 Okay, I think we can get started. So, welcome everyone to the 2022 edition of the ICTP spring school on super string theory. For many of you this will be, or for some of you this will be actually your first in person school post COVID. Before I talk about the school, I want to say just a few words about the ICTP itself. For those of you that don't know it. It was founded in 1964 by Pakistani physicist Abbas Salam, who later went on to receive the Nobel Prize in physics together with Weinberg, and glass off for the construction of what is essentially now known as the standard model of particle physics. It's based on a so called tripartite agreement between the Italian government, the International Atomic Energy Agency and UNESCO. Most of the funds for the Institute are generously provided by Italy. And the Atomic Energy Agency also contributes a significant amount, whilst UNESCO generously provides the administrative structure within which we operate. The reason for the foundation of the center is that Salam wanted to create a kind of home away from home for scientists who come and or reside in developing developing countries where facilities for research are not as may not be as advanced as in other places. And as such it has a mission to foster science and mathematics in in developing countries. So, as a result of that you can see that we have many different, you can for example see on our website, the host of activities and programs that we do. So this school is one of maybe 50 or so activities that take place here in Trieste. And we also have significant numbers of activities taking place in developing countries all over the world. We have a close collaboration with other institutions in Trieste, namely CISA, which is the Italian advanced Italian school of advanced study, the University of Trieste. And there, which is a city nearby the INFN, the National Institute for Nuclear Physics, which is actually a co-sponsor of this school, the National Institute for Astrophysics. There are many, I mean, I've mentioned these institutes that have significant physics and mathematics research activities. Trieste happens to be one of, I mean, a place with with arguably one of the highest densities of physicists on the in the planet. The school itself goes back to the early 80s and has happened every year apart from one year due to COVID since 1980. I think 1981, anyway, the early 80s. And as such has become a standard event in the calendar for young theoretical physicists working in particle theory and string theory and related topics. This year's school is has been organized by myself, Pavel Putrov, who's another staff member here, and Francesco Benini, who is Professor at CISA. And you'll meet all of you'll meet both of them as the week goes on. Four lecture courses happening, three of which are in person with the lectures by Kevin Costello taking place via Zoom at the end of each day. There will be coffee breaks in between so there are two lectures in each morning to each afternoon with a coffee break in between. I think the coffee will either be outside here or on the terrace level which is one floor up. On which you also find the cafeteria where you can have lunch if you want to one floor above that is a bar that has coffee and lighter food and other beverages. Should you so need it will this evening will have a welcome reception and poster session. Some of the in person participants are going to be printed posters that will be on display and I'll be an opportunity to discuss with them. And I think that's all I want to say, Pavel is there anything that I need to add. Well, maybe you can mention that Friday afternoon there will be this session of celebration of the joy of so on. Yeah, if you if you look at the program there's a session celebrating 25 years of the Journal of high energy physics. Which was founded at Caesar, and so there'll be some talks there but then that that event will be followed by a reception which is in in the city of Trieste, and there'll be a bus service. So we'll probably we're providing a bus that will take anyone that wants to go to that downtown but you can also take the public bus. If you so wish to. So with that, with those introductory remarks out of the way, I'm going to hand it over to Pavel who's going to chair the first session. So, because if our first opening lecture, we are very happy to have even monk from NYU will tell us about non invertible symmetries. All right. Okay, everybody hear me. Okay. Thanks very much for an invitation. And it's my great pleasure to visit ICTP and to open this lecture series. My lectures will be on this topic. Now, invertible symmetries. Okay. I should say that during my talk if you have any questions, please just speak up. Let me start with some motivations of the subject in case that you're not familiar with this recent many revolution. So we'll start with some motivations. Okay, as you probably all have learned from physics courses that symmetry is perhaps arguably the most important organizing guiding principle in dealing with physical problems. So first of all, symmetry is ubiquitous, meaning that even messy system can exhibit symmetry properties at long distance. Second of all, symmetry give rise to organization principles for the physical observables. Okay, and the examples of that take the form of selection rules, or the usual kind of laws you can derive from charge conservation. Certainly, symmetry can often give rise to non trivial constraints on dynamics of the quantum systems, such as constraint on the randomization group flows in quantum field theory, as well as constraints on the possible IR face diagram of a given UV description. Okay. So one of the nice things about symmetry is a very natural step that given a physical system to first identify. Therefore, if any physical system a natural first step to solving that system is to first identify the symmetries of the problem, and then deduce the dynamical consequences. So one of the nice things about symmetry and how we use symmetry to constraint physical systems. Let's review a brief history of symmetry or the success of symmetry. As I said, the recent history of symmetries in quantum systems. Okay, quantum mechanical models. Okay, so the first, the first the list in this historical list is not so recent. In the 1930s. Okay, that's where the foundations. That's when the foundations of symmetry in quantum mechanics was laid down by pioneers like Wigner. Okay. In those cases, the symmetry will be associated with a group, which I know by G. Okay. The power of symmetry was pretty used in a study of column field theories. In that context, the central subject will be this continuous symmetries such as the space time Lawrence symmetries, as well as the say the extra magnetic magnetic symmetry as well as the gauge symmetries in the context of a center model. And then their power have been used in the context of studying studying the quantum field theory in the 1980s. Okay, produce many amazing results. And then shortly after, perhaps not. Well, appreciate it at that time is the discussion of discrete symmetries, which is generation of this continuous symmetries which are associated with continuous group to the case when the group is generally started from the paper. Of digraph and Witten around this time. Okay. This was not as much appreciated in the high energy community when it was first put out, but it has spurred activity development in kind of matter. In the kind of matter community, in particular, kind of matter theory community, because this discrete symmetries naturally show up as symmetries of ladder systems. Okay. And people in the kind of matter community have used the results from Tiger with them to understand faces of these systems that are described by lattices at some cut off scale. Okay, and this, this will around 2010s. And then there's a interesting, the interaction between the high energy community, coming from this, the, you know, the first description of this, this is the symmetry anomalies from this paper to the customer community doesn't go one way. There's also a feedback from the customer development for discrete symmetry back into the high energy community. So one prime example is the work of Gautel, a post in fiber. And well it's in 2014. Okay. And there's this in this paper discuss generalizations of this discrete symmetry, which have received active development development can as matter theory. So you can visualize it, one step further. Okay. And in particular, a key result of this paper is this generous notion of symmetries, viewed as about topological effect operators. This is a notion that encapsulates or captures all the previous notions of symmetries and allows further generalizations. And the further generalization came seeing two branches roughly speaking. Okay. So there are many further generalizations things around 2015. One type of generalization is to still insist on having some group associated with the symmetries. This leads to the higher form symmetries, which is already discussed in the original paper. And this is generalization of the old well known center symmetry in Yamio's theory. Okay. So generalization is to still having some notion of group, but consider a higher version or generalization, and this give rise to the higher group symmetries. Okay, and all these guys have the feature that they're associated to do some underlying invertible structure that is characterized by a higher group in general a group or higher group of generalization. So there are steady progress today to date to understanding the symmetries in various physical systems and anomalies associated with symmetries, which can be used to reduce dynamical consequences. But that is not the focus of this talk of this lecture series. Okay. Meanwhile, around the same time, another branch of development in generalizing symmetries come from come from generalizing the notion of group to something more drastic. So this also happens around 2017. This gave rise to the notion of non invertible symmetries. Now, if you look into the literature, this symmetries come with many names, but in this lecture to avoid any confusion, I'll stick to the name of non invertible symmetry. But in the literature, you may also see people referring to the symmetries as categorical symmetries fusion categories symmetries and so on. Okay, or higher category symmetry. The definition of symmetry which you will read, we will discuss in detail in the lecture series relies on this, this perspective treats symmetries as topological defect operators. So the non invertible symmetry will be associated with topological defect operators, but will be associated with this general topological defect operators, which are not associated with group. In other words, they're not invertible. Okay. All these notions will explain detail in the in the main parts of lecture. Okay. And why do I say this is drastic departure from the previous notion of symmetries. That is because from the foundations of symmetry that's laid down by Wigner in 1930s, a crucial assumption made for symmetries is that it should define a unitary operator on your hubris base. Okay. But that is closely related to the notion why symmetries discussed in the past always associated with a group. Okay. The consequence of non invertibility is that the corresponding symmetry operator, the topology defect operator will not become a unitary operator acting on hubris base. So that's a very drastic departure. Okay. So, let me just. So you may you may have some immediate questions on your mind. Okay. Since this is something so drastic. The first question is that even though this is some objects as allowed to buy this perspective. Okay, just insist on topological defect operators but without associating ways to boot. Do they really exist in interesting quantum field theories. Okay, or it is just a mathematical fantasy. Okay, a generalization that has no physical origin or any, and furthermore no physical interpretation. Okay. Yeah, yeah, so I'll come back to that. Yes, yes, yes. So so when I say, when I when I when I discuss this I do not mean that the symmetries are new the symmetries always exist. Okay, it was just not putting to this universal framework. And indeed there are examples which I'll mention shortly. Okay. As this is true for any symmetry in quantum theory they obviously exist. They just waiting for us to discover them. Okay. Okay. The second question is, if you can generate, we can identify some examples. Do these examples produce new dynamical constraints. Okay. In other words if they only produce constraints, which are which can be re derived using ordinary symmetries, then it's not very interesting. Okay, so why do we need to develop this new machinery. So do they produce new dynamic constraints. Okay. And to the, the answer is yes, as you may have imagined. Otherwise I will be, I wouldn't be giving this set of lectures. Okay. So let me now get to some examples of this nine invertible symmetries. And along the way, I'll give some references for which the lectures will be based on, and some additional references for for their way being. On this subject, we can think about what I'm going to say as a guide to literature, non invertible symmetries. So, the, the, the first development for nine invertible symmetries comes in two dimensions of space and dimension equal to two. Okay, as was already pointed out in the audience. There are already examples of this non-invertible symmetries in rational couple of field theories, date all the way back to the work of a valente 1988. And furthermore, the work of head cover and Zuber in 2000. In which case, they discuss non-invertible symmetries in rational couple of field theories, such like the Louisville theory. Well, such like the rational models, and they're also generalizations in the Louisville theory. Okay. Which is not really RCFT but it's for certain certain line operators in that certain to logic with the fact in that case they can be described. And thanks to work of valente. And there are a huge starizations of this original work by frolic. Fox. Run call shrieker in the sequence of papers starting around 2006, which essentially fully specified the set of to module defects in RCFT that preserve a car algebra some symmetry that's beyond the various or symmetry. We'll get to that notion later, and this will be typically referred to as some kind of valente lines or some suitable generalizations when RCFT is defined by non-diagonal module. And shortly after this work for like frolic folks around co shrinkard. There are also realizations of this non your non-evertible symmetries or this non-evertible to module defects in lattice model in two dimension. Okay, so classical lattice model, statistical lattice model in two dimension by the work of Bong and Bentley. Okay, the sequence of two papers. Okay, which spells out concretely at the level of the lattice model, this non-evertible symmetries for the special cases like the icing lattice model. Okay, and there's. Let's discuss some transformations to I think paraphernal theories. So, before they were all always examples in this RCFTs and lattice model with related RCFTs. And what I was saying here. Okay, was the recent development that put this previous development into a more general framework. Okay. And that started with the work of this gentleman, and also around 2017. Okay. So, who puts this discussion of the module defects or non-evertible symmetries in two dimensional quantum field theories into a very general axiomatic framework framework and consequences for such symmetries for consequence of such symmetries for, you know, our chief flows in two dimensions. What is discussing many other papers including this, this one, as well as this one. I'm drawing work with Ryan Thorgren. And as well as this nice paper from Kamagosky and company. And that's the development in two dimensions so far, as you can see, this is very recent very rapid developments from 2017 all the way to 2021 and today. Okay, but our thoughts are fortunately running out of space, but there are further developments in two dimensional in discussion of two dimensional invertible symmetries, even this year. Okay. But now I have discussed the two dimension. The natural question is to ask if such non-evertible symmetry also exists in two dimension. Okay. And the answer is also yes. Okay. And there are many exciting work in this case from starting from last year. In particular this work from Troy Cordova. Xing Shao. Oh, lamb and Shao. Sorry. Okay, which will discuss a version of topical defects which generalized those discussing these papers to the case of four dimensional gauge theories. Okay, in particular. Okay, I'm going to talk about the Yang-Mills theory with orthogonal gauge group. Okay. And similarly for super Yang-Mills theory, and for super Yang-Mills theory with unity of age group. Okay, turns out that such non-invertible symmetries show up in those contexts as well. Okay. And there was also a related paper. This time discussing similar and also different topological symmetries, non-invertible symmetries from topological defects in four dimensional theories, as well as in three dimensional theories from the paper from Kaidi, Omori, and Jen. And in particular, these papers. And we also kind of generalized the results obtained from these papers on constraints from this non-invertible symmetries RG flows to the case of four dimensional theories. In particular, they can prove statements like that in the four dimensional gauge theory with orthogonal group, for example, where this non-invertible symmetries are present. You can argue that any RG flow preserving this non-invertible symmetry has to be has to end at a non-trivial IR phase. So saying for non-trivial constraints RG from this instances of non-invertible symmetries. And in some paper that just just appeared, I think a week or two ago, there are further generalization of all the framework that are already reviewed over here. There's a question you can ask here already here is that there are two kind of two branches of generalizations which I mentioned. One generalizations from this topological operators is considered high form symmetries and higher group symmetries we're all we're still in invertible. Okay. And our generalization is to talk about this non-invertible symmetries. And you can ask if there are some kind of a more general structure that put this together. And this is some working that's work still being developed, but there's already a nice paper on the subject. That's started to discussing this general framework from these others appeared a few weeks ago, which discuss this kind of larger framework that's known as higher category version of the non-invertible symmetries. Okay, so that's a brief historical walkthrough for the developments in the field of non-invertible symmetries over the past decade. What is the general lesson that we learned from all these developments. The first lesson is that quite on the contrary. Before you may expect because this is such a drastic departure from the usual group symmetries which is ubiquitous in quantum systems, you may think that these symmetries are exotic and rare. But quite on that contrary, this non-invertible symmetries are also ubiquitous. So let me give you two kind of intuition for why they're ubiquitous. The first intuition is that almost all known CFTs in D equal to 2 have them. Okay. In fact, the only 2D CFT, which I know that does not have a non-invertible symmetry is the SU2 W start model at the level one. And there's a very nice recent work from Tombatakis and Devnachary, which explains that in dimension bigger than 2, the non-invertible symmetries are as common as non-anomalous higher form symmetries in these theories. Okay. So in other words, if you already accept that the higher forms of symmetries are interesting objects of study, then we are behooved to study this non-invertible symmetries because there's universal constructions as shown by these authors that gave rise to non-invertible symmetries by the mechanism of condensation. So as this authors put it, higher gauging, which is a way to gauge this non-anomalous higher form symmetry on some hierarchical dimension, some manifold space time to generate non-invertible symmetries in the same theory without changing the theory. Okay. The higher gauging is different from ordinary gauging in the precise sense that it does not change the theory. Okay. So I want you to recover, to discover that you have this non-invertible symmetries in the same theory. Okay, once you have this non-anomalous higher form symmetries. Okay. And because of that, it has been an active field of research. Okay. And in many efforts, again, identifying non-invertible symmetries in quantum field theories. Okay. Particularly non-quantum field theories. And deducing the dynamical implications. Some of which we will review in this lectures. And then over this kind of game, it's what we played in the context of symmetries, even before we talk about for the ordinary symmetry, even before we talk about non-invertible symmetries. So it's really running the old game of constraints from symmetries using this new tool, this non-invertible symmetries. Okay. But more crucially, from these exercises, we want to find some universal patterns, such as a notion of anomaly, or this non-invertible symmetry. This is a very powerful universal pattern that was observed in studying constraints from group-like symmetries on phase diagram of quantum field theories and RG flows. Okay. Having a non-trivial anomalies greatly constraints where the RG flow can end. So you want to find similar patterns in the context of non-invertible symmetries that you give rise to some kind of notion of anomalies, and there are, as we'll see in this lecture series. Okay. That is the motivation for these lectures. Let me now give the outline for the rest of these lectures. First, we'll review this notion of symmetries as this general notion of symmetries as topological defect operators. Okay. And finally, we'll graduate from this general framework and focus on the case of non-invertible topological defects, which define non-invertible symmetries. And we'll see that the mathematical structure that replaces group in the context of non-invertible symmetries will be a fusion category. This is precise. This is precise in the case of one-to-one dimension, and in the higher dimension it will be some higher category. Don't worry about this word if you have not heard about the category before. We'll make all these words explicit, and if you don't want to refer to this as a mathematical object, you can just refer to them as non-invertible topological defects. That's equally good. And thirdly, we'll discuss or I'll provide for you a toolbox for identifying topological symmetries to identify non-invertible topological defects, which give rise to this non-invertible symmetries. Which is essentially the tools that were used in these papers to discover many of the non-invertible symmetries in well-known theories, well-known CFDs, and was further generalized in these papers to identify similar symmetries in higher dimensional theories. Okay. So the tools that we'll discuss will apply both the two-dimensional case and the suitable generalization will apply to the three and the four-dimensional case as well. Okay. Lastly, along the way of the discussions, from time to time, I'll discuss dynamical consequences of the non-invertible symmetry that we identify. Okay. And I should say that, given this list of recent development, I think most logical way for me to proceed is to focus on equal two for the generalities. Because this is the case where the axiomatic framework for this non-invertible symmetry is mostly well developed. Okay. So the axiomatic framework for this symmetries are still developed, still to be developed, higher dimensional case, although they're already hints for some structures. Okay. But to introduce this symmetries in a more concrete setting, I'll focus on equal to two. Okay. So I'll introduce these examples in equal to two-dimensional theories, non-invertible symmetry in two-dimensional theories to illustrate the main ideas, the general structures. And hopefully this will provide you with the basic knowledge to follow the recent literature that discuss the higher dimensional cases. Okay. And I think this is the moment I should also emphasize the references I'll be using. Using mostly in these lectures, I'll be this paper and this paper. Okay. But the other paper that's included in this historical outline provide other references for further reading into the subject. Okay. And most importantly, what I hope through these lectures is to kind of motivate you to study the symmetries. And hopefully the knowledge that you gain from these lectures will be sufficient for you to start your own adventure in the realm of non-invertible symmetries. With that, let's jump into the first part that is to review this notion of symmetries as two logical defect operators. So let's start by, let's start slow and recall what conventional symmetries look like. In quantum mechanics from the eyes of Wigner conventional symmetries in quantum mechanics. Symmetries in quantum mechanics are defined as transformations of states in the hubris space. Similarly of operators, which define your observables. Okay, such that transition amplitudes or probability amplitudes are invariant meaning that things like this from two states and operating insertion is invariant under this transformation. Why should it not be invariant up to an overall phase up to some. The amplitudes. Okay, so if you're just acting on the states that in general, the states is defined physically up to overall phase, and that will be related to something I'll mention next. Okay, thank you. But the key is that this phase is universal for the hubris hubris space. So when you look at this bracket, it will cancel. The phase is known as the protracted space space. I think we're as a refinement of the original discussion of Wigner. Okay, this is what I'm going to say next. So this is the definition of the symmetries. But Wigner in 1930s axiomalizes in terms of axiom, axiomatized the notion of symmetry in terms of certain operators in quantum mechanics. That is, this symmetries. Okay, they're equivalent to two notions of two kinds of operators. It could be either associated with a unitary linear operator acting on hubris space, anti unitary, and also anti linear operator on the same hubris space. Okay, and the latter case is not important for this lectures will be relevant if you include, say time reversal symmetries and because symmetries. Thanks to Wigner is associated with this unitary operators. Okay, focus on the case without time reversal activated. Okay, associate with unity operators. Naturally, there's a notion of group. Okay. In particular, this unit operator, let me call them UG. Okay, where G is element of the group. And the group law is realized by the product of this unit is your operators at the hubris space. Is this part visible to for G and G prime inside the group. And this is just the group multiplication law for this group elements. And this unitary is acting on the hubris space. Okay, or states in the hubris space. In general, as follows. Okay. And acting on operators by conjugation, and the related to the question that was asked before, the, the representation of this symmetry operators acting on sorry the symmetry operators acting on the hubris space. So this does not form a representation for this group represent linear representation for this group law, but instead a project representation that is related to the face ambiguity associated with the states in the hubris space. Okay, so far we're just talking about general quantum mechanics, we haven't used any fact about locality, something that's crucial to the quantum field theory. Okay, let's move on to quantum field theory. We have another theorem. The constraints, while the structure of the symmetry is more stringent. In particular, the north of the theorem, north of the theorem implies that if you have a continuous symmetry. You want symmetry, which is a charge symmetry associated with the electron, for example, it will imply the existence of the special operator in your quantum field theory, namely, another current, which is conserved, and the unitary operator that we can are introduced now apply for general quantum mechanics can be reconstructed from this concept current. In fact, there's something even more elementary can define that is the notion of a charge. So for example for the UN case, you want symmetry of electron, and this will be measuring the charge associated with the electron, and they can be defined as follows is an integral over the special direction at a constant time of the there was component of this current. So it measures the charge of a state that is specify your quantum field in your quantum field theory by quantizing the by quantizing the theory on the constant time slice. And this object is T independent. As a consequence, as the conservation law. And in terms of this chart operator. Unitary is just given by the exponential of these charges, the group element. In this case, is determined by the symmetry transmission parameter alpha in this way. Okay. But alpha can be thought of as a le algebra in this case trivially for the one associated with the symmetry group with nurse unitary excellent operators. Okay, so similarly in the quantum field theory. The symmetry action operators is also captured by these charges and the unit areas. Okay, in particular acting on local operators, the symmetry variation acting on local operators at insertion. Oh, sorry insertion x. This is determined by the equal time competition of the symmetry charge q with this operating insertion. So we saw that in the quantum field theory framework at least for community symmetry we can recover all the ingredients that we can are introduced for quantum mechanics will have something more. We have this local object. This is conserve current and we can define these charges. We can go one step further for a focus in the case when the quantum field theory is relativistic, which is equivalent of the week rotation to your clean end. On the field theory, with in particular rotation symmetry, in which case you are free to orient your choice of constant time slice, you can free to choose your direction of time. And consequently, there's a generous notion of the charge operator. A social is a general, not necessarily a constant time slice, but a general called the mention one manifold in your space time, you know by sigma upper D minus one of the host dual of the current one form. So here the current one form is defined as follows. Okay. The host dual satisfy the host dual give you a D minus one form that's why you can integrate over this D minus one dimensional some manifold, and the current conservation is equivalent to the closeness condition for this for this host dual for this D minus one form. Okay, this is from conservation. And similarly, the unitary is given by the exponential of this chart operator as before. Okay. And the way these operators, this charge and unity operators act on that your local operators is by enclosing it. Okay, so for example, if you have a local operator over here with subscript q, which will define its charges under this you want symmetry for some case to be explicit for the case when the symmetry is a you want. You can enclosing it by the charger operator. Okay, for example taken to be the sphere of D minus one intuitively this will measure the charge that's associated with this operator source by this operator. Similarly, if you encircle the same operator by the corresponding unit three. Okay, this will implement the symmetry transformation on the corresponding operator. Okay. In other words, you can shrink this, this ball and replace the left hand side by the symmetry transformed operator. Now crucial property of these integrals, the charge operator defined this way, and the unity operator defined this way is that they're topological. Meaning that even though in this pictures have drawn such that the the encircling codimentary wonderful is a sphere, it doesn't really depend on the metric on the sphere or the shape of the sphere only depend on the topology. Okay. And that is a consequence of this being a closed B minus one form. And consequently, that this this kind of integral only depend on topology of this of the manifold interpreting over as a consequence of the Stokes theorem. Okay. So what's the implication of this topological property. So this, this objects. Okay, this objects, this unit three defines the prototypical example of a topological defect operator. The notion of the logical is what I have just explained. Why it's called a defect is because it involves, it has a dependence, even though topologically on the hard dimensional manifold it's not inserted at the line, you said at a point, but it's wrapping some hard dimensional some manifold. Okay, as opposed to a point. That's why it's called a defect. In fact, this topological is closely related to constraints and correction functions in the following way. So imagine some general correlation function that you're studying the quantum field theory. Okay, with various insertions. Okay. And potentially symmetry topological defect operator, which I'll call you G inserted somewhere in this correction function. But in the neighborhood of this of this topological defect, we have an operator insertion which we make it explicit and call it five. This, the fact that this objects topological means that if you focus on this region, right over here, save some space. We are free to deform the locus of the symmetry defect or the topological defect operator. Keeping the location of insertion fixed. For example, we can deform it all the way to something like this. Okay. And this equality means that we are free to replace this configuration locally by this configuration without changing the correction function. This is called isopropane variance. Furthermore, if you have more than one topological defect, more than one topological defect operating certain your correction function. There's the fusion. So the characterization of the group modification of social ways the unitary operator that we can are defined in the, in the context of the two in the quantum field theory. Okay. In this case, just to be concrete, you can think about the diagrams and drawing is the projection of the higher dimensional diagram down to two dimensions. And if you're not comfortable with that just focus on the case when space time is two dimensional. Okay. The line manifold, the sigma is always just lines for circles. In the presence of two such topological symmetry defects, a topological defect operators, they're all mean the same thing. If there's nothing inserted in between, because they're topological, you can zoom far away. Okay, and they look like a single defect insertion, which is labeled by the product of the individual group element enters here. Sorry, the same sigma. So I'm so I'm taking the limit where they collide in which case the same sigma and given by this. Okay. Ah, right. So, Right, so But the orientation is related to how I label this defect by group elements, in particular, if they are not oriented. Sorry, if they're oriented in the opposite way, like this. This will be equal to the trivial line, meaning that there's no insertion. The reason is that. So, so this is the is the with the bar. Okay. In the opposite orientation. The reason is that you can connect the two ends of the line. In each case, you just form a circle with no insertion. And that will be the case that you are detecting a charge of the identity or charge of the vacuum and that's trivial. Okay. So that this is the, this is how the two orientations are related. Okay. Now, once we understand the fusion, we can go one step further in this picture. The recall the usual way you, you determine water entities is to reduce the insertion of the substance, reduce the insertion, like, sorry, reduce the, the current function with the insertion of this unitaries to something that depends on the charges of the insertions. Okay, and then you can deduce consequence such as the charges of the insertions have to add up to zero for this current function to be non trivial. And to do that step, you want to be able to move across the this water symmetry defect away from this insertion. And there's one step involved, which come from doing a fusion of the two ends of this the budget line. Okay, and that's equivalent to doing this. Okay, locally. And as a consequence, what you find is something like this. Okay. Similarly, is equal for a general group for symmetry defect associated with general group as the transform operator in the presence of a line which have been moved across the operating search. Okay. And this feature, the fact that there's a simple way to move the symmetry line across the local operation insertion with no other line attached is actually a feature of this invertible symmetries, and you will see how this generalize in the invertible symmetries. How many minutes do I have, we started late right. Okay. All right, I'm ready to summarize this part of the discussion this review on symmetries that's about defects. Some comments. Okay, the first comment is that the isotope invariance. Okay, meaning that the deformation of observable in the presence of this topological defect operators is invariant under small deformation of the line as long as they do not, you know, do not cross other operating insertions together with fusion, which tells you how you can further move the line across the local operator. Okay. Any information of how you know defects act on local operators. Okay, there are three ingredients. The three ingredients completely characterized the conventional symmetries as topological defects. So I'll use them interchangeably for continuous symmetry, because the presence of the current, which is something more fundamental. This looks like a complicated machine to produce something that's well known. Okay, so why is this useful. This viewpoint of topological symmetries as the water defect is useful already in the context of invertible symmetries, when the corresponding group is discrete. In each case, there's no north or current, and this topological operators, the water defects should be treated as rather than as the fundamental objects. In particular, you can just like in ordinary continuous group symmetries, you can gauge the symmetry by coupling theory to an entrepreneurial background by turning on gauge connections. In this case, the water defects give you a natural way to couple the theory to a discrete symmetry background. Let's focus on the case of t equal to two just so that I can draw explicit pictures. In which case the water defects labeled by G. This specifies the transition function. The discrete gauge field, a discrete g background essentially based is essentially just determined by specifying transition functions, okay from patch to patch so you can try to relate your space time, and the background is specified by transition function to patch, like from the left to right, particular the transition function, for example in this picture will be given by this g element labeling the topological defect. And furthermore, if you take two topological defects, okay. Instead of doing this fusion compete together because they're topological, you can imagine fusing just one ends of the defect. And this gave rise to, and you do not break the topological property through the way. This is defined for you some topological junction. Everything is topological. And together, they allow you to form a general network, and this is equivalent to a general discrete symmetry background. Okay, so you can think about this as a patch of the full network that you pile on your space time. So here, the basis topological because it's all these junctions you're using this basic building blocks. And, and the very nice thing about coupling a theory with certain symmetry to is to the field background is because it's a, sorry, it's a very nice thing, because this is a useful way to detect anomalies associated with this symmetry, we're precisely the top of the topological symmetry. Okay, so it's not a sickness of the original original theory, which is global symmetry, but as a feature, and this can be detected by gauging non invariance. Once you couple the theory to gauge background gauge fields. In this case, in the three case, this water defects detected by change in the network apology. So the basic ingredients I was talking about here as a topic variance and stuff like that ensures that the virtual network is invariant under small deformations of all these vertices. Okay, as long as they don't cross each other. But if you focus on a patch of this diagram, for example over here. There's a way to change the connectivity of this diagram by performing a gay transformation. For example in this region by doing a gay transformation. Let's say G, G one. And by doing that gay transformation, you go to another diagram that with a different connectivity. So all this internal legs are ambiguous because the because of the world over test condition. The third leg is always determined by the labeling of the of the other two legs. Okay. So the gay transformation essentially removes this line or reconnects G one with G two. Okay, reconnects the topological defects level by G one to G two, giving rise to this diagram. And if there's no no face associated with this, this, this gay transformation. That's the equivalent of saying there's a normal now. Okay, so once again normally is diagnosed by coupling the theory to the symmetric to the symmetry background. The gauge field background for that symmetry, in this case, that work of topology defects. And if the, if there's a face coming from doing a gay transformation. Okay, then that signals a normal. In this case is equivalent to changing the topology of the network. And this face is generally denoted as this. Okay. And this as a as we're saying in close anomalies. And this face is none. It's not arbitrary, just like usual anomalies are subject to Western middle consistent conditions. Similarly, this anomalous face also subject to a finite version of the Western middle consistent condition. Western middle consistency. And this leads to the constraint that this face is something classified by the group co homology h 3g with you and coefficient. Okay. The co cycle condition is coming from the consistency condition. And the fact that you're coaching the out by by exactly cycles is related to that. And at this vertices, there's some choice of gauge, there's some gauge freedom. Okay, and that gauge freedom, allow you to remove part of the faces of the face, the part of the face which can cannot be removed by this ambiguity is over here, which is related to local counter terms in the context of continuous symmetries is precisely this anomaly. Okay. And that's captured by this country will go homology classes. Okay, and they will see generalizations of this for non invertible symmetries in the later lectures. And we end here and. Okay, let's thank you from first lecture. And now we can move to our discussion session where you can ask questions. So, note parts of pens, you can take them here. Okay, let us resume. In the second lecture, we are happy to have Lord on a, and she will tell us about still still amplitudes. Thank you very much. Can you hear me. I hope the microphone works well enough so that you can hear me at the back. No, it's not working well. I didn't touch it. I think it is working but it's just maybe not close enough. It's okay. Mateo I mean, but I should speak louder maybe. Okay, so the microphone is for online and Okay, if you can raise, put the volume of the microphone higher, the better so that I don't have to shout. Okay, well let's, let's see, let's see if you can hear me I will try to speak out. So thank you very much. It's a great pleasure for me to be here and to see you all here and also. Hello, people online. I will be telling you about celestial amplitudes to celestial amplitude I will present them as the observables of quantum gravity in flat space times, which lives on which live on the conformal sphere at the boundary of which is called the celestial sphere. And celestial amplitudes are the root of a recent program, which is called celestial holography, and which has been developed, being developing very fast in the in the recent years. So, so the goals of this program is to address to which extent we can generalize the holographic paradigm to space times, which are more realistic, namely, in this case which are asymptotically flat. So, so celestial holography proposes a holographic approach to quantum gravity in asymptotically flat space time. And I will define what I mean by that. And so it's a very precise proposal, actually, which is rooted on this celestial amplitude. So celestial amplitudes in this context will play the role of the observables. So they will be actually scattering elements. But written in a in a in a convenient way. Such as to exhibit the manifest conformal transformation law under the action of the Lawrence group. They will see the observables of quantum gravity, and they will be living on the celestial sphere. Now we'll recall what is the celestial sphere and what is an asymptotically flat space time. This, this, this story and this program is actually now pretty old. It builds up on on the on many observation that have been made in the last years. Basically, the main observation is that the infrastructure of gravity in flat space times is much richer, more subtle, and also much less understood that that we thought it was it was. It has to do with the some kind of very deep realization that several aspects of of physics in flat space time are actually deeply connected to each other. Sometimes we will see that the so-called asymptotic symmetries of general relativity can give a symmetry principle for very known a theorem in computer theory, which are Weinberg one Berks of theorems. This is just to mention one of the relation that we will go into more details. So it's a quite long story which builds upon physics did that belong to different topics. But as I find that this is what is very interesting to actually relate all this, all these things together. So the main goal of this, of this formulation of celestial amplitude has to do with attacking this program, this problem of addressing a holography in space time which is not anti the sitter, but instead, describe in a more realistic way, the kind of space time we live in. So, motivation for this program. Basically the first is to understand the holographic paradigm for more realistic kind of space times. So, in particular, I mean, we know that the holographic correspondence has given us this in its most concrete realization is the ADS anti the sitter. So the theory correspondence, which establishing which establishes that quantum gravity in a space time of anti the sitter so with negative cosmological constant can be equivalent equivalently described by a theory without gravity a conformal field theory that lives at the boundary of the space time. So this has this paradigm has been proven to be extremely powerful. And we would like to, to, to know how we can approach this problem for more realistic space time namely for that space times which have vanishing cosmological constant. Of course, we know that we live in the universe with a positive cosmological constant. But the approximation of looking at flat space time vanishing lambda is a very good approximation for a huge amount of physical applications from collider physics to astrophysical physics up to scales which are of course smaller than cosmological scales. So, we know very well how the holographic correspondence works for in in the DS, but for flat space time. Somehow, the problem is much less understood, and it has to do with several feature that are actually inherent to the to the to flat space The fact, for instance, that the boundary of that space is not a time like Lawrence in boundary, but it's telling not boundary, where there is no natural notion of locality or time evolution. So these present new challenges that one is not encounter to to face in a DSCFT. But I don't think that nobody thinks that holography is just a peculiarity of, of this kind of space time but that is instead a very rich and broad, very broad principle for gravity. So we'd better understand to which extent and what we can make in for for more realistic kind of space times. So that's the first motivation is holography basically is nice. The second one, independently of holographic motivations. I think this program of celestial amplitude and celestial holography is very interesting. Because it, it conveys new and deep connection between several subfields in physics, as I have already mentioned. And just a few of these topics. And this is what we will start with today is the topic of asymptotic symmetries, which is a purely general relativity classical topics. In general relativity gr. We will also deal with quantum theory, especially soft theorems that were discovered in the 60s by Weinberg and others. And this has to do with more pragmatic topics as it actually all this, the story is beautifully connected to observables in gravitational wave physics. And these are the so called memory effects. So I will not have time to talk much about memory effects. But roughly speaking, these effects are if you want the physical consequences of the of the fact that we have an infinite amount of symmetries at the boundary of last space times. And then, as we will see celestial amplitude. Since we want to relay this to holographic techniques we will have to also build some connection with the 2D conformal field theories. So this is for the motivation. Let me define maybe a little bit more concretely what what we will do here and what is celestial holography. So this will be a duality. And yet to be established between gravitational scattering in four dimensions so I will be working in four dimensional asymptotically flat space times, and the yet to be fully understood dual theory, which will be a two dimensional field theory, which I will refer to as a celestial conformal field theory, or celestial CFT. Sometimes, I will denote this by CFT. So what is a celestial CFT. Well, I don't know. But we will discuss we will go along this, this, this path together. And we will see that a celestial to this is CFT shares many feature we with the conventional conformal field theory in two dimensions, but also as we will see very properties and puzzling aspects, which has which have to do with the fact that we are in flat space times. And people now it's a it's a rich program which involves many people coming from many different subfields from from scattering amplitudes from gr. And also people coming from the CFT, CFT background. So people are really actively trying to come with a better understanding of what is a celestial CFT what are the constraints of this theory, what are the list of properties that they should obey. And how it encodes gravity in flat space times. So most, most, most concretely, what will we do, what we'll do is, what is a celestial amplitude is a scattering amplitude, but written in convenient basis, which, which exhibit manifestly SL2C and variance I will explain this all that detail but just so that you have an idea so we will describe the four dimensional scattering process. So the duality proposal. In terms of a very different thing, which is a correlation function in this so called celestial conformal field theory. So if you want is a 40 bulk to the kind of duality, and I will explain why this is an interesting and somehow natural playground for holographic in flat space time. So now on the left hand side have as I have scattering elements in in 40 bulk. On the right hand side, I have correlation functions, which involve a bunch of operators living on this on this on the celestial sphere. And these operators will be labelled by these quantum numbers. And the two dimensional in the 2d skin, Jay. So right, I will explain you how this holographic map works in the in the third chapter of the lecture. That's roughly speaking, what we will do is that. We will consider mostly massless scattering of massless particles, and massless particle can be labelled by a momenta, which involves the energy omega, and what we will do is we will trade this energy for this conformal dimension delta via a precise integral form, which is called a melon transform. And the 2d spin, on the other hand, will be simply identified with the 4d helicity of the particle. Let's call it L. So this will just be identified. So we will trade the energy of the particle for this conformal dimension delta, the spin of the correlation correlator of the operators will be the helicity of the particle. And the Z and Z bar will label how the particle enter and exit the celestial sphere so I will explain this in way more details basis to for you to have a broad outlook of what we're going to deal with. So, why is it a good thing to do and what is this, the new, the new let's say the new because they have been several attempts before to, to, you know, to to get some flat space holography intermediate safety. So something that works, but most of the things didn't. And the new take of this program is to actually use the huge amount of symmetries that are the boundary of that space times, which are the so called BMS symmetries that I will review today. So basically, the new powerful tool of that we will use. So in the first lecture, symmetries are extremely powerful. As they are strong, they will strongly constrain the problem and give an infinite amount of conservation laws in flat space time. And actually they are. Now people are slowly understanding that they are much richer. And also more subtle than than expected. The goal is to use to a full extent all these symmetries to constrain at maximum this problem and and the celestial shift. Actually we have an infinite amount. We have infinite amount of infinite towers of symmetries. And these provide for us for a full constraints. And I will explain precisely how this constraint manifests themselves for conformal fifth year. Is there any question on on the motivation. Yes. Yes. Yeah, right. So it's different from the usual co dimension one type of holography where we have the city living in one dimension law. And this is one of the if you want unconventional aspect of that. And I will try to explain where actually so it always has to do with what is the natural place for the dual theory to live in right and and you can, you could come with a different with a different type of holography this is something actually I'm interested in into. But roughly speaking, we will see that the celestial sphere which is the conformal sphere the boundary which is two dimensional. Because of the nature of the scattering process in flat space time will be naturally playing the role of these holographic screen. And it just has to do the fact in two words with the fact that the Lawrence group in flat space time acts as the as the conformal group on the celestial sphere. So if you're recast these amplitudes on the celestial sphere by construction by symmetry construction, these will naturally transform conveniently under SL to see. If you have the conformal group and then you can wonder how to extend it to the local to the local group and and have fully fledged to DCFT. I will come of course to that, but indeed this that's that's a different kind of holography is a co dimension to type. If for the as three space dimension one time the CFT is just space. It will be labeled by this angle Z and Z bar. So it's it's a Euclidean if you want. Okay. So we'll see exactly how we start from from massless on shell particle and then how we can extract the data that they print on this to the to this year. That's just a quick question. Just quick question. So, is this national CFT related in any way from a limit of the DS CFT in four dimension from a flat limit. Yeah, so consider ideas for CFT three is this national CFT related in any way by a limit in that, you know, yes for that setup. So now there is no flat limit process that is giving you the celestial CFT from a flat space limit or large radius limit. We know, for example, the S matrix. In other ways is it can be recovered, like in the interior of ideas where the curvature can be affected. Yes, so, so you, this is a very interesting thing to look at specifically what we can have what which kind of relationship we can have starting from even correlation function ideas are relating to the one is a serious fear story. So people have started to look at this but this is something that is not fully understood, and there are actually many reasons why this part is complicated because actually, to, to, I think that my take on that is that to have a chance that this works, you have to really strongly somehow relax. So this is one of the assumptions that we usually do in ADS CFT and one of them is the so called boundary conditions and the boundary which are bouncing here we will have to allow for upcoming flux I will come to that when we'll talk about BMS symmetries but we have to allow for way more relax boundary condition in ADS to be able to do this map properly. Yeah, as you said that flat space time is a good approximation for our universe in some regimes, but the conformal boundary between positive cosmological constant and zero cosmological constant is not really different. Yes, that's a very good point. That's a very good point and indeed, if, if I once I will tell you what I can, what I know about this, it's, it's far from obvious, how will it can tell us about, you know, DS CFT correspondence. So this is another actually interplay that we might be able to come at with the, at some point because indeed, the serious space time is also very peculiar. But actually there are very reminiscent we will see there are very reminiscent feature of funky things that happened in the theater also appearing in celestial holography so there are some some some stuff in common. There's another question. Yeah, I will start writing and don't take it bad but also, would the CFT live in the asymptotic infinity in what sense in the sense that does it live on the special infinity I zero or on some fixed slice of scribe plus or spy minus. Yeah, it will live on the celestial sphere and now I will precise explain what all this location you're talking about special infinity the non infinity the celestial sphere, and this would be here. Thank you. Thank you. Thanks for the question. There is anything else let me know so if you have a question online. Let me know. So 2.1 well let's start with the recording all this the structure of that space time. And in particular what is an asymptotically flat space time. So I'm supposed to stop at 1245 right. Yes. Okay, let's let's start with something you will know which is exactly flat space time, namely the minkowski metric in four dimensions, have a T, the time coordinate the radio coordinates and the sphere angles. Let's take for the moment to time fire. So this is just minkowski now I'll be I will be using a lot in this lecture. So time coordinate. Let me write how this keeps some retarded time coordinate. You, which is simply the difference of the of T minus r. And very importantly, I will be also using complex. So I will be using the metric coordinates in place in place of instead of the street and triangle, which are the Z and Z bar coordinate which will pop up a lot in this lecture so they can be obtained by taking it e to the high five co tangent of T to over two. The Z bar is the complex for a conjugate of Z. So it's e to the minus high five co tangent T to over two. So in these, maybe somehow unusual coordinates, the minkowski line element. Just take this following form minus the square minus to the UDR. So this is just the unit square square is gamma Z Z bar thing here, the Z Z bar or gamma Z Z bar is just the unit sphere metric here, but now written in complex coordinates two over one plus Z Z bar squared. So this is just the unit sphere metric. I will draw a very important diagram which is a Penrose diagram for minkowski space and was diagram is bringing bringing the infinities of a space time to a finite distance and an important thing about this diagram is that light rays and massless particle always will propagate along lines of 45 degrees. So, so time is sorry. Time here are. So this is the trajectory of, of, of a massless particle in this diagram. This particle follows not geodesic and basically let me, let me draw here. So what is the big weird like constant constant our curves in the space times are like that. So these are constant radial coordinates curves and constant time slices. Here in blue. These are T equal constant in this diagram. So I'm not recording how you make the change of coordinates to and you make the conformal compactification. By the way, you can a good reference for for this, this lecture is from in your lecture notes, which is, which are on archive. So in this diagram, the null coordinates. You is going in this way. Very good. And so in each, in each actually each point of this diagram is actually a two dimensional sphere. Well, it's, it's not exactly it's a half a sphere which is mapped to the, to the other to the other side but let me be a bit sketchy and I can ask more about that. After if you're interested but each point in this diagram is a two sphere, which is level by this angle that and that bar. Very good. So there are different locations in this diagram so one, I will just ask is this. This place is called special infinity, I zero special infinity is the place you reach. When you take are going to infinity and T constant. You have past time like infinity and future time like infinity. I will not talk much about that but basically. This is a location where massive particle. And their lives. So, a massive particle will have this kind of trajectory in, in that space. And it ends its life here at. I plus, which is future time like infinity. So, which is obtained when you take T. But since I will be dealing mostly with masses scattering of massless particles, I will be especially interested in this null hyper surface here, which is a null hyper surface. Future null infinity and denoted by this letter calligraphic I cry plus this is quite plus because this is the future, but there is a past analogous of this loop of this hyper surface cry minus, which is past null infinity. I hope you can read something on this diagram. Something is not clear just just let me know. Good so what is the solicitors here and where where will our theory be living. So let me erase this minkowski. In element. So, so as I said, each point in this diagram is a two sphere. And so topologically future null infinity is a real line, which is spined by this coordinates you times a two sphere. And this fear is called the celestial sphere. Like I write this CS to maybe sometimes I will write this CS. Let's let's keep it like that. I don't promise I will be consistent all the way through but so it's really, it's really the, the sphere that you can see when you look at the night sky approximating that we will live in flat, flat space times, but this is this is nothing but the the celestial sphere is just a sphere you can see at night. So that's flat space times. This is a very formal compactified version of this flat space time with this pen rose diagram. So the important thing about that that I want you to focus on is that masses particle propagated along 45 degrees and end their life here on this null hyper surface which is called future null infinity. And the masses particle also come from another location which is the past. So I would like to draw this to draw this diagram sometimes people draw the version which is a triangle but I found this one is this one is less confusing. So this is just minkowski now we'll be interested in more generic kind of space times, which are the so called asymptotically flat space times. So I'm thinking an asymptotically flat space time is, as you might guess, some space time that looks like minkowski in from very far away. And this is indeed what it is. But there is a precise definition of what we mean by looking like minkowski from, from far away. The definition is giving us by the seminal work of bondy. Vanderberg metzner and Vanderberg is always dropped for some reason. Let's, let's put it back for once on the, namely, BMS. So these guys, they make a very important piece of work in, in the 60s so these guys are general relativistic general relativity people. And at that time, they were people were, it was not very clear whether gravitational waves in GR actually also actually existed. So they wanted to make some work that were to prove the existence of gravitational waves at non linear level. And this led them to define the so called asymptotically flat space time. What is that well in the first approximation, this is a flat space time so it will be given by the minkowski line element that I have written here. This is just minkowski plus some correction, which will be tamed away as R is very big so this will be a large radius expansion. And there is a precise prescription for for this. It's very important. So, so we are doing a large or perturbation of that space time. I'm writing some stuff in in in green and in red. And then I will explain what these are. What I'm going to tell you before I forget is that this big D capital, well with an indexed of the bar is the covariant derivative with respect to the sphere metric so gamma z bar, which raises and lowers Z and Z bar indices. So sometimes, you will catch me using this notation big big a just to denote collectively the angle Z and Z bar. So, we have a few terms here, but remarkably not so many, not so many terms that we need to add. And this will be good enough for for these lectures. This is easy of easy easy. So I'm writing here just one holomorphic component. But I have the complex conjugates DCC means complex conjugates, where everything I've written is copied but putting that bar instead of Z. So we have plus dot dot dot plus dot dot meaning some sub leading corrections in this one of our expansions. Okay, so what is that and what are these functions in green and in red, and we explain that, and then you can ask me some questions. So this, this function M here is called the bandimas aspect. And this, this NZ here there is also NZ bar is called the angular momentum aspect. This is well, roughly speaking, as their name suggests, and just in cause the total energy of the system you're describing, and NZ has to do with the angular momentum of of this of the thing you're describing so if you take a care metric, just a care metric black hole solution. So if you look into these coordinates, you will see that this M here is just nothing but the mass of the backhoe. So in this case it's just a constant but here I'm allowing for more generic kind of space times, because I'm allowing this function to depend arbitrarily on the retarded time and the angles. And this aspect will be of course related to the angular momentum of the of the care black hole. If you were to expand this solution into these coordinates. But you see I'm considering a much more generic kind of setup, which is this BMS asymptotically flat space time. And this function here in red is the asymptotic shear of null geodesic congruence asymptotic here for short. And it's very important because it, it's, it's present is telling you whether the system you are describing is emitting gravitational waves. In particular, you should define this so here I'm using AB is collectively for Z and Z bar. This object, which is the retarded time derivative of the shear is called the news tensor. And so it includes outgoing the presence of outgoing radiation. So this is very important. Why have I written this in red and why this in green well, the difference between these two things is that Einstein's equation implies some evolution equation or constraint equation on the boundary mass and the angular momentum aspect so roughly speaking if you write Einstein's equation, order by order in R. You will see that the time derivative of M equals is constrained to be something. And similarly for N, Z, and it's a bar. On the other hand, the CAB here is not constrained so it's really a data free data that you put in the theory. It's relatively different than than the other two things. And so basically, roughly speaking it includes the two polarization mode of the of the strain measure by a gravitational wave detector at very large distance. So just just then then I will stop writing thing and I will take questions but just to explain a little bit more physically what's what this metric mean. So, there is this very famous formula, which is called the bond formula, but actually was also found by Trotman. I heard recently, before bonding. This is very important formula, which is explained a little bit what's what's going on here. So it's telling you, basically, that the integrated the version on the sphere of the mass aspect decreases in time. And the reason why it's decreasing in time. So there is gravitational waves that which is escaping the system so it's very easy to understand you have some some gravitational system, which is emitting gravitational waves. The wave is escaping from null infinity and as a consequence the energy decreases. So this formula was the first actually theoretical evidence. For the existence of gravitation waves. And on a linear level. Is there any question on this diagram on the definition of asymptotically flatness on these functions. It's a trivial notational question so just be sure so CAB the non zero components are just CZC and CZC bar CZ bar Z bar. There's no CZC. It's symmetric and trace less. Yeah. Okay. So indeed, we'll have CZZ or and said C Z bar Z bar. So yeah. So these are the encodes the two polarization degrees of freedom basically. So why there is no energy coming in from the past. Oh yes, there is I am sorry they can be in going. So everything I've I'm writing can be also written for incoming incoming waves. So in this case, thank you for the question so in this case, we will use a different set of coordinates. We will use an advanced coordinate V, which is now t plus R, which is running here along this now personal infinity. And everything I have written can be repeated for incoming incoming wave in terms of these advanced coordinates so here I'm really doing all the analysis at one boundary. And we'll see later how actually, when we will want to talk about the scattering problem, how the data the past and in the future are related to each other. But it is I'm focusing on outgoing gravitational waves for just the sake of, I could you could repeat everything in advance. And in the lecture note of Andy you can find the, the relevant formulas. Yes, is there any specific conditions on Riemann tensor that asymptotic flatness, this asymptotically flat space time should follow like what are the conditions on Riemann tensor. I would say that this is flat basically that you will you will, you can solve the Einstein's equation and you will see that. No, no, but after Einstein equations, there will be some conditions coming from this Riemann tensor right. So, like, there's some this electric components or magnetic components of sorry while tensor I meant while tensor so yeah there's some specific conditions. Yes. Basically here. This expansion, the fact that I am making an expansion in our in the terror expansion in our. And you see for instance I didn't include a log of our, I could have put some log of our in principle, but basically this expansion is in. Basically because there might be some Saturday arising but basically is is almost equivalent to the peeling theorem in GR so which is telling you that the violence or has certain precise fall off in in one of our. So this thing satisfy peeling theorem. Now if you talk to a mathematical GR guide they will tell you yeah but we know there are solutions that will not satisfy peeling and so on and so forth. But and this is a good comment but here basically I'm, as you see I'm, I'm assuming there is a conformal compactification holding and all this. But this is good enough, and actually very excellent for what we want to discuss so they're precise fall off on design. So I'm not allowed to use time of the discussion or I'm not. Okay, then I will have to tell you what our BMS symmetries in the few minutes remaining. So we have introduced most of the thing we will, we will need, and sorry for the GR D tour to amplitude but is, if we don't do that we will not understand where this theory is living with our discord in a sense. Why the why where this the constraint on the celestial safety come from. We go to BMS symmetries. So what we want to look for. So we want to answer the question what is the symmetry group of asymptotically flat space time, the symmetry group of flat space time we know it's, is one carry. But now we are looking at a bigger, or if you want more relaxed kind of version of flat space times. So here what are the symmetries that would preserve such an expansion. So what we want to look for, we will look for infinitesimal vector fields, this type. I, I'm just writing that all components. Nice. So we want we want to look for these kind of vector fields. We want to preserve these asymptotic expansion. So this is the leader of active alongside of the metric. And we will ask this not to be exactly zero that would be looking for a killing vector. We don't want that we want something that preserve the asymptotic structure so we want to ask two things. The rules of the game want to place we want to preserve the fall off conditions, and the gauge fixing I didn't talk about the gauge fixing there are some, it's actually important there's some gauge fixing in the metric. But if you're interested in that you can ask me. So, so this is what we cannot do is to mess up with this, this part of our here. We don't want to introduce the art to the tool here or something like that we want to keep the expansion like that. But what we can do is we can change this function. In green and blue and red. So we can change the body mass angular momentum aspect and the shear. This we can do. When you follow this this game, what you find as a solution is the following vector field. So it's given by a certain function T, depending on the angles. You over to divergent of the, of some why so what is this why, why is that is what is basically the Z components of this vector field. So why only depends on that I would explain that there is another and a similar story. And there is an expression for XIR, but this is not important, what it is, what is important in this is that this T is an arbitrary function of its argument. Why, why a so which is why, why is that bar is a conformal feeling vector on this on the on the celestial sphere. So you can write a conformal equation for this vector field. I will not. So I want to tell you here is trying to convince you that these symmetries are very rich. And this is what people call. So the fact that T this function is arbitrary is what let people to call these symmetries BMS super translations. So I know this is the super string school, but this super here has nothing to do with that. This super just means the following. It just means that we have an enhancement of symmetry. Where the four global translations of one carry, including particular this, these four translations. But also, so they are enhanced. There are many, many more, they actually an infinite amount of them, because this thing is an arbitrary function of the sphere so you have an infinite way to generate them to these infinite dimensional super translation so the super just means that you have an infinite amount of translation on the celestial sphere. So this is actually very, it was very, very surprising for, for bonding message when they found this symmetry structure, because what they wanted to find they wanted to recover the primary group, namely the isometric group of that space time. That wouldn't have made sense. They were stuck with with the appearance of of an arbitrary function here, and there were really pieces okay what is this function we don't want that we want this to a span just the four global translations. So, and they really tried hard to kill this function. They really tried hard by imposing stronger boundary conditions, but what they realized is that as soon as they wanted to kill this function, what they were doing actually, it was to kill, or gravitational radiation. So, they came to a conclusion okay I mean, it seems that if you want to include radiation we have to allow for for the symmetries. So people really didn't know what to do with that. Until in the early 2010, where Andy storminger and collaborators realized that not only these symmetries have to be there, but that they encode in a very powerful way. They were known from a totally different perspective in quantum theory, which was known as a theorems, and I will tell you about about that. Tomorrow, how these symmetries are actually extremely powerful. And I don't have much time to talk about this conformal killing vectors here because I'm running out of time. So, let me just say in one words that if you ask this what these wise to be globally well defined, you will find a six Lawrence transformations, three rotations and three boosts. So relax this assumption, namely, you allow for the wise to be meromorphic functions. So you allow for local singularities. Then you have a similar enhancement of symmetries, where the six Lawrence transformations. Well as you know in CFT two are enhanced to a local version of that. Which are the local transformations, and which span two copies. So why and why bar span two copies of the Viral zero algebra, or more precisely the wheat algebra which is the zero algebra without central extension. Because we are at the level of the fields of the vector fields here. And sometimes people this super rotations. And this was, these were advocating much later, my Glenn Barnish and Cedric to start in around 2008, I think. So this is a much more generous, the recent story that is motivated actually from holographic. From a holographic point of view and I will come back to that so let me stop here for today and take your questions. Hi, I was curious how one could describe.