that one approaching the two other. So 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 safety. So, I'm bit running out of time here. So if you have questions about that, you can ask me in the discussion session. I'll just mention a few results on that. So if you look at gravity, want to derive gravity, you can do so. Let's take for simplicity to positive. Very tall. So these are represented by two operators with J1 plus two, they can depend on each the carrier conformal dimension. So these, the computation is being done in this paper. I want to look at. So they are basically two ways to derive the OPE that I'm going to write down. The first method is brute force. Namely, you just, you just do your main in transform and you do your colonial limits, and you can compute it. So one is force, namely, start from one to amplitude, look at colonial divergencies, they are formulas for that. Do the melin, do the colonial limit, and you will extract this structure. So this has been computed. If you have mixed electricity is a bit more tricky. And now people are to know how to deal with things. And this is what you get where beta is the other beta function. So this is a computation, you can do. And this, the second method, which is a bit more cute is to just just use symmetries. So that's, that's one of the very great thing of these infinite dimensional symmetries that they are so powerful. And you can, they fully constrained the OPE coefficient. So you can actually don't do any computation but just if you remember that you have translation invariance and the constraint implied by the leading soft gravity on theorem. You can see that the symmetries, actually, you, a little asterisk you also have to know about the sub sub readings of great on theorem I didn't talk about. But the main message I want to convey is that symmetry, the symmetries, symmetry constraints. So you don't have to do this computation. They are powerful enough. And they have to, they actually imply some sort of recursion relations on OPE on the OPE coefficients. Before you knew that it was a little better function to us to imply a new vehicle coefficient, actually to uniquely determine, determine the OPE coefficient. So you need, in this case, you need translation invariance, you need the leading sub graph on theorem, the sub readings of gravity on 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 sub bleeding sub graph on theorem that I didn't talk about. So you can derive these, you can derive glue on OPEs, you can compute all sorts of things. And before I go to the summary. Let me mention some, some comments about which is related to, to these OPE and these currents. This is the most more recent literature. And with that, you, you almost, we have almost covered the main features of celestial amplitudes. There is an observation that, so, so that delta equal one and zero up to, what do you call this shadow transform is conformal is sub limit. To celestial currents, but our recent work have been showing that there seems to be an infinite tower of currents where delta is running over all negative integers. And very recently, Storminger managed to show that if you look at this very complicated structure in implicated implied by these infinite towers of current, you can nicely by a clever field definition. You can nicely recast this constraint in terms of a single simple algebra. Well, not so simple, but which is the so called w one plus infinity algebra, which is an algebra which appears in very different context in, in, in higher spin but also in twister space so now they can interplay between celestial holography and twister 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, in more details. And I will just, I think, go to, to the summary of what we have learned in this, in this for 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 have, I hope I've tried to convince you that you will remember that the infrastructure of gravity as something that space time is very rich and very subtle. We have this infinite tower of symmetries of conservation laws that any consistent quantum computer quantum gravity theory should should obey. And infinite amount of symmetries which are constrained, constraining 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 construct to constrain the holographic dual for the gravity. So, which again, the main claim on the main thing we are looking after the fear of quantum gravity in flat space time before dimension, which I recall, describe the real world to some extent could be described as a celestial to the safety. And this, there is this promising dictionary which involves, which involves this malin transformant equal, which makes this SL2C transformation manifest. So we have to summarize. To finish, we have a similar feature that we are used to in usual CFT2. So the whole point is to understand what is, what is the celestial CFT2 this doesn't make sense as a theory. How can we push this and how can we constrain this, this, this quantity. And this, this right hand side here. 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, we have 2D currents. We have catch muddy currents. We have a stress tensor, which is far from obvious. We have primary operators. So 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 is a super, the super translation symmetry, give an infinite amount of concern that we are not to and not used to in usual CFT, because they shift the weights of the operators by one as I've written in the formula before. We shift the conformal dimension by one, and the other thing 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 on the equator of the celestial sphere. Just a constraint from kinematics. And this is something that we don't have in the usual local unitary to DCFC. There are also weird thing happening with the conformal block expansion. That that people are trying to treat the best as possible but this is also something that is not so fully understood so far. Last minute, so let me tell you what are the big question 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, you could have is, what is the relationship, or is there some way to, to use what we know from a DSCFT. So how could we relate the celestial correlators to the one of a DS, but really celestial holographies are very different from a DSCFT. So in a DSCFT, we have, we have this nice time like boundary boundary of a DS. So we have this gravity in flat space we have this annoying null boundary and this is what all the difficulty comes about. Because in a DS what we're used to it were used to put some reflecting Dirichlet kind of boundary condition, where we said that a DS acts like a box. But now in in flat space time. As you can see, 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 the first point, the second point we already discussed but it's almost to say what is a celestial CFT. Namely, can we come up with the intrinsic definition rather that's 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 want to top down we want, 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 these nice symmetry structure that is emerging has compelled us to reconsider many aspects of, of gravity in flat space. We had to look deeper at new kinds of memory effects new kind of observables. And so we had to relax very much this bond bond bond in those 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 wall, 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 space time that space time, which, which contain a horizon, which have black hole inside in the presence of a black hole. So all these stories are way more complicated. But eventually, I think that this, this, this paradigm can, can really learn, we can really learn something new in terms of conservation laws, what are the global conservation laws that black hole space time have to obey. Before I finish, I wanted to thank you for coming to the lecture and for your very interesting questions and comments. Thanks. Hello Kevin, can you hear me. Yes, I can read. Okay, good. So, let's wait a couple of minutes, people is still coming in. And then. Okay, great. Okay, so welcome for the last lecture by Kevin. Thank you very much. Thank you very much. So, please, Kevin. You can start. Okay. Thank you very much actually. So for this lecture, actually didn't get a chance to prepare size and advance some of the right as I go. Yes, today is how to do some computations with wooden diagrams. Just like in traditional ADSEAT, but in this topological string context. So what we're going to do is we'll build a chiral by by studying the topological spring on SLTC. And so far, we've related in the dimensional symmetry algebras on the gate theory side. And our graphic tool. We've seen that these are the same. And the main result will be able to do today. So the first thing we need to do to understand this is to understand how to build the holographic to the holographic. So we'll be able to show that we can match all of the OV coefficients. And in particular, all correlators. So the first thing we need to do. To understand this. To understand how to build the holographic chiral. So we started discussing this a little bit yesterday. So we call yesterday that SLTC has a compactification is CP1 times CP1. So we'll give an explicit description of the geometry near the band on a path near the boundary. We have coordinates in C. This will be the chiral of the plane. So this we thought it was the boundary of ADS 3 W in CP1. And also 10. It is normal to the boundary and really lives on a line bundle over CP1. This is explained in a second. It's related to the hop vibration, which means that N has a pole when W is infinity. In other words, N transforms like the W. These coordinates we're going to use are related to the coordinates on ADS 3 times as 3 as follows. If we write N equals 4 e to the i theta, then Z, Z bar or coordinates on ADS 3. So we're taking an ADS 3 and the boundary is or equal to 0 W, W bar and theta coordinates on the trees here, which we're seeing as a hop vibration, the theta coordinate parameter is in the hop fiber. So the goal is to build a chiral algebra living on the Z plane by studying gravitational engage theory we have in the book. So to get there, we need a few more details of the geometry. Z is not a homomorphic coordinate. There is a Bultrami differential, which looks like mu is factor of N. We've got a rocks and factors of pi N squared. So this may look a little funny, but if you can remember that N transforms like the square root of BW. So N squared transforms as W. This expression here includes a natural volume form in CQ1. So it's a natural expression. But also, because there are Bult here, we're going to be focusing on the open string set because it's easier. And the Bult propology is the homomorphic trance time is important. So that involves a homomorphic volume form in this geometry. So we need to write that down too. It has a whole order of three. I'm just going to look at my notes. We calculated in these coordinates looks like this. So it's important to note that if we look over here, we can, there will be a piece of this which looks like d theta d times the volume form of the two sphere times N. So from this, you see that the integral. Over the three sphere of omega is equal to N. As we would expect. Okay, so far we've just set up the coordinates. Are there any questions? No question of the speech. Okay, I actually can't see you guys anymore. So no big deal. So recall. The homomorphic trance time is the quarantine. We're going to have a gate field. It's a zero one form in this geometry. And it looks like an integral trace. A d bar a plus one of the sixth place. Hey, hey, hey. And this whole thing is here for the homomorphic volume. So to describe it, the holographic type of algebra. We have to first. The boundary conditions at n equals zero is on the boundary of our geometry. And then states violate the boundary condition. Some value of Z. The natural boundary commission, which is a little time thinking about it. It's a little subtle. But we noticed that omega has a pole order three. And we said that asking that a has a zero is a good recognition. So we asked that a goes like n near n equals zero is a good foundation. Wait, so now let me ask, what is the station? And the answer is, of course, we're going to modify the boundary behavior at some value of Z. For example, we could ask that a goes like delta function. Zero plus all n. This is going to be the simplest lowest line state. So in a moment, I want to match, I will match these with the states in the chiral algebra. But for now, if one's face ordinary ADCFT, he might say, oh, what you really want is a solution to the equations of motion on the bulk geometry. Which satisfies this boundary condition. We can find such a thing. You can show that there is a unique gate equivalence solution to the linearized equation, which is just the deep arc equation, which satisfies this modified boundary condition. So this is the explicit formula is a zero. Okay, let's introduce a color index. Plus factors. And the constant factors of five is going to look like this. So the reason this satisfies the equations of motion, we should recall that there was a boundary differential so Z wasn't, wasn't homomorphic. The delta function fixed value Z. That doesn't satisfy the devour equation. But this adding all this correction from, from makes it so it does. What can draw a little picture of it by what this is. Yes, three. This is the boundary radius three. My state. Sorry, Kevin, the second term is T is multiplied by T or what. So what this looks like. As I approach the boundary, the boundary is an equal zero. As I put the boundary, it decays rapidly here with zero zero as I put the boundary. And if I move a little bit away from the boundary, it also decays. So this is the kind of thing you want for a state right on the boundaries localize a particular point. As you move away from the boundary spreads out away from that point with some particular decay rate. The decay rate of course will be related to the two point function. This state is related to the current and the two point function of the current goes like one of us. Okay, so we're going to propose that a a zero gets matched to the state. J a that said zero in the car with this is the S away current. And that's where it was. And then there's some. Or an S one from one to eight and it's anti symmetric or S. So we'd like to also build to build the states correspond to or to these to these symmetrize faces. And what we'll find is that the ways of modifying the boundary condition. At this point match precisely with the states in the CFT. So let's think about the most general way we can modify the boundary condition. So we could say a is like delta function. That said equals at zero. But there's nothing stopping me having more singular behavior in N. So I could have one over N disempower. But once I have that singular behavior in N, you could also think well, why can't I have a function of w two and the remaining terms are going to be terms which are less singular and inclusion will be forced by solving the equations of motion. Never. What possible values do we have here for these indices. Well, I can have an arbitrary pole in N. But we know that N. As a pole, that w was infinity, which means one over N to the K. As a zero over K, having a pole in the W plane will be bad because it would prevent my field configuration in satisfying the equations of motion. So we need that the index L ranges from zero to K. Well, this is perfect because these are exactly states with the white quantum numbers to match the states we see in the chiral logic. And this corresponds to. To the. The factor. Corresponds to. Or X one. L X to K. S. In these states, the same quantum. So I assume everything is crystal clear. So now we've described the states on the holographic regional side. We can ask. How can we compute. Their, their end point options. Let's say the two and three functions to start the endpoint functions of these states computed in principle, I wouldn't. Let's start with the two fun. So, if I take my state. A zero. Let's go and square. I take two such states. They're two point function is obtained inserting into to the kinetic term of all morphogen science. It's going to be integral. The coordinates of. A zero D bar. A B zero and then when we contract indices, there's going to be a. So, since these states correspond to the S away current my current on top of these have two point function goes like and. Over. Square. So this is how we would like to purchase. The factor of one of his not minus one squared. You can see this. Because it's a kid, that's the standard central extension of a cast movie. And it connects a cast movie at level and just because that's the number of fields we have. And then we get from these guys are like, I, I, I, I know X is the middle. And when I contract them, we do two week contractions. And there's a loop in the middle. So the loop in the middle contributes and the two with contractions contribute one of his zero minus one squared. And we have the crucial point to introduce the volume. So let me scroll up a little bit so we can see the formulas for these expressions. Each expression is pretty much the same. Here there'll be like a delta C one. And here there'll be the same kind of thing with the one over C one. So what I'd like to do is can quickly compute this integral explicitly. And we're going to be a little loose and I'll drop some terms, which are also important and and just focus on the relevant terms. I can take first one. I'll drop the color factors to this would be a delta function. We're going to put D bar. A one. And the volume form looks like the n, n cubed C. This is the relevant term in the volume form. Because our geometry is non-compact, we're going to integrate over Z and W, or N is, we're going to do some cutoff near the boundary. Now, if we do integration by parts. This is a pure boundary integral. We're going to get a bit of the D bar. And we're going to find something like this, and N squared. DW bar. And plus to the square squared. The N, or N cubed. DWDZ. And I forgot a crucial term, which is one over Z minus C one squared. And this term here really appeared originally over here. Okay, so all of them, you notice the integral is a total derivative, so you pick up the boundary term, and we just insert all the terms and we get something that looks like this. And then we see we get a really nice answer. Because this will correctly reproduce the chiral algebra 2.1 function. Why is that? The interval over W. So DWDW bar. One plus W squared squared. This is just a natural volume form of CP one. So this is, this is basically one, but it's a multiple of pie. So that part goes away. The integral over Z, because I have a delta function in the integrand. And that produces one over Z zero minus C one squared. And that looks good. And finally we have an integral contour integral over N. And we notice up here, we would n squared, then we would n cubed. So we get counter over N of the N over N, and that gives you no 2 by I, which is also important. Indeed find the holographic 2.1 function. Any questions about this calculation? Yes, there is a question. All right, Kevin, I've just got a little bit confused. Is the integral, if you go up just before you did the integral. Is this an integral over CP one times CP one, or is it over S two times S three? Right, up here. This was an integral over SLTC, but I've removed the neighborhood of the boundary. So here when I go down to the boundary integral. This is really an integral over an S one bundle. So this is an integral over CP one times CP one. Yeah. So where's the fifth coordinate then? It's N, you see, because N, the absolute value of N is fixed. So we're left with the angular value of N. And that appears when we, you know, it's an integral so I can just remove. I can just compute it in a dense open set as long as it converges. So I don't worry too much about the fact that it's a non-trivial sort of bundle. So in which case the integral becomes an integral over C times C times S one. And the integral over the S one just gives us this counter-interquery. Does that make sense? Yeah, thanks. P three point function. Then we will play the same game. If I take three states, we're going to get like F, A, B, C integral over the six dimensional geometry. Freezing it like this, though, it turns out to be a little tricky. And I personally prefer, well, in any holographic computation, I prefer to use the OBE rather than the three point function. So how do we find the OBE? Well, the OBE, the object which when inserted into the two-point function gives, with AC, gives the three-point function. So what is that? This is just the definition of the OBE in general. But there's a nice formula for it. The OBE is given by D bar inverse. This expression is, and you can imagine that as follows. There's my first state. And then here D bar inverse is the propagator coming off. So why is this the OBE? If I insert AC B2 times D bar, D bar of that, well, clearly the D bar's cancel is the three-point function. So the OBE is something that's actually easier to compute than the two-point function. So let me quickly explain the technique for doing that. So the technique we're computing the OBE is, well, AAC0 looks like this. And it turns out that this is D bar, TB times D bar of one over C1 of C1. So I can remove this state by a gate transformation. In the presence of the other state, gate transformation has a non-trivial effect. It sends this other state, C0 to one over C0 minus C1, F, A, B, C, T, C, delta, C equals C0. Now here we see the correct cascading OBE. And of course we can play this game with all the other states. So it's a little more complicated. There is a question, Kevin. In the last equation, how does the B index get soaked up? Because it remains free on the right-hand side, but on the left-hand side, it's just AA. Right, so the B index is present in the gate transformation. The gate transformation is like B minus C1. I'm looking at the effect of this gate transformation on this state. Thanks. Okay, so in this way, one can, in principle, just by working really hard and thinking about standard written diagrams, try to compute all of the correlation functions of the tiring algebra. However, that's really hard. And there's a shortcut. The shortcut is that we know, we've already checked, that on both sides there's this really big symmetry algebra. We can strain everything really tightly. And that will really help us to cut down the problem to a much, much simpler problem. So let me very briefly say why, how, you know, it's already kind of obvious that this symmetry is present on the holographic side, because it's just the gate transformations. But it can be helpful to think about the mode algebra from the holographic point of view. So just like a local operator is obtained by modifying the boundary conditions at a point, we get a field that looks like a delta function at a point in the boundary. A mode will be a field configuration. For example, some power of z, a delta function that's where z is on the circle. And maybe there might be some negative powers of n plus some other stuff. The negative powers of n are localized on the circle. So we need to see, you know, commutators of modes can be seen, can be computed using the same gate transformation trick we just mentioned. So let me briefly explain. Why does functions on SL2C, these are homomorphic functions, tensor SOH live inside the mode algebra. The answer is very simple. If I take f, which is a homomorphic function on SL2C, f times ta times delta function, and z equals one, this expression satisfies the field equation plus a, even though z is not homomorphic. So geometrically, what we're doing is we're taking our homomorphic function and multiplying it by a delta function on a five manifold, which add infinity equals to this circle. This circle where z is one of that. And the reason this satisfies the field equation is just, this is closed, you know, the delta function, any five cycles close one form, and drf is zero. Because it's a homomorphic function. So in this way we see that this global symmetry algebra really lives inside the modes of our chiral algebra. How does that help us? We know that the commutators of modes of our states are given by function on SL2C as a way. If I take in any chiral algebra the commutator of modes is some expression, is an expression in terms of the ecoversions. So this relationship, it turns out to constrain OBEs, and so correlation functions uniquely, except the two point function of the stress energy tensor, which goes like this. So in other words, once we know this result, then we have this huge symmetry algebra, and once we can identify the quantum numbers of the states, everything else is determined by a single grammar. And the homographic algebra is isomorphic, the planar. So I think I'll stop there. Thank you. Thank you very much. So let's start the discussion session. Yes. Hi. At the beginning you mentioned something about a whole vibration.