 Okay, so first of all, as usual, let me thank the organizers for the invitation to speak here. It's always a pleasure to be back at the ICTP, where I was a diploma student. I won't tell you how many years ago, but it was many years ago, okay? And I was told that this is a very relaxed conference that I can take two hours, so no problem. So today, in fact, so we have a lot of things to cover because today I'm going to be telling you about the S-metrics. And the S-metrics is an object that is even older than quantum field theory. So we'll have a lot of things to do to cover a lot of history. So but instead of starting from where the S-metrics was originated, I decided to start from one of the textbook definitions of the S-metrics, which is something that we all learn in quantum field theory. So the standard definition of the S-metrics starts by giving you a Lagrangian for some fields, then you compute a correlation function of operators inserted at various points in a space time, and after you compute this object that has lots of information, you take the Fourier transform, but before, by the Fourier transform I mean you just multiply by wave, plain wave functions for particles that are incoming and for particles that are outgoing. And then using general arguments in quantum field theory, you argue that this object must have poles of this form where particles become on shell, and then you compute the multi-residue of this object, and that's your S-metrics. So you start with this object that has lots of information in it, and then you throw everything away, almost everything away, and you only get something that is a multi-complex dimensional residue at the location where all the particles become on shell, okay? So that's the standard definition of the S-metrics, and this definition assumes the presence of poles in the correlation functions. This of course can be formally proven for massive particles, but here we will assume it's also true for massless particles, and I'll be working at fixed orders in perturbation theory where this assumption is valid. And in fact in the rest of the talk, we'll only be dealing with the S-metrics of massless particles. So people dealing with S-metrics theory in the 60s or in the 50s were mainly concerned with massive particles, and part of the reason was that they were afraid of this analytic structure that was going to be completely wild once you bring together branched cuts to poles. But we will pretend that that problem is not there and just continue. Now the definition I gave you is basically a story of interactions in a space time. So the way we compute the S-metrics is as a sum over possible interactions that can occur in the interior of our Minkowski space time. And here I've drawn for you the Penrose diagram. This is scribe minus and scribe plus. These are called also the null infinities. And this point here is a very singular point. This is a spatial infinity, okay? And this is r equals to 0. And the story goes as follows. Some particles are incoming. Some particles are outgoing. Here these are massless particles. They come from scribe minus. They do their things. And then they go off to scribe plus. And we describe these by using Feynman diagrams. Now Feynman diagrams are excellent, right? The reason they are excellent is that they make the two pillars of quantum field theory manifest, locality and uniterity. So Feynman diagrams lead to these expressions that make manifest locality and uniterity because they come explicitly from local interactions of the theory. So it's not a surprise that Feynman diagrams give us something that makes this manifest. But since the 80s, starting with the work of Park and Taylor, it became clear that scattering amplitudes are much simpler objects, way simpler than what Feynman diagrams wants to give you. So is this a hint of something? Could this be a hint that manifest locality and uniterity are not the basic properties of a formulation of the S metrics? So I say manifest because, of course, whatever you construct must be local and unitary. So it's not that we're going to depart from locality and uniterity, but we would like to depart from manifest locality and uniterity. But if we do that, what are we going to replace it for? So are there more constraints, different constraints that are not locality and uniterity that we could use as the basic building blocks? Well, another very important constraint that we're also learning quantum field theory one is that the S metrics has to be Poincaré covariant. And this is very constraining because transformations must be consistent with those of asymptotic one-particle states which are irreducible representations of Poincaré. There is something else that is a very important constraint for massless particles. And this comes under the name of Weinberg's soft theorems. In the 60s, Weinberg realized that when you take one of the particles to be soft, again, this is only something that you can do when particles are massless. So it's a constraint that you will get every time you have massless particles in your theories, such as gluons or gravitons. And you take one of the particles to be soft. Then there is a connection between different S-metrics elements, an S-metrics element with N plus one particles and an S-metrics element with N particles multiplied by a factor that contains the information of the wave function of this particle. In this case, it's a graviton. This is a polarization, in fact, tensor. And the momenta of the particles that are left over in the corresponding S-metrics element. Now, this constraint was so powerful that allowed Weinberg to derive the universality of gravitational coupling, electric charge conservation, no particles with LECD greater than 2. Basically, everything that we know or that we need for quantum field for particle physics or the basic principles can be derived from these soft theorems. So they are very powerful. So once again, these are the constraints that I would like to consider. Irreducible representations of Poincaré or how the S-metrics has to transform on the Poincaré and Weinberg's soft theorems. I want to argue that these are constraints that happen at infinity that can be defined at null infinity. They don't have to require, they don't require you to know about spacetime in the interior. Well, those are two, but are there more constraints? Well, especially in theories of gravitons, in theories in the scattering of gravitons, one could be looking for symmetries of asymptotically flat spacetimes. And this is known as the BMS group. So in the 60s as well, BMS discovered that the group of symmetries of asymptotically flat spacetime is not only Poincaré or the corresponding restriction of Poincaré, but it's actually much, much larger. So let me tell you a little bit what BMS is. So this is again our Penrose diagram. We have null infinity, pass null infinity, and future null infinity. And the topology of null infinity is simply r cross s2. This s2, we can think about it as a Cp1. And sl2c acting on the Cp1 is nothing but Lorentz. Translations of that Cp1 is basically the standard translations, but now restricted to null infinity. And together, they found the restriction of Poincaré to null infinity. But BMS realized that the group of asymptotic symmetries is way larger. In fact, you can take translations of different points on the sphere. So these are the coordinates on the sphere z and z bar. And you can translate each point independently, as long as f is a smooth function. And these are called super translations. So the group BMS is actually an infinite dimensional group. It contains, of course, the Poincaré group, which is finite dimensional. But it's actually infinite dimensional. You have one copy here called BMS minus, and one copy here called BMS plus. So is that useful for something? Every time you think that you have a new symmetry, it must be extremely powerful. Well, could it be that if you take one element of BMS plus and one element of BMS minus, then you can define something like this, or you can impose a condition like this, where s is the s-metrics operator. Of course, B plus has to be on the left, and B minus has to be on the right, because the asymptotic states are on the future null infinity and in the past null infinity correspondingly. Well, this would be ideal if you could do that. But the two BMS groups don't talk to each other. So this formula doesn't make any sense. And why don't they talk to each other? Well, the reason they don't talk to each other is because this very singular point, spatial infinity, so all of a spatial infinity was sent in the Penrose diagram to a point, and that point becomes very singular. So it doesn't allow you to connect the two null infinities, and therefore the transformations don't make any sense as parts of a single thing. But in 1993, C&K, I won't attempt to mention their names, C&K were able to resolve spatial infinity. So now there was a family of spacetimes where basically I0 can be resolved. It becomes a smooth object, and therefore you can connect the two BMS groups into something that can be called the diagonal BMS group. And now you're done. You have something that connects pass infinity to future infinity, and if you're bold enough, you can conjecture that there must be a symmetry. And if there is a symmetry, you can have war identities, and these war identities must have a meaning. And it turns out that the meaning of these war identities are nothing but Weinberg's soft theorems. So you put in here, you compute your war identity with the diagonal element that identifies as the element of B plus with the element in B minus, and you find that this relation is true. This is the S matrix element for M particles, and you can show that the insertion of this operator here is equivalent to the insertion of a soft particle. And therefore, this becomes the soft theorem that Weinberg had. So just from symmetry arguments, you can derive the soft theorems, or you can go backwards as well. Now, sorry, you get excited by that, and you say, well, can I be even more ambitious? Can I even replace the SL to C by a full virus order? Well, people suggested that that could be true, and if you do that, you will find more war identities, or you will suggest the existence of more war identities. And hence, there must be something else apart from Weinberg's soft theorems. So Weinberg's soft theorem is the leading order in an expansion in the soft parameter, in the particle that you're assuming to be soft. So there could be subleading terms, and in fact, there are subleading terms. So this is Weinberg's soft factor. If you see, it goes like 1 over q. The subleading term is now something that depends on the total angular momentum of the particles that you're scattering here. And it goes like you have one power of q and one power of q. So this is order 0, and there will be subleading corrections. So just using these ideas that there are more symmetries at infinity, we discover more constraints. Now, here is a question, or a natural question that one could ask. Can the S-metrics, or can any S-metrics be completely determined purely from BMS representation theory? Well, we know that Poincaré, representation theory of Poincaré restricts very much your S-metrices. But could it be that BMS is what you need to completely constrain S-metrices? And all the S-metrices that we know of are representations, are irreps of BMS. Well, I'll give you the answer, as it stands now. I don't know. But hints that this could be true are the following. So in 2003, we then discovered twisted string theory. And this expresses three-level scattering amplitudes in n equals 4 super-jam-miles as correlation functions on a sphere. So at the time, well, we had this sphere. But what was this sphere? Where is this sphere? Well, could it be that this sphere is a sphere that BMS talks about? And in 2005, the BCFW Recursion Relations were discovered, which has pressed all scattering amplitudes at three-level in terms of only on-shell processes. You never have to go deep in space time. So it's also hinting that there is something of the boundary. And in 2009, this was completed for the computation of all loops scattering amplitudes in n equals 4 super-jam-miles. But of course, you could say, well, this is hardly any evidence, because n equals 4 super-jam-miles is a very special theory. And we know there are miracles that are happening in n equals 4 super-jam-miles. So maybe there is nothing to it. But having the fact that we need a sphere is a very strong hint. And you can start to look for ways of connecting the space of kinematic invariance of the scattering of n particles to the modular space of n-puncher spheres. So here is the space of kinematic invariance. Of course, I'm suppressing many dimensions. I won't be able to draw all the dimensions that I have to. SAB are the Mandelstein variables, just squares of the sum of momenta. And here I'm drawing a single Riemann surface for you with n-punchers. I'm not drawing the modular space, because the modular space has dimension n minus 3. So I won't be able to do it. Instead, I'm drawing for you a single point in that modular space. So what we're looking for is a connection or a map from this space to this space. Well, that map is achieved by using this real function that I call f of sigma. This is something that many of you will see and recognize immediately. You will say, well, I've seen this in a string theory. Or at least if you studied Gross-Amende, the paper of Gross-Amende in 88, you would recognize this as the saddle point action of the string in the high energy limit. Well, if it's a saddle point action, what you're supposed to look at are the critical points of this function. So you compute the critical points by taking all derivatives and setting them to zero. And this gives for you a set of equations that relate the space of kinematic invariance to the space of n-puncher Riemann surfaces. Why is this natural and why is this useful? Well, the reason is that unitarity and locality, which is something that we impose on top of this space, meaning that we tell the space which sub-spaces are important for physics, meaning the spaces where multi-particles can become on shell, is matched by the natural compactification or imposing these constraints chooses a compactification for this modular space, which is called the Liemmann form compactification. Now, you have the space. You have a way to connect the sphere to the space of kinematic invariance. Now you would like to construct scattering amplitudes using that. Remember, I told you that we want to impose two basic constraints. One of them was Poincaré covariance, and the other one were the soft theorems. So let's start with Poincaré. So we all know that Poincaré requires gauge invariance. In fact, gauge invariance comes only from the fact that we don't know how to impose the correct Poincaré representations without introducing redundancies. So it's just a human handicap. So gauge invariance is not something fundamental. So consider massless particles of electricity plus minus 1. These are gluons, say. And for each particle, we have the following scattering data. We have momenta and polarization vectors. Under a general Lorentz transformation, the polarization vector, which is our local way of describing a particle with electricity plus minus 1, transforms in this very strange way, which is precisely a sign that is not in the correct representation. In fact, all we want is a transformation under the little group, which is this little piece here. Everything else should go away, especially this part, which is proportional to the momentum of the particle, which is the part that we call a gauge transformation. So in 2013, in collaboration with Song He and Ellis Joan, we were able to find a construction that makes three things manifest. One is a construction that is an integral over the model space of n puncture spheres, subject to these constraints that I mentioned, these scattering equations. The integral is manifestly gauge invariant, and the un-colored structure of gluons appears as something separated from the rest. So it's something that can be imposed naturally and separated from the rest. And the structure looks like this. At three level, the complete three level S metrics can be written in this form. The first part is the integral over the model space subject to the equations I mentioned. The second part is the part that makes gauge invariant manifest, and the third part is the part that contains all the un-colored structure. So for this talk, I would like to concentrate on this subject, because I think it's the one that has a very interesting story to tell us. So this part is the one responsible for gauge invariance. So let's see how that works. So we're looking for this Pf means a Fafian of a matrix psi. The matrix is a 2m by 2m matrix that is divided into blocks. The first block is the contraction of momenta. Here is the contraction of momentum with polarization vectors, polarization vectors with momenta. And here momenta with momenta. This is a matrix that is 2m by 2m. And the locations of the punctures enter in this form, making this matrix anti-symmetric. Whenever you have an anti-symmetric matrix, you know that its determinant is a perfect square. And therefore, you can define the square root of the determinant. And that's called the Fafian. So this object here is the square root of the determinant of this matrix. Now look at the structure. Polarization vector and momenta enter in very symmetric ways. In particular, what I want you to realize is the following. That if you replace any polarization vector by its corresponding momentum, and you go back for a second, you will see that a column on this side will become identical to a column on this side. And a row here will become identical to a row here. And therefore, the rank of the matrix decreases. And the Fafian being something that came from the determinant vanishes. So Gage invariant is manifest. Moreover, the Fafian is the basic object that transforms correctly under Lorentz transformations. In the case of massless plus minus 1 helicity particles, you get exactly the transformation you want without all the baggage that appeared before. So just purely from representation theory, this object is somewhat special. Now how about for gravity? We succeeded in finding something for gemmules. Do we have something for gravity? Well, the Fafian transformed correctly under this representation. Well, if we want gravity, we need helicity plus minus 2. So why don't we just square it and take the determinant? The determinant transforms correctly. And putting the determinant here is something that makes sense. And it can be shown that this formula computes for you the whole three-level S matrix of Einstein gravity. Once again, Gage invariant is manifest. But more importantly, and I don't have time to explain it today, but more importantly, soft theorems also become manifest in both the gemmules formula and this gravity formula. And these were the two requirements that you wanted in order to have something that has any chance of being representations of BMS, of the BMS group. So this is a general framework. This framework where you can have the structure of integrations over the model space of Riemann surfaces times something else. Well, we don't know yet, but here are some of the theories for which the representation is known to exist. I showed you Einstein gravity and gemmules. But from them, you can construct representations for Einstein Maxwell and Einstein gemmules. And from gemmules, you can go to gemmules scalars. And these theories are very interesting on their own. They have this representation theoretic power, which tells you or constrain, coming from particles with non-zero helicity. But you can also get theories like non-linear sigma model, in other words, the chiral Lagrangian and Galileo's, which are purely scalar theories. And they don't have any constraints from Poincaré, basically. You can also get more infill and direct more infill. I believe these theories are here. And the reason they are there is because they have very strong soft limit constraints. So Poincaré gets replaced by soft limits. So the question is, are S matrices, and perhaps in the CHY representation, the naturally reducible representations of the VMS group or some extension of it? Now, you would say, well, this was all very interesting. But you also mentioned about n equals 4 super gemmules. Here, what I've been talking about is something that doesn't depend on the spacetime dimension. Just like soft theorems don't depend on the spacetime dimensions, the representations that I showed you don't depend on the spacetime dimension. But I started the motivation by saying that n equals 4 super gemmules was special. So we can also start and search for other symmetries. In planar n equals 4 super gemmules, now specializing to that theory and going back in history a little more to 2012, people have realized that this theory enjoys an infinite dimensional symmetry too. In addition to the VMS that we were discussing before, there is another infinite dimensional symmetry which is called the Yang-Yang. And it's made out of the super conformal algebra and the super dual conformal algebra. Putting these two things together gives you or generates an infinite dimensional algebra. So is there a framework? So if you are saying that these infinite dimensional algebras constrain your theories enough so that you can compute them, is there a framework that makes these symmetries manifest? The answer is yes. And the framework is called on-shell diagrams. And I want to spend the last five minutes of my talk, which as chairman is showing me that I have five minutes, or I reminded him that he has to show me, telling you what on-shell diagrams are, because I think they also hold some clues as to where the subject is going. So on-shell diagrams, so one can show that all plane and amplitudes at all loop orders are given by interactions of purely on-shell particles. So remember, Feynman diagrams use off-shell particles. Here, there is a construction that only uses on-shell particles. And therefore, our interactions take place. And here is a catch. They don't take place at null infinity exactly, but they take place in a complexified version of null infinity. But again, no need for interactions in size space time. Now, in n equals 4 superjam miles, everything is constructed out of two basic building blocks, which are three-point amplitudes, which people denote as a white vertex and a black vertex. And the reason you only have two is that you don't have higher derivative interactions like f cubed. You only have f squared in jammed miles, especially in n equals 4 superjam miles. And therefore, you only have interactions of the form plus, plus, minus, where plus and minus are the helicities of the particles, or minus, minus, plus. So when you supersymmetrize everything, you end up with only two kind of three-particle interactions for our particles. Of course, if these particles are all on-shell and they are all real, these amplitudes vanish. But here is where the complexified part enters. So you take one particle from the real space time, and you produce two complex particles, which are still on-shell, but their momentum has been complexified so they can go off in different directions and interact with other particles all happening in this complexification of the boundary of the space time. Now, in 2005, together with Brito, Feng, and Witten, we found these recursion relations that allow you to compute at three-level any scattering amplitude in terms of connections of on-shell particles. So you build a recursion relation which seems to be a loop recursion relation, but remember that all these particles are on-shell. These ones are different from these ones because these are complexified, so that's why I've drawn them with black lines, and the green lines are on-shell particles, but they are real. So these black lines are on-shell particles, but you have four of them, and you have to impose that they are on-shell, and that fixes in four dimensions completely their value of their momentum. So there is no loop integration to be done, and this is actually a three-level object. It's a rational function, and by building something over all possibilities of these diagrams, you build an amplitude out of smaller ones, and that's the sense in which this is a recursion relation. So once you're done with all the three-level ones, you can use them as the seeds for an all-loop recursion relation which was discovered in 2012 where you introduce this source term and now the L-loop amplitude is carrying in terms of amplitudes with less number of loops or less number of points, and by putting all these things together, you construct the whole S-metrics, excellent, you construct the whole S-metrics in terms of purely on-shell diagrams, okay? So here are my conclusions. I hope I've been able to convince you that S-metrices, not only relate states at null infinity, but seem to be described purely in terms of boundary data and boundary interactions, okay? So I've given you two precise examples, one in terms of the BMS group and the CHY representation and the other one in terms of on-shell diagrams. So I also hope I've been able to convince you that there seems to be a connection between the symmetries of null infinity and the CHY representation. Perhaps ambitwistory string ideas will make the connection clear. So in 2014, starting with the work of Mason and Skinner, people have been able to generalize the twistory string construction of Witten that only works in four spacetime dimensions and generalize it by replacing twistory space by another space called ambitwistory space. Ambitwistory space is something that can be defined in any number of spacetime dimensions and therefore you can also try and define a string theory living there and that is string theory leads to these formulas. So it's also a possibility that this construction might hold the key to how CHY representation is connected to the BMS group. And finally, I showed you that the connection of on-shell diagrams and yang-yang symmetry, which I didn't tell you, but the yang-yang symmetry is a non-local one. So the level one generators are generators that connect objects at different points in spacetime. So this is inherently a non-local symmetry, shows that these boundary descriptions are useful and perhaps fundamental, okay? So now the chairman shows me that there is zero minutes left. So I'm gonna close with the question, is there the possibility that a holographic S-metrics theory exists, which completely bypasses the existence of a spacetime? Okay, thank you. The first question actually is related to the question you already posed. Do you think that one can take the flat space limit of the standard ADCFT to arrive at what you have? That's an interesting question indeed. So people have tried, I mean, there is a very encouraging sign, which is all the reformulation of correlation functions in terms of melin in melin space. So people have been able to write correlation functions in ADS in terms of melin integrals, basically doing a melin transform. And once you do that, you start to see an incredible analogy between the integrants that appear in the melin space and some sort of Feynman-Dyton expansion. So you would expect that perhaps there is a way to take the flat space limit of those expressions and then recover some structure coming from the boundary. But I think it's a little too early to say that that's actually gonna work. So the melin parameters are the ones that play the role of the kinematic invariance. And one more question. In standard field here, we have this RG flow. Can you speak a little bit? The RG flow, the Wilsonian picture, how do you think that they would arise in your way of competing? Well, that's also an important problem. So in fact, there are many important problems. One is the RG flow. Another one is phenomena like spontaneous symmetry breaking. These are still phenomena that remain to be understood. Yeah. So what's the status of non-partibodic objects? They say black hole or solid ones in this approach? Yes, that's also a very important question. In fact, I was very happy to see that people were able to use some of these ideas in exploring something called classicalization where they try to see how gravity can become unitary. I mean, even three-level unitary can be restored by using these ideas of classicalizations where the black holes will emerge and then they will be, they will resolve unitary. And one way that people exploring classicalization has one way they have been able to explore the limit is by considering the scattering of two gravitons going into an infinite number of gravitons. So precisely these formulas have allowed people to explore that limit. And if you tune the kinematic invariance of the final particles correctly, you can see a behavior that seems to resemble something that you need in order to study those phenomena. So I think there is future in that. Yeah, okay, I will go back. But then I think it's the same as a non-potability thing like black hole and so on. Similar thing is that you think about QCD, it's a mass stress field and so on. But in the real QCD, so you have to have some mass stress field as a sympathetic state, right? Otherwise you think that's not the problem. So you can't use in the real QCD where it's just really non-potability effective producing mass scale and so on. And then of course, from the beginning it's outside you all. Well, there are connections. There are very, very surprising connections. You would say, well, saying n equals four super yam meals, of course, we don't have confining confinement there, but you would say in n equals four super yam meals, the natural observables, while in fact in any Yang-Mills theory, are the Wilson loops. You would say, well, but why aren't you computing Wilson loops? Why do you compute S-metrics elements saying n equals four super yam meals? That is a conformal theory, even worse, right? Well, it turns out that, what do you know? S-metrics elements in n equals four super yam meal all exactly do all two Wilson loops. So it's, I mean, that example shows that there are connections that sometimes are very unexpected and are just waiting to be discovered. The last one, quick. Since you kind of have an all loop expression, I wonder if you see something like researches, so where you, so I think it's a bit related to a previous question. So where's the non-perturbative part, where's the renumeration going into it, and normally you see something, or in this research, you see something going wrong and you can guess the non-perturbative part by making things right, so, but for you something looks right from the beginning, so I wonder where it's this. Yeah, that's a very exciting possibility that in fact perturbation theory can contain information about the non-perturbative completion of the theory. So the expression that I gave you in terms of the old loop formula is an old loop formula that construct the object in terms of these on-shell objects, but how to compute it is a different problem. So you get a formula that represents the object, as I said, purely in terms of on-shell data, but still some integrals have to be done in the end. So these integrals are still integrals of some phase space which is completely on-shell, but it has to be computed. So people are developing techniques for computing that, but if you mix that perhaps with integrability and the approaches that people have been using for integrability, then you could try and imagine that perhaps at some point we will have control over the series and then use all the developments of resurgence theory to get something, yeah, that's in fact, I mean, I should have mentioned that, but that's in fact one of the best indications that doing perturbation theory is actually a good way of proceeding. Okay, great, let's speak again.