 So let's start with the third lecture by Sasha. Welcome back. So I will start because there is so much already material that I'll just remind all the notations and what we have done. So we're discussing the universal infrared structures of gravitational theories and asymptotically flat spacetime. And let me recall some names. So this is a Penrose diagram of Minkowski space. It has now infinity where the light ends. It's called scry, scry plus and scry minus. They're labeled by retarded time or advanced time, which runs along this edge and point on the sphere, z and z bar. This point, z and z bar, if you look in the usual normal Minkowski coordinates, even though this is the same letter, in the normal coordinates, it's the same point, the same z and z bar on scry plus and scry minus. These are typidally matched. So if it's a north pole, that's a south pole. And that's a relational to usual coordinates. Then we started the question of scattering and looked at the structure of spacetime far away. So when distance goes to infinity, when we approach scry, have a natural 1 over r expansion. Here, the first line is the usual Minkowski space and retarded coordinates. Z and z bar is metric on a sphere. And then we introduce the corrections. There are, say, three of them. Let me remind you, this MB is called mass aspect. And remember, it was related to the mass of a spacetime and the loss of energy as the radiation goes through. So this CZZ does not have a name, but its derivative has a name, which is a new tensor. And it describes gravity waves at null infinity. And then we discussed the so-called angular momentum aspect. And by solving the constraints, we looked at Einstein equations, say, GUU or GUZ component. We found equations of this type. The first derivative of mass aspect and angular momentum aspects are fixed up to integration constants. So we end up with a picture that the data that we have to specify at null infinity is characterized by the new tensor and this three integration constants. Then we discussed a little bit that there is a matching. So this i plus plus and i plus minus, as you go to very late or very early times, on scri plus or scri minus. This is denoted i plus plus, i plus minus. And we discussed the fields here naturally matched, which was this antipodal matching. This is this equation. And then we wrote these charges, which are given in terms of this integration constant mass aspect and this angular momentum aspect. And these charges are labeled by arbitrary function in the sphere here, this FC and Z bar, or some vector field YA. And if we combine this antipodal matching with these charges, we got constraints on scattering data of the type that if we integrate something over scri plus, we get the formula of the type flux plus memory is equal integral over minus flux plus memory. And in the case of this mass aspect, the flux was a flux of energy. So this corresponds to the QF describes energy flux. And QY is related to angular momentum flux. And this memory is, in this case, it's a usual memory. And this is so-called spin memory. It doesn't matter. So this is this infinite set of conservation laws, which relate what you observe, or say this initial data, that you prepare what has to happen at scri plus. Now, yeah. No. Yeah, so let me repeat the question. The question is, if this data is enough to describe massive particles, it's the first part of the question, second part, if it's known how to describe them in general. So the answer is the first question is no. Here, I assume that there are no massive particles. So here, everything is massless. And so massive particles, which start their life here and here, I set them all to 0. So that's the answer to the first question. And regarding how to do, is it known how to describe them? Well, here, they will be encoded. The state of massive particles will be encoded in the values of mass aspect at this. So here, I recall I dropped boundary constants here. So all this, I assume the space time starts here in the trivial state and ends. And in general, if there are massive particles, this will be non-trivial. And I guess a more physical or better way also, one can choose a different gauge and different slicing of Minkowski space to sort of resolve this point, which is a bit singular in this coordinates. So this coordinates not suit very well. Now, having described the charges today, I wanted to talk about symmetries they correspond to and the relation to soft theorems. And then end with further results and comments, which will further results. So, and notice that the paths we are taking is the opposite of the historical one. Historically, this is what was known from the sixties and this is what is the recent development. But I chose a simpler or more straightforward path, which is we first define charges and now I will review this old subject of symmetries in a symmetrical Minkowski space. So if there are any questions about that, that's okay, there seem to be no questions. Yes, and another little comment is that today, some of you might have seen that lectures by Andy Stromanger appeared online and you can find there are many details of what I'm talking about. Actually, very nice and concise version of many of the formulas you can find in the sections two and three of their last paper was Hawking, whereas they just review all these things. Okay, so let me proceed now to the symmetries. The relevant notions that I will be discussing is known as a asymptotic symmetries or a asymptotic symmetry group. And the idea is that as we know that general relativity is invariant on the D.P. Amorphism, as we saw in the first lecture, there is a redundancy in the description of it, of the graviton field, and we can consider D.P. Amorphism, which are generated by some vector field. And then, if psi mu has some complex support, and it, we can imagine that it has some complex support and doesn't act non-trivial and asymptotic data. In this case, this is a trivial transformation. However, we can imagine also gauge transformations which do not die at infinity. And if they do not die at infinity, they can act non-trivial on the physical data of the theory. And this are non-trivial physical transformations. And the idea would be that the charges that we discuss are exactly of this type. A little bit more precisely what we are doing, what we are asking for is the following. As I described, this is a bondy gauge, so we fix the gauge. And moreover, this was by itself a trivial step. A little bit less trivial step is that following all the results, we figured out the follow-ups. By this, I mean, for example, that the fact that if you look at, if you look at the behavior of the metric at large R, then we have, this is simply rewriting of the same thing that it behaves, say, minus one plus four to one over R. And U R is minus one plus over R. G is equal to one. G Z Z is of order R. G R R equal to G R Z equal to zero. Now we can ask the following questions. Ask, find all the vector fields that generate if they are morphisms such that they preserve the bondy gauge and they preserve the follow-ups. This solution, this problem is now completely well-defined. You can take, recall that if you have some metric and you do it with morphisms, the metric transform the linear derivative and you get the formula of this type and coordinate system. So you can now, in principle, go plug everything I told you and find what are the vector fields. You might have heard the famous analysis of this type is that if you, this, the result will depend on which asymptotically approach. So the well-known result is that if you consider some spatial slides and you analyze the symmetries close to spatial infinity, as expected, you recover simply this vector fields that generate one correct transformations. Or even simpler in QED, you can repeat all the same words and you will recover the set of large-gauge transformations just to U1 and this corresponds to conservation of charge, which is asymptotic symmetry at spatial infinity. So the result, a very unexpected result of BMS in the 60s was that if you repeat the same analysis close to null infinity, suddenly you find, instead of just the usual one correct group, an infinite dimensional extension of it. And say, that was very unexpected and for many years it was not completely clear what is the use of this fact. So let me describe this infinite dimensional extension of the one correct group, namely this BMS symmetry. The idea is that if you consider the following vector field, so it is this, well, as you might already guess, this F is exactly the same F that we had in the definition of the charges. And these charges will be nothing but generator of super translations. If you take this vector field and do the computation, you'll find that you'll find that the follows are preserved and for any function of F. If F is equal to zero low harmonics, namely if it's a constant or powers in Z, this adjusts the usual translations. But in general, this can be arbitrary function. Now you can, by computing the literative, you can find how the fields transform under this transformation with the following result. Okay, sorry, I should have four of them. So there is a one here and I think maybe let me write it like that. So this is a one parameter. Then this is a complex number, so it's two. And there is, there should be one more here. I don't remember exactly what, so let me just write it C1 here like that. There might be a factor here, one minus Z bar but so there is one, two, three, four, because this is a complex conjugate, yeah? Well, since we are looking asymptotic at asymptotic symmetries, we only interested in the part which acts on trivially on the asymptotic data and I believe that all the subleading parts will just act trivially. So they was in equivalence class. We can add whatever we like. Okay, so let me repeat the question. How can we generalize Poincare symmetry to something larger if since we have a Coleman-Mandoula theorem? So let me remind you first what Coleman-Mandoula theorem is. The Coleman-Mandoula theorem asks the following question. Imagine you have a non-trivial theory with an S matrix. What are the maximal set of, maximal extension of the symmetry group? And the result that the maximal extension is just either Poincare or supersymmetry direct product with some internal symmetry group. Well, there are some technical assumptions but they are completely innocuous. We assume that say there is for each mass there is only finite number of particles with mass lower than the mass. So there is no accumulation point in the mass. And the answer to this question is that we haven't, in if this would be a symmetry of scattering amplitudes that would be a problem. But these generators actually, they do not annihilate the vacuum. So it's not a symmetry of the S matrix in a Coleman-Mandoula sense. So there is no problem. And the implication of this, well sometimes people say that, okay, this charges that we consider here, they do commute with S matrix but they do not annihilate the vacuum. And in Coleman-Mandoula I believe it was important to assume that the charges, they not only commute with S matrix but also annihilate the vacuum because the idea of the Coleman-Mandoula is basically that if you have a too much of a symmetry you can imagine you have two particles that scatter and imagine you discover some great symmetry which relates scattering like that which allows you to move wave packet. Then you can say that the scattering head to head is the same as scattering far away but scattering far away is zero. So scattering is trivial. But here what happens is that, and it was important that the symmetry generator relates two to two scattering, two to two scattering again. Here, since these generators do not annihilate the vacuum, the relation is different. It relates two to two scattering to a different scattering amplitude where you have a different number of particles which is that you have an insertion of a soft credit. That's the answer. Yeah, you can say that. That this, no, it's a little subtle, so I say this is spontaneously broken, but we, let me leave it that way even though you can probably complain. About this statement. Okay, now let me write how this transformation, how this transformation acts on the fields. You can compute the transformation of the fields. And it's something like that. It acts as a derivative. And here this maybe is the most, the only things that you should maybe remember from is most important is this. So remember that we discuss this asymptotic data in terms of NZZ and this integration constants. So if you start with Minkowski space where NZZ is zero and you do a super translation, NZZ stays zero, but you get a shift of this integration constant. So it moves you in this space of integration constant. Well, and there is some transformation of mass aspect. Well, again, it's maybe the important thing is that if you start with some Minkowski space and where mass aspect is zero, you still get zero. So if you act on the vacuum, you get a shift in this integration constant of CZZ. If you act on some scattering data, you get different physical scattering data. Things arrive at different times. So notice that if you imagine that the usual translation, the usual translation translate the whole space time here, we can translate independently along this time U at every angle independently. So it's sort of independent translation at every angle. That's the effect of this transformation. And we can expand F in this function F on the sphere in a set of harmonics. And this all this vector fields they commute with each other. So we get an abelian group of super translations and what in the sixties people found that there is this BMS group is a semi direct product of super translations times SL2C, I guess over Z2, which is just a Lawrence group. This is a usual Lawrence formations. Okay, so this is a simple extension. And as we will discuss a little bit, then what happens is that now we have a asymptotic symmetry, which is a large gauge transformation which acts on trivial and asymptotic data while preserving all the fall-offs. Then if we formulate our theory canonically, so if we try to quantize it using usual language of phase space, symplectic structure, et cetera, there will be associated charge which acts while acting on the fields through the direct bracket generates this transformation. And the result is that this charge QF that we discussed, it exactly does the job. Yes, that's the next subject. So this we discuss now super translations. They are, they are, so they generate SL2C. So they extension of SL2C. So this is a normal BMS and this wise it's an extended BMS and this wise it's what's called super rotations. That's what I would like to discuss next. So this is what is called super translations. So you can think of it as this angle dependent time translation, it acts on the field the following way. In the following way, most important is there is a shift in integration or maybe not most important, but there is a shift in integration constant quite curiously. And again, we will further see that this, the word identities due to this charges corresponds to Weimberg's soft theory. Yeah, I think there is a reason way and well, people did it, I think, even some, but I don't quite know what would be the invariant meaning of that, so. But in principle, I would imagine that one can try to do that. Yes, there was this papers by Joffrey Compair. They were discussing and they were discussing that if you try to extend this transformation to the whole space time, there are some defects, but I think the story is not completely settled, I would say, maybe you wouldn't agree. That's my impression that yes, we can try to extend them, but it's not completely clear what is the meaning of that. Even though I guess in the context of semi-classical space time when we have a black hole or something, that one can try to do that and discuss what does it affect. Any other questions? Okay, these were super translations and now as, yes, in the canonical formalism, this super translation will be generated by the Dirac bracket with this charge QF that we discussed. Or in quantum theory by a commutator with this QF. That's the relation to the previous discussion. And the other charges, they are a little bit more exotic and much more subtle, which are related to this YA's. And similarly to this story, let me maybe start here. You can consider again another vector field which takes this time the following form. One plus U over two R, YZ, DZ. Plus, not very illuminating probably, but the only intuitive connection is that that's how boosts and rotation look at null infinity. If you rewrite boosts, well, there are other terms, but let me spare you from this. That you can look at them up in many places. Well, so it acts on the sphere. And the interesting thing is that, so remember the rules of the game is that we have a bondage and we have a fall-offs. And if you do this exercise, remember we have this ZZ bar component, which is R square. And this is fixed by the asymptotic structure. We don't want to change that, say. But if you do this transformation, you find that the ZZ bar component translates like that. ZZ bar, DZ Z bar. It's very simple. You see, the important fact is that it's R square, and we have a DZ over YZ bar. So this is something that we would like to set to zero for to satisfy these conditions that we preserve the fall-off. And that's what BMS did. So as you can imagine, that you can have any holomorphic vector field here. But what BMS also did is that they said, okay, we would like it to be regular on the sphere. So if you compute the norm of it, you don't want it to blow up. And if you write down all the vector fields, that's this type, you find exactly six of them, which is, let's trivial a little bit. And these are just rotations in the boosts. Now in 2009, in the paper by Barnitian Tuasar, of course, it's inspired by this fact that in two DCFTs, we also have a holomorphic transformations. And we know that it's extremely powerful and they completely govern the quantum theory. Everything transformed in the representation of the erasora. We have word identities, basically. It's a signature of the quantum theory. Let's say, okay, why don't we think locally and just find the solutions, all the solutions of this equation locally. And now you see that you can take any holomorphic function you like. That's locally satisfies this equation. And they say, why don't we, well, actually, I don't know if they say that, but the idea was that maybe we should take it seriously and maybe there are some trivial implications. And even though probably after some time this, it's not exactly the way you should think about this, but it happens that this is right and this charges, this charge is labeled by why, or exactly the charges that generate this symmetry with holomorphic y. So that's the interpretation of the other charges. Now, let me a little bit elaborate, write similar formulas as I wrote here. How the field transform under super rotations. And let me write only one formula. But again, some things which you might forget, but there is a piece which you can probably keep your memory so there is a transformation of this type. And most importantly, there is this homogeneous space. So there is this importantly third derivative of the y. And well, there are a couple of exciting things. I guess in certain coordinates you can rewrite it and it looks like transformation of a 2D stress tensor. And people were wondering what it means. And recently there were several papers where people tried to construct out of this news object which transforms like stress tensor and whose insertions look like what it is 2D stress tensor but there are many issues with that or some issues with it. We can maybe discuss it during the discussion session. So important thing is that if you take the yz to be just the Lorentz transformation, this third derivative is zero. And if you start with a vacuum where nz is zero, you end up with a vacuum. On the other hand, if you start with yz which is part of a super rotation, then you remember something that was important that we were saying that the phase space that we're considering is such that nzz is some function of u but it falls at large and early times as a function of u so that we can integrate it. And the limit of czz, remember czz was the integral of du of nzz, it exists. But here we see that if we start with a vacuum we transform nzz becomes a constant. And if you integrate it with u, czz blows up. So this transformation takes us away from the phase space. In other words, simple exercise to see what happens is that you can take, if you take a cosmic string which is a solution of Einstein gravity with a defect, it is a topological defect. You have Minkowski space. And the only difference is that you have, if you go around this cosmic string, you have to identify phi plus two pi say minus some delta. There is a, we cut off the slit of a spacetime. When you compute, you can take a metric of a cosmic string and put it in a bondage. And you will find that the newest tensor is holomorphic as here, but it is singular. So it has this kind of form, one over z square. This is an example of a spacetime which if you wish is super rotated and it indeed it's not asymptotically flat spacetime. It's what is called asymptotically locally flat spacetime. So if you look, if you go everywhere on the sphere it looks like Minkowski space, but there is a global effect. Well, super rotations, they have this feature that they act not in the space of asymptotically flat spacetime, but in some extended space and which presumably, but this was not worked out or anything. So at the moment, this is the way you think about the super rotations. We study, we have the charges and then we study the word identities of these charges and they do correspond to this so-called subliding soft theorem. That's, I think that's status at the moment. Moreover, notice that why another comment is that again as we're discussing that there is a set of charges and they correspond to soft theorems and now we hear, we discuss the super rotations and then there will be a corresponding soft theorem. One of the, as we discussed the thing about this triangle you learned some connections and you can try to extend it and what people realized quickly that this subliding soft theorem, it actually exists in any number of dimensions. It does not only exist in four dimensions but in any number of dimensions. Whereas this story was allomorphic transformations of the sphere, they only exist in four dimensions and I'm telling you about this super rotations and how they were discovered. But in higher dimensions, clearly we don't have a two-sphere, we don't have this homomorphic transformation. I think there was a paper by Campiglia and Lada who identifies the symmetries in high dimensions as some smooth-deferred morphos of the conformal sphere. But somehow this connection to two-dimensional, conformal transformation on two-dimensional sphere and two-dimensional conformal field theory seems to be an important inspiration for the subject even though it's not, as you see, there are many maybe questions about that. Now I would like to switch to, so this was a description of symmetries and we discussed the charges and now I would like to quickly describe how to think about soft theorems and write them down and discuss and make further comments. Are there any questions at the moment? Yep. Yeah, so let me repeat the question, it's a very good question. How do we, I wrote an action of a charge as some post on bracket, how do we define a post on bracket? So this is a very long subject, there's many, many papers, many experts and many subtleties, but the idea is that we constructed a phase space and now we should find a symplectic structure and I believe the first it was done in the paper by Tzernkowicz in Witten in 1986 and then Bolt had developed it. So the idea is that there is more or less a canonical way to find a symplectic structure and then you can just follow the set of steps which is pretty standard and find this action. The post on bracket will be controlled by simple, if you construct symplectic structure, you can compute the action. The subtle part about this, what I'm saying is not quite, it's not quite, I think, fair because here the whole, the subtleties of the subject comes from this sort of soft modes and originally when here I write as a phase space this NZZ and this integration constants and I think in what people did until recently this integration constants were not included as a part of the phase space but they're responsible for this soft part of the action of the charge and again from talking to people, my impression that at the moment the way people extend this phase space including the soft mode is basically driven by the soft theorem intuition. So we derive the soft theorem, so we know how things should act and then you take this, your symplectic structure and the way it acts on the fields that say non-zero frequencies and you assume certain continuity things but there is no, I think, principle or theory of how to do that, people just did that and write down more or less this bracket for which generates correct results. I don't know, there should be a much deeper mathematical theory and understanding of that. Well, so we used some of the Einstein equations to identify what is the data. Yes, definitely, sorry, yeah, yeah, yeah, yeah, definitely to construct the symplectic structure you use Einstein equations, yeah, yeah, sorry. Maybe that was your question. Yeah, that would be the idea but of course we don't know how to do it nonperturbatively, yeah. Yes, well, I think perturbatively actually there is, well, it's all sort of, that would be the ideal, yes. So this fantasy of field theory of living at infinity it requires I would say two main ingredients which is locality and unitarity and I think in this story it's pretty clear that unitarity is gone. Whatever 2D CFT it is, it's definitely not unitary and locality maybe there is some hope but even this is very, it's very exotic if you look at it. Ah, yes, very good. Maybe, because so far I haven't written a correlator of which looks like a relate of 2D CFT I was going to do it in next but let's say we managed to write this correlator which is a scattering amplitude in this case and then we compute that and then unitary theory should be decomposable and representations which satisfy unitarity bounds. What people did, they computed the things and what you get is a so-called principle series or completely different representations which do not look like the usual CFT, moreover they continuous, it's not a discrete spectrum. There is an also issue with singularities but when I mean non-unitary, I mean that if we take this as a correlation function on the 2D CFT and we look at its representations that propagates there, not the usual representation, the spectrums continuous, they have complex dimensions. Now let me relate this to something very simple and to this soft theorems. And the basic idea is let me, I like this little fact which is not, at least I haven't seen it in the QFT books which is very simple and useful in this story which let me call it LSE, LSE in position space. Recall that the usual, what is called usually LSE is the way how you start with correlation functions and then you get scattering amplitudes by looking at the residues of the poles. And in this case, we work solely in the coordinate space, it would be useful to make a connection with the scattering amplitude to do something like LSE in the coordinate space. And the way that you can do it is very simple. I will do it in two lines for free fields and then this whole subject works along the same line. So imagine you have a Taylor field. I, pi's are equal to correct coefficients to make this formulas work. I set them all to one or two or something. Then this is a usual mode expansion of field. And A here, they satisfy, let me, I can absorb this factor into the definitions of A. And then we have a usual commutation relation which is, I don't remember if there is this and the three, this is simply the free field, quantization of free field. And now the idea is to take the free field and drag it to null infinity. We consider the following limit, take a limit, are going to infinity, integral over the retarded time with some energy. And we consider the insertion at some retarded time, distance with some unit vector, let's say. The idea is I took this field and inserted, I'm inserting it far away and reading off the leading asymptotics in R and after integrating it with some wave function. Well, you can plug this formula in this expression and it's very easy to see. The picture is that we are taking the field, dragging it to null infinity and then integrating over time with a wave function which has simply e to the i omega u. And if you do this computation, you'll find that this becomes a creation operator. Let me write it as the absolute value of K. This energy, this creates, projects itself out into this creation operator at future null infinity. Again, what I did is I took a field and it takes a limit and now if you plug this here, you get this projection. And the direction of the momentum is encoded into where we drag the field in which direction and the energy is encoded in this wave function. Now we can take a correlation function. We can take a correlation function and start dragging fields to infinity and integrating them with this wave function. And in this way, we will be able to turn the correlation function on the scattering amplitude. In this way, you can think of scattering amplitudes as insertions at null infinity and moreover, you can show that they transform if you choose the proper factor. So derivatives that transform like primary fields of two-dimensional CFT. And you can repeat the same thing for the metric and take it to null infinity and you will get this kind of relations. For example, a new tensor at some frequency omega and z and z bar is related to the energy of the photon. Times some simple kinematic factor and let me write it like that, absolute value of omega and if omega is positive, it creates a photon in the direction z, z bar or gravity in the direction z, z bar with energy omega and if omega is negative, when we do a Fourier transform, it's a dagger again with omega z, z bar. And now we can take, well, this is a LSZ prescriptions that we discussed here for scalar field. Now we can apply it to gravitan. In this way, you can immediately, if you recall the structure of the charge was that it has a piece which is linear in M, ZZ or in CZZ, this corresponds to creation of a soft photon. Maybe I should once again recall you as a roughly as a structure, for example, of QF, it has this DU and here there was a piece linear in NZZ and then there was a piece quadratic in NZZ. Now you can plug this in expressions and you will find that this piece creates a particle on a LHC particle and this particle low frequency because we integrate against DU without any way function. And this, okay, let me proceed and see what happens. I restarted it, we'll see. And this piece, which is quadratic in NZZ, it is a hard piece. You can see that it measures the flux of angular momentum or energy in the given direction. So that's it. In this way, in the very pedestrian way, you can derive the action of these charges. You can check after plug in this A and A daggers that when you act on the fields, they transform properly. Now, let me, in the last five minutes, yes? Okay, it works. Let's see. I can proceed without it. It's, I can, ah, okay. Then I cannot use the blackboard. I can use one, yeah. Oh no, it's the same, apparently. Oh, maybe it was that. Okay, great. So far it works. Okay. Soft theorems. Now we can insert these charges into some commutator with the S matrix, which was zero, if you remember, and see what we get. So we consider some out state commutator S and Q. In state, and this should be zero. And we get first what is known as a Weinberg soft theorem, which is, I already wrote it. Well, it's a statement that if you take a limit of scattering amplitude, as n plus one particle, and you take the energy of one particle to zero, if some, some graviton, some momenta, you get some i, polarization of this graviton, Pi mu, Pi nu, Q, well, let me take this Q. Q is a, omega is the absolute value of Q. So it's a momentum, four momentum Q mu. And times an amplitude of, with n particles, well, that's all known. One nice peculiar formula, peculiar simple fact about this formula is that if you do a gauge transformation, which has this, the invariance, you can plug this formula and we see that for it to be gauge invariant, we need the energy momentum conservation. i should be equal to zero. And for the subleading soft theorem, this is something very concrete and very new, which came out, I guess, of this analysis because people didn't know, even though there was some works, but I think it's definitely people didn't know about the subleading soft theorem, which takes the following form, yeah. It takes the same amplitude, but now it takes a limit, omega d omega. Notice that this operator projects out one over omega piece, which is the divergence here. If you act with this on this, it's zero. Of the amplitude is equal to the following expression. Time over i, epsilon. And this j is, now it's an operator, which acts on a lower dimensional amplitude and it just measures the total angular momentum of a particle. So it is a orbital momentum plus a helicity part. Again, if you do the gauge transformation on this object, you will find that the consistency, its gauge invariance require that the total angular momentum is conserved. This is what is known as a subleading soft theorem. For gravitons. Now let me have to amount of time. And that's a time to finish with some comments. First comment is the following. Imagine we consider some effective field theory. And well, as we imagine, we have Einstein theory plus some high derivative corrections. And you can add matter with all kind of interactions. For example, we can consider r mu nu rho sigma f mu rho sigma, or we can consider phi Riemann square, et cetera. You can do the analysis similar to the soft theorems for most generic effective field theory. And it was done by Henriette Elving and her collaborators, which I don't remember the names. And you can show that this leading and subleading theorem, they're completely intact. So they're completely universal for any theory. First result, whereas if you go to further orders in one over omega, say sub-subleading theorem, this gets corrected from this high derivative terms. This is in accord with the universality structure of the symmetries that we discussed, that they should be, this is a picture that should be valid in any theory. Second is if you consider quantum corrections, then you find that, say, leading soft theorem is always stays the same, the leading soft theorem is not corrected, is not corrected. But you see that the subleading theorem soft theorem is one loop, is corrected only at one loop and sub-subleading is corrected only up to two loops. Now these corrections due to infrared divergence is singular and there is a whole different discussion about how to think about this, how to discuss infrared finance as metrics and what is the meaning of these extra charges, which are sort of anomalous, how to think about them, et cetera, et cetera. Yes, and I guess there are other directions which maybe let me stop here. Thanks.