 Look, okay. See, okay. Today, yes, I will switch a little bit to gear with respect to the other days. Trying to introduce a geometrical way to actually describe physical processes, specifically for the way function that we have been discussing in this in this day. I decided to divide into parts because it's probably the part of the topic which is least less familiar and if I was have been slow for the previous lectures probably will be extra slow for these ones because yeah I think it's good. I prefer that the message goes through then just provide a bunch of information that tomorrow you already forget forgot about them. So before doing this, in any case, let's do a small recap of the lectures of of yesterday. So what were the main messages. The first message was that the there is a correspondence between the singularities of the way function. And the subgraph of the graph associated with sign what what what what what this meant this meant that if we take this this process where again from the vacuum we are getting on the boundary for states. Okay. We can already know what what are the singularities by just by looking at this graph by by send it by associating the some of the energies of the sub process associated to each piece of the sub what does what does it mean this means that for example, if I take this graph. The sub graph of a graph is there the graph itself, which is this one, and I will associate the energy of this process which is in this case the total energy so we know that there is this singularity, then another sub process. That's given by what's happened here at the vertex, and then you associate the energy at this vertex the energy of this vertex is given by the sum of the external and this and the energy of the internal state. So if as usual I indicate this with why we know that this is a singularity. And finally, the same at this other vertex. So we know that this, this process will have a single artist up to this for for points. No three points or. Now, another thing that we learn is that we can actually. map this graph to what I call reduced sub radius graph, because what we did yesterday was actually mapping the way function in a generic effort of cosmology. So again, the contribution to the graph G to a function, the contribution to graph G to a function in an arbitrary effort of the cosmology, what we say that this can be seen as an integral over the external energies at each vertex. If this is the set of the old vertices. With the measure, which depends on the cosmology of the plot space with function. And therefore, which depends on the energies at a vertex and the internal energies which associated to the edge. So this is because when we do the computation, you realize that the energies at the external energies at a vertex enters enter always in the combination of their sum. In this case, it's like saying that I indicate a one plus it soon as let's say x one e three plus e four and x two. And so really conditional x one plus x zero x one. y plus x zero. And so, rather than drawing all this graph, one thing you can do is just erase the external the back to boundary legs and just having this graph where now the label y is associated to the vertex and the label x one and x two are associated to the to the to the sites. Okay, so the, and so, and this is the third point. The general idea is that you can exploit all the structure of the flat space with function, which, given that now access an integral, sometimes I call universal integral integral. And then you decide what your course, what is your cosmology by fixing what is this measure, and you see how the, this part of the integration interacts with the singularities and the structure of this, of this, of this integrant. The general statement is that there is a number of feature of the way function arbitrary for the cosmologists, which are inherited by this from the structure of this universal of this universal integral. And then, for example, for this case. So, doing the computation. This, this, this graph will give the simple answer for this universal integral. And so if you want to extract, if you want to compute the way functions to say to some, I forgot the cosmology what one has to do is to do the integral from big big x which are the actual external energies. Now, if I one thing that. So this, if you remember what's coming from the fact that we were modeling. And so this is actually in this precise form was true for conformal couple scale. The cosmologists. And, and then we were modeling this through an action which is was a flat space a master space action with a time dependent coupling where so it is just some field definition in which you change the you change the position of the work factor of your metric from being in front of the kinetic term in being just encoded into the interaction. So what you had is that this time dependent coupling. Pretty much can have this form, but this is the work factor. And if you choose. And so what we were doing to obtain this form was just taking an integral representation. Okay, and so doing some change of some some shifting in the in the integration variables to get to this form. Now, if we choose a work factor. Of this form. This lambda tilde of Z with here, let me put a eta. Minus eta, but it's a minus eta because in any case this it runs just from minus infinity zero. Then you get to that this this measure will have the form of okay up to some number. Z to some power alpha minus one eta Z where this alpha is just a minus minus minus one. Okay. So if you think, so if actually I substitute this form here, and I shift again the variable of integration such a way that I get your my integral from zero to plus infinity. You get the form of integral that you saw in my Pimentel lecture. So you get that. Okay, so the origin of those integrals that you saw in the German stock is precisely this idea that you can write for conformal couple scalar, your way function in an in an arbitrary for the cosmology as an integral over measure which can specify which cosmology you have an oven integrand, which is the fat space of the fat space with function. Now this picture is a little bit more general. Because, actually, the, for example, other states scalar states, like massless or other light states are related through differential operators to the conformal couple. So, I won't go into details of that here, but then you can take this this this formulas apply some differential operator and get the answer also for other states in this in this sense even if we started with a very specific type of toy model, you can get a larger number of information about a larger class of time order. The last thing I want to specify is that therefore if you look at this type of formula, what we know what you know is that this parameter alpha, which come from this free from this integral representation, really is encoding cosmology is encoded the number of dimension, the special dimension, and it's encoding also the number, the, the, the type of interaction, because this guy is telling you that you have a fight to the k interaction so this was for polynomial for interaction. Now, if you don't have polynomial interaction, if you have derivatives, which just are special derivative, everything goes through you are gonna just gonna get a numerator, which depends on this derivative, but the single structure will be completely unchanged so this formula will be the same with an extra numerator. If instead that you have also time derivative things a little bit different it's the rules change a little bit and the expression is a little bit more complicated. Okay, but yes. That's correct. That's correct. I typically draw that like a five cube because it's, it's easier but in a sense, the good thing is that you know that writing in this parameterization. This x one x two is just related to the sum of the external energies at the vertex so even here you can put a five cube and if I for it doesn't matter the structure is not going to change. In this sense, with just a type of formula you can get information about different type of interaction and so you can draw more general conclusion. So, therefore this reason this is universal in in many senses is universal because the integrand doesn't depend on the cosmology you have to introduce dependence of cosmology through the through the measure is universal because it's show manifestly that the structure is the same and no matter what is the number of points of interaction and no matter what is the number of special dimensions. Okay. And so the final thing that we learned last yesterday was that precisely. If you remember when we computed explicitly for the first time this formula was using the explicit expression of the buck to bulk propagator which is a true term expression you are getting three terms with a single argument and you could you will get in this one term expression. What thing the last thing that we learned yesterday was that precisely keeping in mind the fact that there is an association between singularity and and subgraphs. You could we proved a recursion relation for which if you multiply the total energy of the process for your way function. Then this is just given by a racing and the edge. Okay, and associated to each vertex the energy of the vertex itself by shifted by the energy of the internal state. So for a more general graph, for example, for an higher point, higher vertex graph. This will be given by iteratively erase one edge at a time. So this is one is to this one extreme. So here I raised this edge. Now, you have to some the other term that you will get by racing the other edge. And this is actually true also at loops and obviously again that in the case of loops is going to give you the integrant of the loops in the sense that you don't have just this integration here you have to think that also you have to do the loop integration. Okay. And so this also could be mapped in the in a graphical rules, which was that you will take you take the graph this reduce graph. And then you interactively divided in connected the subgraphs and associated to the inverse of the sum of the energy of the subgraphs, for example, let's do this, this example again. Now the first hub graph is the graph itself. The external energy is some of the energies for that. Now this can be divided into subgraphs just in a way. And I associate again, x one plus y. And finally, the other one. And then you get on the spot. The final answer for this one again you will have to do for this other. For these other graphs, you have to do the same thing. You take the big graph, and you write one over x one plus x two plus extreme. Then now I have two ways in which I can divide these into connected subgraphs. So I have to some about on these two ways. Now, if I take this first subgraph, I do the same job x one plus x two plus y two three. And then I keep the value in this. In this other two. And you get one over x one plus one one two, and one over one one two plus x two plus one two three. You do the same for the other option that you have. I'm going to write it directly. So with this rule, no matter how complicated the graph is, you can write on the spot. This, what is the disintegrated by just applying these rules recursively. So it will rise a very general way to actually write this and this is, if you notice this, this is just has a physical pulse in the sense that you see do not minus is it really have only pulse which are associated with the subgraphs. So for the way that, and this actually turns out to correspond to what people usually called old fashioned perturbation theory, or time order perturbation theory in, in normal in, in a fat space to call it this way. Okay. So, now, the message, instead of the left. So, okay, before going ahead. Is there is any question about this. Well, sometimes they are can be good methods for of computation in case what one of the nice thing of. For example, this way of computing to with respect to of using explicitly the buck to buck propagator is the fact that you don't never get spruce poles. You don't have to do time intervals. Once you have a general proof that this are resources structure exist. You can just apply it, and you don't even have to know that behind that there is another way of computing them is doing the time integration. This way, yesterday I was very I was referring to this type of computation as a sort of boundary computation because whatever need to know it's like the spectrum of the theory and the energies because this is why, even if we call it external internal energy, what is really is is the sum of all the momentum which come from for my vertex. Okay. So really, these are all external data. So you don't ever have to talk about anything that is happen inside. This is true for loops. Yeah, so you can compute for example the bubble. In any way. So, these rules will give you also a two term expression. What if you have if you will do the fine man way of computing it will get a nine term expression with some spruce poles. Okay. If you compute to you, if you compute using the five man, the family rules, if this will give you nine terms. And also this, because you get three term for each, each, each edge. Okay. So three to two is nine. Okay. Yes. The goal. This recursive structure this release hiding some combinatorics, which is actually what will be encoded in this geometrical approach that I'm going to describe now. Well, if there is any other question and I think this is yes. When you say more complicated refer to having derivatives. Yeah, so if you have because in cosmology you can have separately time derivatives and special derivative so if you have just special derivatives, everything goes through the same. The only difference is going to be that you're going to get some numerator which come from the special derivatives. The rules change a little bit because if you were to do the family rules, you also have to consider that the derivatives, the time derivative act also on the, on the propagator. And the point is that when you have, when you have to the derivative of the propagator that have a function that's not in some delta delta function and so you get also some boundary contributions that here you don't have. So, the rules change a little bit. But what does not change is the. The single artist structure. So one thing that you can think to do is some sort of as a new Batman type of approach where you use this type of graph as basis. And going to the single artist and knowing what are the amplitudes associated to the lower point processes, you can fix the numerators, which is actually much more effective than having to do all this, the actual modified rules. So in this case, given that I am you have also this integral, you can even do this pass it in always come and reduction at three level. Not just the loops. Okay, so. So, now, the starting point for figuring out that there is some interesting mathematical structure behind this way functions, and actually you can even use it as a first principle definition for this object is precisely the fact is the fact that we can describe these processes through this use graphs. Let's start from here. Now one thing that we can do is thinking that these labels x1 x2 or y are homogeneous coordinates in projection space in this case it's it's so. So the way to think is that you have vectors in our tree and you intersect them with a plane where some figure leave if you have some some vectors and so any point along this way is really equivalent. In this case, this space is just our tree made out, it's always just made out of our three vertices, for example, if Z is an R3 vector, but then you require that all the points along this ray are equivalent, which means that this ratio has to hold you really have one degrees of freedom left so this condition will tell you that this vector Z will will be inside the it will be will be projective will be in between. Okay, so I guess that this is everything that you have to remember if you're not familiar of projective geometry if you don't are not familiar with it. So everything or this point are are equivalent so there is a class of equivalence of vertices of points, which mean that whatever expression you get to better be to make some sense invariant under this risk. Which is what in a comment that I made during Julius talk is called the Jill one in violence. Okay, so let's go back here so let's say that. So I take this graph, I declare that this variables here are homogeneous coordinate given the three are going to be a homogeneous coordinate in pizza. And so one thing that we we observed is that this graph as subgraphs to which we can associate some conditions which were x one plus x two equals zero x one plus y equals zero and y plus x two equals zero. Okay. If we picture this on in pizza, these are just lines, for example, this can be the line x one plus x zero. This can be the line x one plus y equals zero. And this is the line x one plus y equals zero. If we choose some orientation, let's say, let me call it is vertex one vertex to this intersection, but extreme and I choose this orientation for this line. And this is will be the positive half plain. Identify by this line this is negative post is negative post is negative. So you'll see that the condition, the positive so that these three lines, identify some area. Okay. So in this case they identify a triangle. Identify a triangle. Now, as Julia in his talk mentioned described yesterday. Well, also triangle is a polytope and to any polytope, you can associate a differential form. Okay. Identify by a condition that it, it has the nominators which correspond to the boundaries in this case, x one to x one plus y, y plus x two. And in principle, whether your numerator is a number of is a function is also determined by the gel. So this measure written explicitly will be just the x one, the y, the x two. This is the volume of gel one. This precisely tells you that you need to fix these are homogeneous coordinate. Okay. And then you see the as I said before this has to be invariant under this, this rescaling which will mean under scaling all the access and wise together. But you see that here we have three monomials, you have the measure if it has also made by the product of three stuff. So here you know that there is this number should be here. Okay. So this one. One thing that you recognize now is that precisely this, this rational function that appear in front of what's typically called canonical function is precisely the way function associated to this, this graph. So the general statement is that for each graph G, there exists a unique, there is a resist a call it up. Such that he is a canonical form as the canonical function, which is precisely the way function associated with us. Okay. You map the problem of computing and studying the way your way function into the problem or computing and analyzing this clinical function or canonical form. Now, just to be clear. Just mean that you're taking the determinants of these three vectors. So what you're saying is that you have some extra IJK, YI, DYJ, DYK. Okay. Where these are the indices that tells you that the example in this case you are a vector with three components. So this formula in a complete invariant way, a gl y invariant way can be written in this form where this one two and three are the vectors associated to this to the intersection among these these lines, which you can compute them to be Z1, 1 minus 1, 1, Z2, 1, 1 minus 1, and Z3 minus 1, 1, 1. And the number here I wrote this way is again the determinant of these three vertices. So it's an actual number. And you need the precise number because I said before, you want that if I rescale for example Z1 by some lambda. You want the full form to be invariant. So here I have two power of one. I need two power of one at the numerator. The same for two, the same for three. And so this is unique things that you can write. Now as somebody commented during Julia's talk in principle, this form is fixed up to a sign here. But the sign is just given by the orientation. Okay, you decide your intention and you fix the sign, but this is an arbitrary choice. Okay. The orientation means the arrows here. Yes. Well, you might decide to choose some presentation but then if something is positive becomes negative so it will change the positive condition, it's better to choose some something which is oriented surfaces so that you don't have this problem and everything boils down to just plain positive conditions. So now, one thing that you understand from this idea is that if I take a bigger rough. Okay. Also, the dimensionality of the space where the object that gives you the way function lives, it's bigger. So here I could draw a triangle, but in general you cannot do the drawing so one is to learn how to study these objects without having to draw them. I mean in triangle triangle that is nothing to study. Okay, so one general things that you can regress is the fact that, given that my space where this object live has homogeneous coordinates which are given by the the labels associated with the graph. Then if I have some arbitrary graph, the homogeneous coordinates in the space that you can associate to the space is just the least of all the access. The list of all the wise. Okay, which means that the point up will live in a project space. So as I mentioned number of of vertices of the graph, plus number of the edge of the graph, minus one which is the projective condition. Okay, so they are there are vectors in an s plus any, but then they are projective so the project space and s plus any minus one. Okay. Now, before I'm going to how, for example, yes. I would say it's in R and s plus. No, you seem imagine that you have that when you have homogeneous coordinates. Even if you want to think in terms of our homogeneous coordinates means that you have your expression, which is written in terms of some polynomial. Okay, of some degree delta. But then the polynomial degree delta is all the time all the terms as products of all the variables you don't never have different type of powers for example, you can have when you have living in a polynomial you have all the time something like x one plus y plus x two, for example, but you never have something like x one plus one. Okay, so when when you are in homogeneous coordinate, even if you want to think about real space, you're implicitly in in projective space. Okay. You can always fix you if you are in homogeneous coordinate, you can always make some some rescaling is going to have a precise behavior in the scale here if you do some rescaling. Each of the thing behave like, like, like one wants. Okay. And now in this case, the fact that you have a gel one invariant because in principle you have some some sort of covariance you can do the rescaling you can have some, for example, if I will have had some power square. If I have, if I do this type of. Of the of the one transformation, you will get some lambda in front here. And in that case, the problem is that the forms are not uniquely uniquely, uniquely defined. So you this is so the request of gel one invariance in general is the something which is well defined and implicitly from a fit most physical perspective that implies that your object as only logarithmic singularities. Okay. So if you have something like one over x square logarithm singularity means that if I do some change of variable, I can rewrite this thing has the Z one over Z one, the Z two over the two, the Z three over the three. Okay. So, if you that you have just simple pulse when you have simple portion you put a measure if you do some whatever is getting these things in this invariance so in a sense the fact that you think in projective space and think becomes immediately nicer. Is because secretly what the type of structure you look at it as just logarithmic singularities. But going back to the previous session or. Yes, where does this. No. No, no, from the physical point of view, nowhere in the sense that the, for example, you see here, I have my space here my variables here are x one wise next to if you think about the for example loops. Even if you think about this formula right before with this measure. Obviously when I put the measure which is given by some choice of the cosmology. This jelly one invariance is done so here, the claim is not that all this give you also the correct measure or which you have to integrate the claim is that this picture will allow to compute this. The integrand. And then, when you want to actually do the integration, you have to put some adequate measure which is going to break the joy and why and so physically the full. The full FRW cosmology type of a function doesn't have this this property, but you discover that the integrand, if you attach to it, the measure of projective space, it does. So we use this to actually understand the integrand, and then you can actually do the. Now, this is something that I was planning, I will try to say tomorrow if if I have time, but even so there is for certain type of cosmology for example if I choose to do to extract the function for the sitter in one plus three dimension with the five cube interaction. Now, with respect to this access. This has still some logarithmic singularity so you can use this point of picture, even to do the integration. Okay. So in some cases, for which this picture is useful is useful not only to compute the integrand, but also to compute the integrand, the integral. Okay. Yes. This is. That's correct, yes. No, no, but this is not the actually just one symmetry doesn't have anything this gel one symmetry doesn't have to do with the sitter because you have it even for flat space with function flat space so you don't have to do that. Yeah, but I think not. Yes, no, no, no. There made your sense. Yes. No, I agree but what I'm saying is that if I look just to this guy doesn't have to do with the conformal doesn't have to be conformal symmetric doesn't have to doesn't have to have any of these properties. And also when you do embedded space, really, at least the way that you understand it to you typically do it in a position space so you uplift, and this is everything in energy space. That's correct. No, but it just something that you realize that you can have precisely when when you have. I'm not embedding anywhere actually I'm, I'm, I'm just using the very same number of degrees of field I'm not going to hire a higher dimensional space I see is used to do for doing that in many space. So, you know, so you see the fact that you can use this trick is due to. So, when you have something which has just simple pool. You can, you can hope that I mean typically okay so let me go on what's the back. One thing that in we noticed is that if first this thing or written in this energy space. It's written all the time in terms of homogeneous polynomial. We said that okay what's happened if we are gonna you organize this variables of this of this polynomials into a vector to give me some some space. So you say that you have these three and then if you attach this matter this this this metric which is the typical metric of to realize that this is jelly one invariant so we say okay let's use this this this factor, and then we are. And actually even backwards. Once you realize that you have some something like this, then the natural measure to put the natural way to describe this point of precisely through through project space so because here. This are not telling you about distances about vertices just telling you about. Lines, in this case, in terms of homogeneous coordinates, okay, and that's it's enough to think that. And then it's enough and then and then as Julia pointed out before you can say that okay, I can realize that if I put rather than equals zero some positive condition. They define some in this case some convex object from some positive some positive object, and then I know that to this positive object, which is described in this homogeneous coordinate. I can nicely describe it in project space and attaching this this this form so then you say okay but this is not the form on which you are going to integrate later. Fine, but this picture will be enough for me to describe from the physical perspective to have novel ways of computing the integrand and novel ways to actually analyze it for example analyze it single architecture analyze for example, you might be interested in what's happened when you go to when you take residues when you take multiple residues. And so, to infer a little bit what is the. The structure of the way function that this the project is measure is not to measure in which you're going to get neither in general in this to get to the FW with function or even to get to the loops because it's not even the measure to get loops. Fine, you can do it in another way. Yeah, but just to make sure I understand this is going to come to the surface of this. No, no, no. Let's say that the one. Well, the volume of this point of this is of. Okay, this actually computes the volume of the dual of the point of I didn't want to talk about about this but yeah. Why, why, at all I'm interested if I'm not going to be honest with you. I know okay no no fine I mean here on that's a super legitimate question because I am not because for the moment the only thing I did, it's taking a simple example show that. To this example, you get a triangle that you can associate the canonical form and this canonical form is the way function then you. And then you say okay but we computed the same object before with these rules so what do I need this. You need this because when you have free sample. This object for example, you can still apply for some reason you're interested in this process for some reason you can apply these rules. Okay, and obtain some among those expression, but then you want to check that your answer makes any sense for example you want to extract some information some some process. You want to take residues, or you want to do some some operation on on on on on it. Good luck. Okay. So, the point is that this, which is a hope I have time to talk today that there is a nice way in which, without actually doing the geometrical equivalent of these rules, you can look at the picture. You want to extract the information but just some graphical operation without to do in any art competition then write the final answer or gain formation and to extract space in particular. One thing that we could prove. Okay let me. I have just 15 minutes so probably I will list the things that you can prove with the the methods that probably I will have described more of it and not today. So, for example, I emphasize a lot of time that the way function contains the flat space amplitude as coefficient of the singularity of the total energy singularity. Okay. Now, for example, one thing that we did is. First, we asked the question okay we start from an object to this way function which is not neither has manifest unitarity, nor it has it is lowering its invariant. It is when I say that it doesn't have manifest unitaries that even if your evolution is unitary. It is an object which is live just on a space like surface so in principle, I'm now we know the cut the cosmological cutting rules, the cosmological optical theorem that Erico formulated, but in principle, if you if you didn't know Ericos work. If I give you that to a function that is unitary is not something manifest. Okay. So you have an object which is not, which is not Laurence invariant. But you know that the scattering amplitude, especially if you have, if you're studying the way function was limited to expect to be and Laurence invariant amplitude. So the question that you can ask is, how do you see Laurence invariance emerging. I do you see the fast pace cut the rule emerging. And this is something that with the, so you can prove with the few lines exercise on doing operation on, on, on this point that you have how Laurence invariant of the amplitude is encoded. Unitarity of the, the, the amplitude is encoded. Now there is something in in amplitude which are called stymian relations, which tell you that the double discontinuity across across partially overlapping channels in the physical region or to vanish. That means to all these words. This means that if I have a, if imagine that I have some loop, a six particle process. If I take a discontinuity along this channel, let's say 123456. And then, after doing this, I take the double discontinuity along this other channel to three, four, five, six, one, they are partially overlapping because this blob contain just all the blob contain part of the, of the, of the particles which are on the other side. Then, these two are incompatible in the physical region so you expect that there is no, the, the, this double discontinuity is zero. And this is actually this factor has been used in the amplitude literature to compute or to my concerts about higher loop amplitudes without having to actually do the group integration. Now, so, and the, the proof of these relations are from papers in the 60s and 70s, because they were formulated before for correction function and then mapped to, to this metrics. And let's say that they are not the, at least for me the existing to read with this technology even you could prove for flat space for the for for the amplitude. This time and relation with a few few line argument with a combinatorial argument. Okay, this is okay I'm saying all things about flat space, but we want to know something about cosmology. Then, and so the important thing of this time and relations that this time and relations in flat space are supposed to go side. So you can ask the question, are those things valid also for the way function. So we asked this question, and we actually show that yes, it's the case so also psi satisfies time and relation. So, so you know that if you have some arbitrary complicated process that you want to compute. Maybe you can use also this type of conditions to try to make an answer so without having to actually do the computation. So why should you be interested in in this from a practical point of view from a practical point of view, because it gives you tools to understand the energy properties of the way function without having to do the hardcore computation. Because any question that you can ask on the way function get translated into the question which are combinatorial geometrically imagine that you want to ask the question about the symmetries. How much symmetries can have this way function maybe there is some hidden symmetries that we don't know like, you know, it was found the dual conformance symmetry and the young young for for a certain amplitude and certain graphs in the flat space you can ask some similar question and looking directly to an object and making general statement this is a general symmetry of the theory or it's a symmetry of a graph can be not easy. And this gives you another way of looking at the problem. Okay. Now, from a theoretical this is from a practical point of view for my more theoretical point of view. This gives you a different first principle way of defining an object. So I could, if I imagine that we were an audience of mathematicians rather than physicists, I could just have skipped all the physics part and I could have talked about this object because started from a first principle definition. And which means that it can give some ground to have a different way of looking out of defining this and this problem, because then what you do is you define this mathematical object. You study and then you talk to a physicist and you will discover that this is computing something that physicists. are interested in. So one to 50 or the end of the day what's it is also some sort of mathematical theory, mathematical theory that helps you to do some to study physical processes. So, in cosmology, or in general in physics, at least, I mean, now I'm talking from a very motivational point of view for me is that doesn't have what I'm saying doesn't have for the moment any practical value. But the point is that when we study some problem and we found difficulties understanding also it seems that our understanding is breaking down. Very likely it happens because the language that we have been using at some point is not the most suitable. So the question is, then finding some. And for me, I have my personal bias that quantum field you're in in a curve background. This is not really the most natural language to describe this type of processes. So the question is, how can I find a different language that can be more suitable. So the first thing that you have to do when I made some proposal, you check that I can recover the known results the results that are known to be to be correct. And then try to show that try to see whether I can go beyond in a much cleaner way so from a more theoretical point of view, the general aim is to see whether these geometrical languages that has been used a lot are more difficult and now, with this point of start up yours in this other context, could be any more useful to answer some other question that might be, maybe you can answer already with the quantum field here in current space but my just more difficult and more natural. Okay, so. Yes. Yeah, you made a good three vertices. Okay. That's correct. No, you actually have a six pulse there. Oh yeah I mean I can write. So I can write the this differential forms in in a very in in a very depth way also because, for example, so let's let's do it in the text sample. Let's say that I have you want to compute. This is the example you want to compute right. Now my space is again as you said that as five homogeneous coordinate. So this leaves in before. So actually I can't draw before. So, okay, one thing that you can observe is that this graph. You can think about this graph as a combination of two graph of this type. Okay. Let me call X to prime X to second extreme. Now, if I look this separately obviously to each of them that is associated a triangle. So these vertices so if, so, in this picture, if I take as basis over three X, what X one and why an extra. Sorry, if I take for basis over three one, zero, zero, zero, zero, one, zero, zero, zero, one. I can actually indicate the vertices, which originally wrote as one, one, one, one, one, one, one, one, one, one, one. I can actually write it. I can actually write them. If I call this vector X one bar, Y bar and X two bar as X one bar minus Y bar plus X two bar as one bar plus Y bar minus two bar minus X one bar plus Y bar plus X two bar. Why I'm saying this because then you know that when you see this, you can associate three vertices, which are, which have this, this form. So in this case we have X one bar minus Y one subar plus X two prime bar X one bar plus Y one subar minus X two prime bar minus one bar plus one subar plus X two prime bar and the same here. Now, in this graph, this X two is the same X two prime. So really, you have a set of six vertices, which are given by X one bar minus Y one subar plus X two bar. So you just identify these vectors. I'm not giving the mathematical, the rigorous mathematical definition. I just want to give you an idea of how you can think about this, this, this point of. So these are the three vertices associated to this part of the graph and then there are other three associated to the part of the graph, which they share this X two vector. Now, the point of is the convex hull of these vertices. So if I want to compute this. Okay. You have six vertices. Then what you what you know that you know given that you know that to this you have some subgraphs with associated to the singularities, you can write the subgraph associated singularities has two vertices and you know that there are six. So what you know is that your canonical function has a denominator, which is a polynomial of degree degree six. Here we live in before. So you have a Y before Y. So you know that there is a numerator, which is a polynomial order one. Okay. Okay, and this requires some technology that I will explain tomorrow so I will give you just just a sketch. So if you remember from Julius lecture, one thing that you draw yesterday is that when you have some point up, which is not so simple which is in the in the space so for example in P2 is a square, a square as four boundaries. So it has four denominators leaves in P2 again is why the two wide. So also in this case, you have a numerator, which is the polynomial order one. The question is, I can identify in our case I can identify the nominators looking at the graph. How do we identify the numerator. If you remember from what Julia said is is given by the the locus, which is identified by the intersection of these lines outside of the object. So this is point A, point B. So this line in this case will give you this numerator. So here it happened the same. You can actually write this on the spot knowing which are these vertices to do the actual computation I need some technology that didn't explain today so for this reason I'm not doing it because otherwise will be just just confusing. I'm just giving an argument how you can just write that. So why you need a why this is the case this is the case because otherwise, even that polls are given by the intersection. I'm sorry. Yeah, they are given by boundaries, having this point here will give you also extra polls in the in the form so you need a numerator, which kills them and the numerator is given precisely by these locks. So, there is a giant interpretation of both this which are given by the, let's say the sides for a P2 but they are called facets of the point up and the numerator, which is given by this type of condition which is called the adjoint in a mathematical approach, but it just the locals or the intersections of the facet of the point of sight of it, which gave you this, this numerator. Okay, so there is a now to compute it explicitly. So if you have a polynomial of a higher order for for a polynomial of order one that's that's that's very simple for I order can be difficult. So you can resort to triangulation. So triangulation means that for example, you can compute a square by just the value into two, you can compute more easily, the form of this triangle, the form of this triangle, and you sum them to obtain the canonical form of the full square. So it's not a unique procedure but you can choose whatever you want. So the important thing is that in the context of the cosmological point open with function, different triangulations, give you a different perturbative expansion, meaning you can get the Feynman rules, you can get the first version of interior rules, you get some other rules to compute them. And Martin general, if you give you an arbitrary complicated point up, you have to feed it to a software to know which are the triangulations. However, the graphical rules that I will explain tomorrow gives you also some graphical rules to know how to compute the triangulation, but using these points. For example, you could compute the canonical form of the square, but just compute the canonical form of this triangle and subtracting the canonical form of this other triangle. So what this is easier to do, this is easier to do for the simple reason that it's, it is using just the faucet the lines, which are also boundaries so you're not ever in when you do the splitting in terms, you're not introducing spruce poles because remember the lines associated to this, to this, these lines are associated to the actual pulse of the way function. So if you don't introduce any new lines like this one. It does not belong to the point up. So it's another condition. So if you compute this as the sum of this so you will have a splitting of the of the way function to term with one denominator which will correspond to this line which has to cancel over the sum. In this type of feature, you never do this, but tomorrow you will see how we do this because here I don't have a general rule to do it for an arbitrary point up here actually do. So you look at these graphs, and you can deduce how you can write it using this triangulation this actually another reason for which is useful to do this because this type of rules. You can find some of them, which has less terms than the one that you will obtain by just using those those graphical rules that explain yesterday. Okay, and I can shut up. What is the one on the adjoining. This one is one of the degree of the polynomial. This is a polynomial. This is a six. And because of a scaling argument, you know, you're looking at the subgraph you know that there are six subgraphs so you know that the denominator will have the is a product of six polynomial of the one so it is the six. And before, so respect to why it has so you have a five or your own use coordinate. So you so you should do the scaling. From here you get some lambda to the six from here you have lambda to the five, you want this to be gel one invariant. And then you this has to scale up like once it is. I assume this. Yeah, I mean the gel one invariance allow you to know what is the degree of the numerator. When you have some context in which you can have multiple levels also of people put in those contexts typically you don't have gel one invariance, but you can have one variance but there you have to be a little bit more careful in how to do things because you know, you need to be this is what's well here up to the overall sign, they are the five unique. So, the fact that I've been able to map something to something to something that is gel one invariant is also because you have some sort of uniqueness up to an overall minus which is. So what is the purpose of writing this why before why that's on now here I here I wrote it just to make this kind of scaling argument and otherwise we only work with this. For physical purposes, this is in that you want to continue. But when you work to this space, you have to be careful of the project dimension so to remind that I exist. It's, it's okay. But for practical purposes, you don't need it. And one of the questions is triangulation is just like this factorization in when when even one of the particles become unshared right it is just no. Well, it is like one of the particles goes to boundary because it was pretty simple with respect to the story that Julia said, where, you know, I didn't understand to be honest. Sorry. So. Okay, so then I'm not going to make reference to it. So in this case, so it's not anymore. So, so, so in this case really is, for example, let me because you can just do it in different channels. That's correct. And it is like, for instance, when one to become unshared. No, but you know that here is not the case because to do to have the interpretation of your saying, you need to consider the full process where you can talk about as channel of each other we are talking about now that a single graph as three of what can have multiple Right. Why, for example, when you do this, the only one to get for the only scenario essence one of us. Okay. So, in that case, you can say, okay, I'm doing something as a channel, but in that case you're still looking at the same graph. Sure. So, typically, there is no great interpretation for this because some of this type of triangulation correspond to the final rules using the bulk to bulk propagator so you get this one over the twice. In front of everything. But some other. You get some other police pole that is not clear what it is. Right. But instead in this way, even if you don't have this, this lines where you keep using the same lines, you have all the time. So you are, you really are doing partial fractional right without the introduces for this. This is really as a, as a geometrical way of doing partial. A little question. Yes. I have slides. I have slides. Okay. Turn it to blackboard. No, no, no, no. I don't know. I don't know. Okay. Okay. Okay. Well, I sent it to the link. So I got this link. That said send your talk there. Oh, okay. I have them on my computer. Yeah. Did I use my computer? Yeah. Sorry. Please. Yeah. Yeah. Just closing all my windows. Yeah, the only problem would be that you will have to connect to that. Okay. Okay. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah, the only thing would be there. Yeah, you get it to change the slides. Yeah, okay. Yeah. Yeah. Is it possible to remove that side actually. Full screen. No. Okay. We are very happy to have Andrea. Thank you for the invitation to this wonderful place and it's very nice conference. I didn't know that there would be like such a central huge blackboard otherwise I would have given a blackboard talk but okay too late. If either of you don't see the screen here. I'll give you some time to move over to the other side. Otherwise I'll just start. I mean yes but it's also very far on this side so the people in the center may, you know, be a bit, you know, looking like a reference to this. But okay. All right, so let me start with disclaimer. Which is that I cannot change slides. Okay, let's do that way. Okay, so let's start with the disclaimer. So this conference is about cosmology and cosmological constant in our universe is positive, although it's very tiny. But I will shamelessly ignore that. And in the following set lambda to zero. So I'm going to give you a consolidation which is that we will not be talking about scattering trust in exactly flat space but in flat space one space terms that go to flat space asymptotically but are not trivial in the center of the space time. So we'll be looking at asymptotic flat backgrounds. And there are also some features of what I will discuss which may resonate in some way or other with physics in the center space. Okay, so if we look out the window. And by this I mean not by getting distracted by this beautiful scenery, but if we look out into the night sky into our universe, and we as theorists want to model the different regions of our universe. Here are the three typical geometries by which we model different regions. We've heard already a lot about the city space. There's also anti the city space which is useful for describing highly rotating black holes, and then for scales in between, let's say black holes. And we've heard a lot about the cosmological scales. It is allowed to use flat space which is a good approximation at least so long as you're far away from sources. And a lot of the work on understanding scattering or ultimately quantum gravity in such space times over the last couple of years has gone into trying to understand if we can think about gravitational processes in the bulk of these space times by looking at just the boundary data and just trying to describe it in terms of some way or the other. And we've heard a lot about that already. Now, in ADS, we are in a very fortunate situation because the boundary is time-like and so there's a natural notion of time. This is not so for both the city and the Kopska space. So this is one of the instances where some physics in flat space that I will talk about resonates with the city. So in one case, the boundary is light-like and the other is space-like. So there's no standard notion of time. It's completely obvious from the boundary point of view. And so those are challenges that face both people living in flat space and the city space in trying to understand how we can describe quantum gravity via a sort of holographic principle where we just look at the physics at the boundary. Now, what I will talk about is the flat space context and there has been a lot of work over the last, let's say, half century including very new recent ideas that have led to the proposal that there could be a holographic principle for spacetimes that are asymptotically flat. And that proposal that culminated from these very old, half-century old ideas and some more recent insights into the low-energy and permanent structure of gravity reinvigorated by strome and collaborators. This proposal is that quantum gravity in an asymptotically flat spacetime has potentially a dual description in terms of theory on the boundary. And here we would like to think of the theory that lives on the sphere at the null boundary, the celestial sphere. And we will refer to this theory as a celestial conformal field theory because it has some similarities with conformal field theories. And while I've stated this for bulk spacetime dimensions, we expect a holographic principle to hold in general dimensions. And what this proposal offers you is a new perspective or a different perspective on probing fundamental properties of the S matrix and potentially gaining insight about the structure of quantum gravity in asymptotically flat space. So this is the outline of my talk. I'm going to start by discussing some aspects of this celestial conformal field theories or CCFTs for short. So we'll talk a bit about the observables and then the universal sector of the theory which has to do with symmetries. The main part will be to try to take what we have learned from scattering on flat space in this new language and put it on non-real backgrounds. And here I will discuss, so in the situation where we are in anti-sitter space, we have a lot of insight on quantum gravity. And one key result there was that the boundary on shell action gave you a generating function of correlation functions. So in the first part of this main part of the talk, I will discuss how this extends to flat space. And then I will get my hands dirty on explicit geometries and let's see what the avatar of non-real-bound geometries is from this perspective of the boundary CCFT. And then if there's time left, I will discuss infrared dressings for backgrounds that make them finite. And please ask questions at any point during the talk. In particular, if something is not clear. Okay. So the observables in astronomical flat space time are scattering amplitudes as you well know and they are given as functions of the momenta or more correctly distributions. In the case of flat space which I will restrict to, we describe particles, masses, particles by a momenta that we can label by an energy and by an angle or a point on the celestial sphere where the particle goes to. Scattering amplitudes have very nice features. We can discuss their analyticity and the uniterity and how they constrain the physical scattering. And they have also very interesting hidden structures such as the double copy relation which I believe we'll hear more about later today. But these scattering amplitudes expressed in this energy basis are not very suitable for discussing holographic realities of quantum gravity and asymptotically flat space time. Instead, we can obtain observables that are more suitable by integrating out all the energies and introducing a new parameter which I will call delta which is the conform a boost weight on the celestial sphere. So if we do that, what happens is that our scattering amplitudes take the form of what looks like correlation functions in a two-dimensional theory on the sphere and why are the nice observables in flat space holography, the reason is that they transform nicely on the conformal transformations on the sphere and the conformal transformations on the sphere come from the bulk of the transformations. So they make different symmetries manifest than the usual momentum space amplitude. So you trade manifest translation symmetry or manifest Lorentz symmetry which on the celestial sphere is conformal symmetry. So that's why they're very nice. They have skew some properties but make manifest others. So for example, they're very nice for understanding what kind of symmetries constrain amplitudes. And here in particular, we have actually infinitely many symmetries that were sort of more manifestly uncovered by going to this new basis. And the first of those is the famous Vonley-Thunderberg-Matzner and Sachs asymptotic symmetries which are asymptotic symmetries of asymptotically flat space times and then there is an asymptotically flat space. Yes. So we have to think about the amplitudes in a different way because so I will get to this in a second. Once you integrate over all the energies, for example, you no longer have the nice like Wilsonian paradigm where you have a the coupling of the UV and the IR. So these things sort of go out the window in this new prescription. You can see this as a bulk but I would rather see this as a chunk of amplitudes in a very different way and try to answer different questions. I'll say more about this in a second when I'm on the slide. But it's a different way to think about things. Now you no longer have definite energies. You have definite boost weights. And you have to rethink a few things that you have understood for amplitudes in a different way. And I will discuss one of those things that we have to rethink of in order to describe them. Maybe you can ask again, yeah. So just to finish here. So any kind of hidden structures that are fundamental should survive a basis change. And so some of these hidden structures like the double copy that I mentioned also survive, but now you have to generalize them to in this case operate the value of expressions. And this I think is the second instance where there's some sort of operator value structures also appear. And I guess Arthur will answer this question later in the day. So let me say a little bit more about these observables. So we like that these amplitudes look like correlations functions in the CFD. Those are things that we can deal with. They are obtained by just taking the S matrix and putting them in a boost basis. In the simplest case, we start with plane waves. Let's say mass or scale of plane waves. Where the momentum are given by the energy times some null vector q that points to the celestial sphere and the plus minus for inverses out. And then the way to go to the new basis is by taking these plane waves, these external states and melanchorming them from zero to infinity over the energy and then introducing this parameter delta. And this way we obtain a wave function that's now no longer dependent on energy, but on this boost wave. So we have new wave functions transform as conformal primaries in this two-dimensional theory on the celestial sphere. And I committed it, but there's a plus minus i epsilon prescription which keeps track of inverses out states. So we take our plane waves, we map them to conformal primary wave functions, and then we have what we call celestial amplitudes, which are these correlation functions. But this celestial conformal field you're living at the point where you're not your garden variety conformal field theory. And this is in reply to, or going in the direction of the question. So there's no Wilsonian decoupling because we integrate over all energies. And so celestial amplitudes are only well defined if the UV is sufficiently well behaved. So you could see this as a buck because most amplitudes that we would write down will have infinite melon integrals, but you could also see there's a feature in that in string theory for example. So you could use this as a condition that a good UV theory should have a well-defined celestial amplitude, and maybe that helps us in sort of constraining from a more bottom-up perspective what kind of UV behavior we can allow. There's another feature that's rather non-standard, which is that the celestial amplitudes, they are not exactly vanilla correlation functions on the celestial sphere. They actually have distributional support. They have distributional support in the coordinates on the celestial sphere. And this comes essentially from the energy momentum conserving delta function from the bulk. The non-flat don't real backgrounds that I'm going to talk about have the nice behavior that these two issues go away. So yes. So we have understood a little bit about loop amplitudes in this business, but this has not been an extended study. So I can tell you, for example, certain loop amplitudes, let's say in young mills, super young mills can be written as an operator acting on a tree-level amplitude. There are infrared divergences and they are encoded in this exponential operator that acts on the tree amplitudes, but there has not been an extensive study of these kind of questions. Yeah. Yes. Yeah, the two-point function instead of one over z1, two to some power, it's the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point function of the two-point be a distributional support that sets the conformal cross ratios to be real. So that's that. So it's non-standard from the point of view of CFT, but it's what you get by writing amplitudes in this different basis. But yeah, when you put them on backgrounds, then since you break some isometries and in particular translation symmetry, that will go away and that will make the amplitudes more manageable in some sense. Okay, so now let me finish this part of the talk by saying something about the universality, which has to do with the low energy sector of the theory. So as you well know, an important notion in QFT is that of the energetically soft limit, which gives universal factorization property of the amplitude when the energy of an external mass as particle is taken to zero. Now we're integrating over the energy, so we lose the notion of an energetically soft limit. But we can introduce the notion of a so-called conformity soft limit. The dual variable to energy is the boost rate. And so special values of the boost rate do actually correspond to the different terms in the low energy expansion of the energetically soft limit. So this starts at the value of the boost rate being one. So there will be a pole at this boost rate that leads to factorization of the corresponding celestial amplitudes. And that is the celestial analog of the leading soft theorem factorization property of the amplitude. So while there is no Balsonian decoupling, we still have a sense in which the low energy physics gets encoded in some very, very distinct explicit properties of the corresponding celestial correlator. And the knowledge of that there should be something interesting going on at these sort of integer values of the boost rate starting with one, gives us actually a bit more insight. And let's us actually, let's say start to unravel this infinite tower of symmetries that govern the S matrix in asymptotically flat space. So here I've talked about the leading soft theorem, but there's also a sub leading and a sub sub leading in gravity. And then there's in principle an infinite tower of terms that you can write down in the expansion. And one question is, can you make sense of these other terms? The infinite many, do they, what do they do? So this is a question about what are all the symmetries, which is of course interesting in its own right. But if you want to construct holographic dual pairs, that's one of the first things that you have to match between the bulk and the boundary. So here we want to ask, can we use the knowledge of soft theorems and some CFT tools, since we are trying to describe some putative dual CFT to classify the symmetries. So this is what we've done in work with my postdoc Emilia Tavisani and with Sabrina Bostelski, and then more recently with my student Yoko Pano. So what we wanted to do was start with the soft theorems, write them in this boost basis, where we have correlation functions of operators with special conformal dimensions, and then we want to run a CFT machinery. So we want to ask, so these operators are special, we want to classify what are all the water identities for these operators, because that is related to symmetries. And the tool that you use there is conformal representation theory. So now you just use the knowledge that in the soft limit, you find some special operators. And then you use a CFT tool, which is very powerful, conformal representation theory, to write down all the possible water identities for these special operators. And then if you use some standard CFT tools, plus in general dimensions, it's a bit more complicated, but you can do it. You can then construct not the currents in this boundary theory, write down the charges and determine what the symmetric rope is. Now I don't have time to go into the details of these constructions if you're interested, ask me later, but let me just give you the result, which is that in two dimensions, so four dimensions to celestial sphere dimensions, there's an infinite dimensional symmetry group. And that resonates with the fact that from some sort of asymptotic symmetry generations in GR, we know that the artist born being met in the Thunderbox sucks symmetries, which enhance translations to infinitely many symmetries. There's also an extension of that called super rotations. And these super rotations are on the two dimensional sphere, they give you an enhancement of global conformal symmetry to local conformal symmetry. And so they are very important for this sort of holographic setup, or the idea that there should be some underlying holographic principle. Okay, so that's consistent with the GR analysis and what we know from two-dimensional CFTs. In high dimensions, again, putting together the knowledge of soft theorems and conformal representation theory, and being sort of a quantum field theorist with a conservative view of what you mean by symmetry, you find that the symmetry group is finite dimensional. So in gravity, that's just translations and rotations. And that's again consistent with a CFT person would have told you. But then it raises the question, what is the role of high-dimensional BMS transformations which have been discussed in the literature? Okay, so now I've finished the part on the different aspects of celestial CFT. If you have any questions now, would be the time. Otherwise, I'll move on to non-true. Yeah, okay. So what I want to emphasize here is that you just write down, okay, there's infinitely many terms, but they're not universal. I mean, once you go beyond the sub-subleading in gravity, they're no longer universal. And things mix up. And so it's not very clear. What I'm saying is that if you map this to the boost basis, you can just go in general CFT, do conformal representation theory, and ask, what are all the special operators? And you will find that for all the conformal dimension, which would correspond to all these infinitely many sub-leading terms, there is an operator that's special. And I can, okay, use more technical jargon to explain more what's going on, or if we can talk afterwards. But there's a sense in which you can classify all these operators, which the name is prime. You look at what all the primary descendants are, and they are there. Now, the next question is, which relates to what you're asking about what, how things are independent, how they are actually constraining the S-metrics and so on. That's a question about what also from this celestial CFT perspective, what do all these operators, what's their constraint on the amplitude? Yeah, but just from representation theory, I can just tell you there's something special about these operators, and it can be classified. Okay, so now backgrounds. So ultimately, a litmus test for any holographic proposal is if you can keep track of non-predebative processes, namely from the boundary theory, if you want to discuss non-predebative processes in a bulk. Now, that's a very difficult question. We can ask a simpler question, which is, what is the imprint of bulk geometries in this celestial or conformal field theory, and in particular on the corollaries. If you have a non-trivial background, they break as symmetries, and so what happens to the structure of these correlators in particular told you manifest Lorentz symmetry is something that we want, because it gives us this conformal symmetry of the correlation functions. And we would also like to understand if we can describe bulk geometries just from states in the boundary theory. And this will make use of some old and new insights that relate classical backgrounds to scattering amplitudes. So this is in particular work of the last couple of years. So here we should remember that if we want to describe amplitudes on backgrounds, the classical field, phi, that's produced by a source J, is the generating functional for tree graph approximation to the corresponding quantum field theory. So the classical limit of the generating functional for connected correlators, it's dominated by the classical solution to the equation of motion. And if you stare at that, you will find that there's this relation, that the classical solution is the functional derivative of the generating functional by the source. So that means that if we differentiate n minus one times the classical solution, it will give an endpoint correlation function. And I've written this here in a momentum space, so the bar here denotes the Fourier transform. Okay, so that's the tool. Now the first question that I want to ask is, and which somebody who is very familiar with ADS50 may come up and say is, well, in ADS50, the boundary on shell action gives you the generating functional for correlation functions in the boundary CFD. What's up in flat space? So that's one of the first questions that we want to ask. So if we start with the action, you can write it, massage it into something that looks like the equations of motions, which are zero on shell and the boundary term. And the boundary term is what we will focus on. So in asymptotic or flat space, this boundary would be the union of the past null infinity and the future null infinity. And we're going to look at this question for the simplest case, which is a complex master scalar that's minimally coupled to gravity. So if I write down the wave equation, I can put it in a form where I have something that looks like the flat space wave equation. And then all the the source and the Christophels, I all put into this effective J. And then I make the answers that I have some incoming field and some outgoing field. The incoming is just a plane wave. The outgoing, I will solve via usual Green's function methods. And then what I will find by computing the boundary, computing the on shell action, but push to the boundary. I will get the following result. So if I push it to large R going to this region that's highlighted in blue, in such a way that I keep either of the boundary coordinates, so the advance time or the retarded time fixed, then I obtain the boundary on shell action. And what happens in this process of taking R large is that I can use a cell point approximation and all the intervals that are involved in this procedure, they're localized. And so what we'll find is that the boundary on shell action localizes on the Fourier transform of the effective source, which is evaluated along the incoming incoming momentum. So that's a very simple result. Okay. And now I want to relate this to the two point functions. Or in general endpoint functions, but we did this for two point functions to make a relation between amplitudes and the boundary on shell action. So from this we'll run method for computing amplitudes on backgrounds. I have endpoint functions, so this is just the two point function that relates the classical solution to the two point amplitude. Now I plug in the result that I obtained when I tried to solve the equations of motion that I showed you before this one, which relates the classical solution to the effective source. And then in the final step, I will put in what I just showed, which is the large radius, large distance limit, which tells us that the boundary on shell action reduces to the essentially the effective source. And so in that way, you can show that the two point amplitude is indeed generated by the boundary on shell action in asymptotic flat space. Okay. So this is something that we better hope to be true because it's true in ADS-CFT. And if flat space holography is anything like that, that's something that we expected to work. Okay. So now I want to talk about actual backgrounds and compute scattering on backgrounds. And here we will focus on what we call particle like backgrounds. Technically, what that means is that these backgrounds can be generated by three point amplitudes with off-shell core here and emission of messengers such as photons and gravitons. And more in the simpler terms, these are backgrounds that are sources of mass and charge that are, for example, the Coulomb field of a static or speeding point charge or Schwarzschild on curd geometry, as well as some of their limits, including ultra boost limits, which give you shockwave geometries. What all of these solutions have in common is that they take the form of Kerr-Shield backgrounds. So what are those? So Kerr-Shield backgrounds are gauge or gravitational backgrounds, which are characterized by some scalar function B that solves the free wave equation. And a null vector is called the Kerr-Shield vector K, which is null and geodesic with respect to both the flat background and some non-trivial pool background in the case of gravity, where you write the metric as the flat metric plus some... it looks like a perturbation, but it's actually an exact solution to Einstein's equations. So it's not just a linearized result. Okay. So those are Kerr-Shield backgrounds. And now we can sort of capture the scattering on top of all these backgrounds that I just mentioned by just understanding what happens for Kerr-Shield backgrounds in general. So we'll do this again for complex massless scalars that are mainly coupled to gravity in the presence of some source. And what we do is to get the two-point function as I've described is we solve the wave equation and we do that iteratively in the coupling, which in the case of gravity is Newton's constant. So when we do that, there's a piece that's a contact term, and then there's a piece that's the thing that we want to focus on, which encodes the non-trivial part of the scattering process. Okay. So what we want to do is we want to take massless scalars. We want to scatter it off some backgrounds, in particular off of black holes, which will either be Schwarzschild or Kerr geometries, but we can also do this for the Coulomb fields of charges. Then we have obtained the two-point amplitude by the method that I just described. Then we'll go to the boost basis and see what the imprint of the geometry is in the celestial conformal field theory. And what turns out to happen is that celestial amplitudes on backgrounds are much nicer than in flat space. So again, in flat space we have this finite delta function support on the sphere, but also in general these mailing integrals where you integrate over all energies are divergent. On non-chill backgrounds, this is not necessarily true. So first off, the amplitudes that you get are supported everywhere on the sphere. And second, if the background has classical spin, such as Kerr or its ultra boost limit, the gerotone geometry, then the spin seems to act as a UV regulator, where these energy divergent integrals get modified. There's this hunker function appearing, and this has finite support. So that's very nice because this gives us correlation functions that look more like those in normal CFTs. Now we can do something else, something that's interesting, which is the ultra boost limit of black holes, which gives us icobook sexual shockwave metrics. And there is an analog engage theory. And what's nice about shockwaves is that they turn out to be generated by conformal primaries in this two-dimensional celestial CFT. So to be more explicit, if you take scalar shockwaves, so they have some support. So x here is the spacetime vector, q is a null vector that points to the celestial sphere. They have some distributional support, that's a shockwave, and then dressed with this logarithm of the spacetime vector squared. And you can check that this shockwave geometry or the scalar shockwave is actually transforming as a scalar primary with definite boost weight delta. You can construct spinning shockwaves via a double copy of the scalar ones. And also in those cases, you find that these backgrounds are actually conformal primaries. And this is very nice because now the background itself is described as by an operator in the CFT. And so you expect scattering on top of those backgrounds to also be very nicely behaved. And indeed, this is the case. So for the electromagnetic and the gravitational shockwaves, if you compute two-point function on the background, you expect to get a three-point function, where now one of the, what the third operator is now the one that generates the background. And what you see here is that the now three-point function in this theory takes precisely the form that you would expect from CFT. So this is now the vanilla CFT three-point function with no extra delta function in the angles. Since I'm running out of time, let me just flash here also the result for gravity. Also in the case of gravitational shockwaves scattering on top of that, you got something that looks like a normal CFT correlator. But there are some asterisks to it, which I think I don't have time to discuss. But the bottom line is that if you describe scattering on backgrounds, you get more similarly looking, you get correlation functions that are much more similar to those in normal CFTs and therefore more amenable to further calculations. Okay, so I think I'm out of time. So I will skip the dressing for the background story. This is let me just summarize this in one sentence, which is that we know, so here we solved the wave equation to leading order. But if you go to higher order in the coupling, you will have infrared divergences. Normally for particles, we know a way to get rid of infrared divergences by dressing them with part of foolish procedure. Here we can do a similar thing for this particle like backgrounds. And there's a nice interpretation in terms of the celestial conformal field theory, which if you're interested, ask me later. So to summarize, I've told you about a new perspective on the asymmetrics by the celestial amplitudes. We like this because we can use CFT tools. And I've sort of sketched for you that we can make use of this powerful tool to classify symmetries. There are more curious features about these amplitudes, which we still have to poorly unravel. Backgrounds are nice because they get rid of some of these more funny features, but they're also important for actually understanding a full holographic picture of us in the flat space. So let me thank you here. And let me sneak in this. You can do that too. And it has been done in the literature. And then you also get nice looking correlators in the CFT. So actually, the way that the Lorentz symmetry is broken is very mild. We still have in all the amplitudes, I didn't show you to them. So normally you pick for the background, you pick a reference direction. But here you can just replace. You can just call it reference direction something. And then that something will appear. And everything will look, you know, it will take a nice structure in CFT for the shock waves. That's actually what I did. So there was this null vector Q, which is this shock wave. Let me go to this one. Oops. So there was this null vector that described the shock wave here, this one. And you see, so I picked some coordinate set shock wave on the sphere. And then when you actually compute the explicit expressions, that's how it shows up. So it's a sort of very mild way of breaking it. In this case, it's very nice. In the case of black hole backgrounds, which are not described, general black holes are not, we don't know yet how to describe them just from this 2D boundary perspective. Then it will look a little bit more messy. But yeah, you can just characterize that breaking. So once you integrate all the energy, what does that become? Yeah, exactly. I mean, that's, yeah, I mean, that's what you would do. And then if that's something that either becomes a mess and then it's not used. Yeah, that's a very interesting question. Yeah, I've not been studied. Yes. Good, good, good. So, okay, I have not, I don't have the details here, I guess. So there is, I brought this here, there's actually some constant here that depends on where you put the sources, whether you use the retarded, the advanced or some mixed greens function. And, but it turns out that there is no, like, if you're talking about the large R issue, that isn't, there is none. It seems that, so you would expect maybe you have to do some renormalization, some holographic renormalization to get rid of the vertices. On the nose here, we just find this result. So super simple. And this could be zero for one of the four kinds of propagators, but it's not zero for the other three. So there's maybe something more that has to be understood. But on the nose will end on something that looks perfectly finite, which is maybe surprising, but. Oh, so here what we have done is we have just, we are agnostic about what the whole, what the geometry is. So that's all in this shape effective, right? So yeah, that would be in here. Okay, yeah. Yeah, thanks for this great talk. I have a curiosity about the paradigm that we had at the beginning. So, no, that's various works on mainly transforming glistening to it. And thank you, to see what could be a putative CFD. So my curiosity is what happens if you want to take some specifics in B and try to go the other way, you would have to understand that people buy this. Yeah, very good. So as you are correctly saying, a lot of the work he has been bought them up, taking some amplitude, many transforming, see if we can decode something about a putative CFD. There has been recent work that looks at certain young mills amplitudes on translation breaking backgrounds. So this is in reply to Schroder's question. And they, the correlation functions that you obtain in that way look like those of a leewheel theory. So you could conjecture, it has been conjectured that the 2D theory that's dual to this young mill theory on that particular background is a leewheel theory, a 2D leewheel theory. But this is still a perturbative conjecture, right? So that's maybe the sort of most advanced statement of the four-dimensional, two-dimensional relation. There's also been work in lower dimensions. So there are two-dimensional S matrices and there have also been like proposals of what the dual zero-dimensional theory is. You could go and pick your CFD and then look at the observables and do the inverse million transform and see if you land on some amplitudes in some theory that you know. You could do that. Before knowing enough about the CFD, it seems like a bit fishing in the dark. So I think that's one of the reasons why we sort of worked with this bottom up picture. But there is other interesting work that goes top down, which is this, I don't know if you're familiar with this twisted holography story where Costello and Paquet and others, they are looking for exactly mathematically rigorously dual holographic pairs of certain special theories with a special amount of symmetry or supersymmetry. And there you can identify exactly dual pairs. Yeah. I also have a question, a general question that I also have to answer. So if he started with the flat-based amplitude, then many transformation is quite complicated in particular cases and the function that we get on the other side, even for simple variables are complicated. And also we have this problem with QB, so often it's just in distributional sense. Now, if you really think about exploring celestial beings from this way, generating data, are we asking the right question or are we looking at from the right point of view? If you want to find some simplicity, it doesn't seem to be there at the moment. There are other questions that you asked about software and other things, not directly touching on one. But what would you say about that? So I think different questions have simple answers in different descriptions. I think the real power here from going to this boost basis is because it makes symmetries manifest. So I would like to think of like scattering of three bases that have very distinct features. One is, of course, the energy basis that we all know and like, and they are very powerful and so on. Another one is in position basis where we have memory effects. So this is particular for the infrared physics. But then we have this boost-weight basis, which makes very clear the symmetry interpretation. And so I think the main power comes from there. And I don't know if it would have been like understanding that all this infinite tower of symmetry, so it was found that they actually form an algebra like this w1 plus infinity algebra. And okay, this algebra was known, but from a completely different perspective of FISTA theory, I don't know if this would have been found in working in the energy basis. And so this is some very basic result about the structure of, let's say Einstein gravity in Asimdoli flat space. And so I think this is where the power comes from. Now, once I've understood everything about the symmetries, maybe for some questions, it's not the best basis. But you know, neither is the position basis we're talking about. There's lots of different questions that you can ask, and some bases are more suited to it. For me, the power here is that we can use CFD tools. And yes, the correlation functions, they have this, in flat space, have these funny features in on backgrounds, not so much. So they're much, much nicer behave. And so as I mentioned, this proposal that there could be a Liouville theory that corresponds to some particular bulk theory, I mean, I think this is this statements of this kind could be very powerful. Yeah, but I think it's worth exploring different I mean, different questions on different basis and whichever one is the most suitable should be used. So yeah, let me sneak in this last slide here since there's also cosmology conference. So there should be some or hopefully some connection between, you know, celestial goods rev and cosmological goods rev and maybe we can learn from each other about different tools. So if you're interested, let's talk. Thank you. Right, that's it. Thank you. Thank you so much for the talk too. Yeah, it's in the And not in the You saw that I put it right here. This one is like a point or something like that. Oh, yeah. Okay, great. So let's go. First of all, thank you to the organizers for the opportunity to do this talk today. So as here the title says, I will talking about this recent work with Johannes and Shun about integrating native geometries in the AVGN theory. So let me begin with an outline of the talk. I will begin with a short review of how can we build integrants from native geometries. In particular, I will focus on how can we build integrants for the logarithm scattering amplitude. Then I will turn to our computations and I will show how we can compute perturbatively these integrations in the AVGN theory. I will discuss how can we use these integrated native geometries to compute the Casper Reynolds dimension of the theory. And I will discuss the conformal invariance of the gene singularities of the integrated results. Finally, I will give some conclusions and I will talk about possible future directions. So as many of you know, over the last decade, we have gained a very new understanding of scattering amplitudes in a very vast amount of theories. And in particular, in this context, the amplitude is viewed as the volume of a certain mathematical object. To be more precise, the amplitude is obtained from the canonical form of what is called a positive geometry. So to be a more precise example, let me first begin by a short introduction to what is momentum-twistor variables, in particular in four dimensions. So if we consider the standard spin-off velocity variables in four dimensions, we take the dual-coordinate, which are defined as the difference between two consecutive momenta and the scattering process. We can introduce momentum-twistors as this combination between the spin-off velocity variables and these mu variables that are defined in terms of incidence relations. So in particular, momentum-twistors are very useful because they naturally give non-conformal invariance. We just have to contract momentum-twistors with the suitable Levy-Chivita tensor. So having introduced momentum-twistors, let's consider the region in momentum-twistor space that is defined by these constraints. In particular, let's pay attention to this far constraint, which is the positivity relation between the lines, momentum-twistor space that are associated to loop momentum, because lines in momentum-twistor space are points in dual-coordinate space and vice-versa. If we consider this region in momentum-twistor space, we get what is called the four-particle L-loop amplitude-hydron of N-glass force of N-mills. And how can we get the amplitude from this region? Well, this region is a positive geometry and then has a canonical differential form. And we can get the amplitude from this canonical form to be more precise, the amplitude is proportional to the canonical form of this positive geometry. Let me introduce this internal notation that will be useful for what is next. We will be using a node for each one-loop amplitude-hydron that is for the region defined by these first two constraints. And then we will be using a dash line for each positivity relation between loop variables, that is, for each of these constraints in the far line. So, in particular, for example, for the five-loop four-particle amplitude-hydron, the geometry is given by five nodes where we connect all nodes but all possible positivity conditions. Very recently, about some months ago, a new formulation of the N-glass force supinean-mills amplitude-hydron was given in terms of what is called neodygeometrist. So, here we have to pose a negativity condition, which I will denote with this red thick line between nodes. And crucially, if we introduce this concept of neodygeometrist, the sum of the neodygeometry where the corresponding positive geometry gives the coupling of nodes. So, if we take into account this the coupling of nodes, now each positive geometry, which is given by relating all nodes but all possible positivity constraints, is given now by an expansion by all possible ways of connecting the nodes by negativity conditions. And this is just an expansion of all possible combinations of neodygeometrist between the nodes. So, as usual, if we now take the logarithm of this expansion, we will get an expansion over all possible connected graph with neodygeometrist. And this logarithm is the logarithm of the canonical form that was associated to a positive geometry. And this logarithm is precisely now what will be related to the logarithm of the amplitude. To be more precise in this affirmation, let's introduce now the interim for the logarithm of the amplitude at L-loops, as it is here, where from now on I will use the amplitude normalized by the three-layer value. And the precise statement is that the interim for the logarithm of the amplitude at L-loops is directly proportional to this omega tilde form. And here the normalization can be fixed by requiring that this interim satisfies certain limits. So, what happens now if we take this interim for the logarithm of the amplitude and we perform all but one of the loop integrations? Well, surprisingly, this formulation in terms of neodygeometrist allows one to say that this integration will be IR finite. And in particular, for the fourth particle case, this integration is given by a function of a unique prostration for dimensions. So, as I mentioned before, this result of integrating all but one of the integrations gives an IR finite, because all the divergences concentrate on the last loop integral. So, let me give a review of these many interesting results of having found for this F function in n equals 4 sub n mils over the last years. This function has been computed up to L equals 4, that is for three loops for the F function. It has been found to be IR finite and to have uniform transcendental width. And very interestingly, it has been used to compute the Casper norm of the dimension of the n equals 4 sub n mil theory and the QCD theory to a full four loop order. By full, I mean that it has been used to compute the first non-planar corrections. And this is done by applying a certain functional over this function F. I will give the details of this later. Also, the lean similarities of the integrated results have been shown to be conformal invariant. And there is a very interesting relation between all plus Yang-Ming amplitudes and these integrated results. Also, if one takes the expansion of these interiority netting geometries in terms of netting geometries, there has been some non-verdubative sums of diagrams in n equals 4 sub n mils. And finally, this analysis of these interiority netting geometries has been extended to higher points, that is to scattering processes with more particles. And in this case, in the general case, with n particles in the scattering process, this F function is now a function of 3n minus 11 cross ratios. And this is exactly the same number of variables as QCD amplitudes. So the question that we wanted to answer in our work is, can we generalize these results that I showed you before, but to other theories, ultimately aiming for QCD? Well, given the similarities with n equals 4 sub n mils, the AVGN theory, that is the three-dimensional n equals 6 char simons matter theory, is quite a good candidate to start this program. So let me introduce what is the AVGN theory. The AVGN theory, as I mentioned before, is the three-dimensional n equals 6 char simons matter theory, where the gauge group is now u n cross u n, and the r symmetry is su4. The field content of this theory is given by two gauge fields, both in the joints of the, I mean, a mu is in the joint of one of the u n's, and a hat is in the joint of the other u n. There are also four complex colors in the v fundamental representation of the gauge groups, and four dark fermions also in the v fundamental representation. Schematically, the Lagrangian is given by these four terms, by a char simons term for the gauge fields, the standard kinetic terms for the matter fields, and then the interaction terms, which are quartic, in this case, which is schematically given by this quartic term between the scalars and fermions, and this term of order 6 for the scalars. Interestingly, this project, this positive geometries analysis has been extended to the VGN theory in the last years, and to do this, we have to take the positive geometry associated to the amplitude hydron in n equals 4 superannuals, and we have to impose these symplectic constraints on the external kinematic data and on the loop variables. And then, after imposing these constraints on the amplitude hydron of n equals 4 superannuals, we get the all loop amplitude hydron for FBJM, in particular, for the four particle case. So how does this analysis, in terms of netting geometries, looks like in FBJM? So in FBJM, now, apart from taking a connected graph to describe this logarithm of the canonical form, we have to also take B-part type graph, that is, graph where, after we assign a direction to each edge, each node now is a source or a sink, and this has been used now to compute this canonical form omega tilde up to five loops. So let me note that this is a major simplification in the number of graphs in the expansion. So before giving our explicit results for FBJM, let me make a very small parenthesis to introduce the five-dimensional notation to describe the kinematics in 3D. So this five-D notation basically consists of taking the embedding of the three-dimensional and because of its space into the five-dimensional icon. And why is this useful? Well, if we take this embedding, now, do I conform an invariant expressions in three dimensions become Lorentz and scaling variant expressions in 5D? So what happens now if we take the interact at L loops for a logarithm of the amplitude, and we perform all but one of the loop integrations? Well, now do I conform an invariance in FBJM, constraints the result to have this formula is here. And here we can note that there is a parity event term quite similar to the one that was in n equals 4 superannuals, but also we have now a parity odd term given by this contraction with the epsilon-layer treatment. And this is quite similar to what was observed for the five-particle case in n equals 4 superannuals. Here in FBJM we are starting the four-particle case. So what are the perturbative results for the integration of these n-id geometries? From the epithelial theorem, we get that the one loop integrant for the logarithm of the scattering amplitude is this one that is from here. There are some subtleties with an idea of continuations that we have to take into account, but we can get to these results. And from this one can read that the parity odd event term is vanishing in the integrated n-id geometries, but the parity odd term is trially constant. To go to loop orders we have to use some loop integrals, in particular we use this triangle integral that can be obtained by this link with Feynman parametizations. And we also use this five-layer integral with an epsilon numerator, which is much less trivial in the triangle integral, but can be obtained by iterating Feynman parametizations. And it's expressed in terms of this truss and delta-weight two function. So how is this integrated n-id geometries at one loop? Well, we have to take the two loop integrals for the logarithm of the scattering amplitude, which from the amplitude here is this one that is here, and we have to perform only one of the loop integrals. That is, we have to leave one of the loop integrals frozen. The loop integral that we perform is only a triangle integral, so we get this result which is here. And from this we read that the parity even term now is non-trivial, but the parity odd term vanishes. At two loops, we use now the free loop integral for the Lorentz-Moscat-Reynab loop, and performing the two loop integrals that we have to perform and leaving one of the iterals frozen. We get now that there is a parity odd term non-vanishing, and the parity even term is vanishing. And the parity odd term is expressed in terms of this transcendental way-to function that I showed you before. So what can we say now about the Casper-Normal dimension of ABJM in terms of these integrative results? First let me introduce what is the Casper-Normal dimension. So if we take a whistle loop now with the casp, with the light casp, there are divergencies in the expectation value of the whistle loop that cannot be absorbed by redefining the constant, the coupling constants of the theory. Instead, we have to introduce now a renormalization factor in the expectation value to renormalize these divergencies. And as usual, when we introduce a renormalization factor, we can define an anomalous dimension, which is in this case the Casper-Normal dimension. In any case, for superior mills, this integrated net degeometry has been shown to be useful for obtaining the Casper-Normal dimension. In particular, we have to apply this functional that is here over the integrating net degeometry. And then we get the Casper-Normal dimension. And as I mentioned before, this was used to compute the four-loop Casper-Normal dimension in n equals four superior mills and in QCD, including the first non-planar corrections. Interestingly, in the planar limit of n equals four superior mills, there is an all-loop prediction for the Casper-Normal dimension that comes from interoperability techniques. And as usual, when one performs an interoperability computation at all loops, the coupling dependence is encoded in an interpolating function of the coupling, which is not fixed by interoperability. But crucially, it was shown that this interpolating function in n equals four superior mills is quite simple. It's basically the square root of the coupling to all loops. But in AVJM, we have this proposal also coming from interoperability, that the Casper-Normal dimension should be basically the Casper-Normal dimension of n equals four, but replacing this interpolating function of n equals four superior mills, but for the interpolation function of AVJM. However, in AVJM, this interpolating function is much less trivial than in n equals four. And there is an all-loop construction that it should take this formula as shown here. If we use this construction and this proposal for the Casper-Normal dimension of AVJM, we get that perturbatively it should look like this. And let me mention that recently, with my advisor, Dio Correa and Iconic collaborator, we managed to compute the Casper-Normal dimension of AVJM to one loop, using an interoperability approach, in particular a thermodynamic better-answered approach. So, can we obtain the Casper-Normal dimension of AVJM from the knowledge of f and g? So, the answer is we have to use the whistle loops scattering amplitudes duality, which states that the logarithm of the scattering amplitude is identified by the logarithm of the whistle loop, in particular of the tetrahonal whistle loop in which the vertices are located at the corresponding dual coordinates. And this duality is at the level of the integrands. In AVJM, this duality holds for the four-particle case, but it has shown to file for more than six particles, at least at the level of the bosonic whistle loop. Then if we use known results for the renormalization theory of whistle loops, in particular now the logarithm of the tetrahonal whistle loop is expressed at the lead and virgin order in terms of the Casper-Normal dimensions. We can use now the duality to say that this is the logarithm of the scattering amplitude, but the logarithm of the scattering amplitude is expressed in terms of the neodygeometry. So, we can relate the integration of the neodygeometry with this expression for the lead and virgin order of the logarithm of the whistle loop. And from this, we read now that if we define these functionals, the Casper-Normal dimension of AVJM is obtained by applying these functionals over the F and the G functions that come from integrating neodygeometers. These functionals can be computed using Mellon-Barnes and we have recovered, using this, the leading order for the Casper-Normal dimension of AVJM. Let me comment very shortly about the transcendental weight properties of our results. In summary, we have obtained that the integrating neodygeometers look like this, and from this we see that at L loops, we have transcendental weight L, and this is different to what was observed in any of our source experiments, that in that case the transcendental weight at L loop is 12. This behavior in AVJM is exactly the same behavior that was observed for scattering in amplitudes and for the Casper-Normal dimension. Finally, let me discuss the symmetry properties of the leading singularities that come from integrating neodygeometers. And from leading singularities, I mean the rational functions that multiply the transcendental functions in the result of integrating the neodygeometers. For this, it is useful to consider the leading singularities in the frame in which the unintegrated loop variables goes to infinity, and we can do this because of the conformal symmetry of the results. Also, it's useful to normalize the leading singularities by the three-dimensional particle factor, and by this I mean the factor that multiplies the delta functions in the three-level amplitude. Doing these normalizations, I'm going to X5 to infinity limit, we get that at least up to the loop order that we computed. The leading singularities of the integrating results look like this, and crucially these expressions are invariant under the conformal generators. So, we can say that in the X5 to infinity limit and normalizing the leading singularities by the three-dimensional particle factor, we obtain conformal invariant expressions. So, let me give you the conclusions. We study in the EVJM theory the result of integrating the logarithm of the loop integral for the logarithm of the criterion amplitudes up to l-minus one integrations and for the four-particle case. We perform explicit integrations up to l-free. We find IR-finite and uniform those integral functions. We found the presence of parity of terms in the integrating results. Moreover, we built a prescription to compute the dimension of the theory, and we tested this prescription at the leading non-trial order, and we got a positive result. And we found the conformal invariance for the leading singularities of the integrative results. So, future directions could be in first place under some of the relations of the integrative results with the logarithmic insertions. And what do I mean by this? In any of the four supermeals, the integrated results were identified with the expectation value of the corresponding tetraionale whistle loop with the Lagrangian insertion at an arbitrary point, and divided by the expectation value of the whistle loop without the insertion. In EVJM, this is a little more less trivial because now the Lagrangian in EVJM is not a gauge invariant. What this gauge invariant is the partition function. Also, it would be interesting to generalize these two higher points, that is to scatter in processes with more particles and to higher loops. In particular, maybe this could be done using the differential equation method for finite trials. And finally, it would be very interesting to see if we can sum not perturbatively some subsets of diagrams in EVJM as a dual sum for the n equals 4 supermeals case. So, thank you. Well, we are doing that with Johannes and Schoen right now. Yeah, basically, so far we have managed to compute the differential equations at one and two loops. And the next step would be to obtain the differential equation of three loops and to use this basis of master intervals with the new fault and filter weights to obtain the interactive results at free loops. And maybe that would give some insight of how to then sum not perturbatively energy geometries at all loops. Also, this relation between kappa and n equals 4 and EVJM. So, that works in general, right? Yeah, exactly. This relation, this one, is obtained from integrability and is non-properative. The problem with this is that you have to have an expression for the interpolating function, which is quite not trivial, but there is this current proposal that is also competitive. Yes, exactly. Strong coupling is gained by half-interpowers of the coupling. Yeah, very well. Yeah, very well. Okay. Yes. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Yeah. Okay. Okay. Okay. Okay. Okay. Okay. Okay. The only sharp problem was, I mean, there was no sharp problem that wasn't already kind of the sharpest problem was they find her an insult. That's, I mean, that's a good problem, you know, but it argued me in hand. Okay. Okay. Okay. Okay. Okay. Okay.