 So, as I said at the end of the last lecture, I'll just sort of illustrate some of the general features through a couple of examples. I mean, I'll just sketch the details. I won't go into too much of the details of the calculations. They're fairly straightforward, but I'll give you some idea. So like I said, we look at four-point functions in sort of three-level corrections to Feynman diagram corrections. So these will be the leading corrections, so three-level corrections to a CFT by marginal operators. So it can be a free field. Of course, a free field is the simplest example. So example of free field CFT, perturbed by interactions, marginal interactions, we'll imagine that there are some cubic couplings, some cubic couplings, or some quartic couplings. And so we won't really worry about the details of the theory. This will be just to illustrate. So we can imagine a quartic coupling of some operators, a quartic coupling of some operators who have a dimension in the unperturbed CFT, who have some dimensions delta i in the unperturbed CFT. And so a diagram like this will be in position space. So we can imagine x1, x2, x3, x4, and some u, small d. And if there are some operators of dimension delta i, you can imagine wanting to do an integral like that. And we can ask, what is the sort of melon space representation for something like this? And if you try to do this in position space, it looks, of course, and it is somewhat complicated. But the melon space answer, as we'll see, will be very simple. So just to tell you how this is done, I'll just indicate in this particular example how you normally tackle the sort of how you write amplitudes like this in melon space. So basically, you use a sort of Schwinger parametrization for you use a Schwinger parametrization for each of these propagators and write it up to some overall factors we have. Yeah, so this is the position space amplitude for this Feynman diagram. So there's an operator delta 1, delta 2, delta 3, delta 4. So there's some quartic coupling of some operators, phi 1, phi 2, phi 3, phi 4. And so the position space correlates. So you can imagine you have a CFD in which you have operators of dimensions, various dimensions. And now you're perturbing that by, let's say, marginal operator, something like g phi 1 or 1, or 2, or 3, or 4. Let's say I'm just keeping it very general. So I have a, yeah, I should maybe call it phi 1, phi 2, phi 3, phi 4. So let's say you have some fields in the theory. And then these fields in the original unperturbed theory have a two-point function, which is 1 over xi minus u to the 2 delta i. So in this perturbative diagram, we'll give you a position space contribution like this. So I'm going to, so this is a four-point position space contribution. And by the way, yeah, so conformal invariance, if you want this operator to be marginal, you can check that the conformal invariance condition is that summation delta i is equal to d, the dimension. So that's, of course, the condition of marginality that the sum of all the dimensions in this perturbation is d. So this is a conformal invariant amplitude. So just like we started off by saying we can rewrite this in terms of a million representation. So I just want to illustrate that, in fact, it will turn out for all these, and there'll be a sort of relation, the written diagram, and there'll be a sort of a correspondence that these contact interactions will be very simple in melon space. Though they look maybe more complicated, but in melon space they are, in fact, just a constant. So that's not obvious from the way you define these amplitudes, but it will turn out to be the case. And so the way, so yeah, I'm not necessarily just, it could be that it breaks down. At this moment, all I need is this sort of marginality condition that will be sufficient to write down a melon form of this. So the way you sort of do these kinds of integrals is to introduce Schwinger parameters for each of these propagators. So basically, you use the identity that 1 over x squared to the delta can be written as an integral. So basically, this sort of Schwinger representation for the propagator, this is one of the standard things that you use even for Feynman diagrams, and you use it similarly for the Wittem diagrams as well. So it's a sort of a standard procedure for evaluating these sort of diagrams. The advantage of this Schwinger parameterization is that now you can do the u integral. It's just a Gaussian integral. So if you do that integral, you get, again, up to various factors. You're left with only the integrals over the Schwinger parameters. So to do the u integral, of course, you have to complete the square and so on. I will spare you all the details. So you get something, a Schwinger parameterized integral, which depends on the xij square, the position. So you've done the position integral. And this form, you can sort of process it further. But the main thing which is useful is to write it in this melon form. What is useful is to minus 1 over summation sigma. So that's the same factor. All sums and products are from 1 to 4 for each of these Schwinger parameter integrals. So now the way to convert this into a melon space is to use another identity, which is very useful. In this context is basically the melon transform of the exponential function. So the fact that the gamma function, the exponential function, so you're familiar with probably this thing the other way around. But this is also true, as you can also see by evaluating the residues at the poles. I mean, the gamma function poles closing the contour. You can see that you get this. So in any case, using this identity, for each of these e to the minus x's, you can write e to the minus xij square. You can write it in terms of, so you get something like sigma i to the delta i minus 1. And then you have a product over all these sigma. So times various factors of summation sigma i to various powers, d by 2 minus d by, so well, I think I shouldn't be going into all the gory details. You have to do some rescaling of the variables. And then you rescale the sigmas by an overall factor. And then you get something like this to the power summation delta i minus d. So the details are not very important. What's important is that you can easily convert these. This is a general way in which the Schwinger parametrized integrals can be converted to a sort of a melon form. You see that you already get all the gamma functions that are part of the definition of the melon integral. And of course, there's an integral over the ds. So you have the Schwinger parameter integrals. And then you have, so this is the part which is what you had in the definition of the melon amplitude. So we have written it in terms of these auxiliary variables, these sijs, the melon variables. And now if you impose the conformal invariance condition, this piece just goes to 1. Because if you have this marginality condition, then that piece goes away. And then each of these integrals, the sigma integrals, are just independent integrals. And these s's, of course, are complex over the contour, the usual vertical imaginary contour. And so then these are actually oscillatory integrals. And what they give you, so these sigma integrals, essentially just give you a product of delta functions of delta i minus summation jsij, which are exactly the constraints in that we had the melon amplitudes obey when the conformal condition is satisfied. So these give you the measure. So this is, so here actually this is the usual dsij. But after you do the sigma integrals, basically you get something which involves the usual measure over the melon variables. And basically times 1. OK, there are some other proportionality factors involving various gamma functions and so on. But so you see that the melon amplitude is essentially a constant in this particular case. So you have, so this, that's the bottom line, even if you didn't follow all the steps. This kind of contact interaction, in fact, has this kind of contact interaction, just has a melon amplitude, which is 1. So it's very trivial. And this is reflecting the fact that there's no single trace operators being exchanged over here. It's just a contact interaction. So there might, there would be multi-trace kind of operators, but those are the ones that are given, as we saw by the gamma function. In this particular case, it's sort of a free theory. So the gamma functions capture the operators here. So the melon amplitude is very trivial in this particular case. For something more non-trivial, we look at an exchange diagram. So we again look at something with delta 1, delta 2, delta 3, delta 4. And yeah, if it was not. No, this is, so the beta functions would come from the loop corrections. So here this is just a classical, it's just a correction, which, yeah, so here, so it's just a, no, it's not a loop integer. So this was the first case. So it's a little trivial in melon space. The second one is one where you have something like this. Oh, by the way, yeah, there's one thing I wanted to just make a comment here. So supposing we didn't impose this condition, so you didn't have the conformal invariance, but of course you still had the scale invariance because this integral is scale invariant. But supposing we didn't impose this condition, so then this term would still be there, you can actually still do the sigma integrals. So for summation i delta i not equal to, I keep using the different symbols, not equal to d. Essentially what you get is you get one delta function which imposes, there's one delta function, and then you get gamma functions of this delta i minus summation over j sij. So there's an overall delta function which is actually the sum over i minus sum over sij, i and j. But instead of these delta functions, you get a product of gamma functions. So recall that this was the thing that we, in terms of the Mellon momentum, this was the constraint that delta i plus pi square as I called it. So if I think of sij as pi dot pj, this condition was the fact, this was what I call the on-shell condition, so these were the on-shell conditions that conformal invariance imposed for you. But if you relax this, if you relax the conformal invariance, what you find is that the Mellon amplitude, instead of being one, the Mellon amplitude becomes a sort of product of gamma functions. So it's basically, so this is gamma of delta i plus pi square. So going off-shell is sort of replacing the delta functions which, if you wish, were sort of amputating your external legs, you're replacing them by gamma functions. And recall, gamma functions have poles exactly. So gamma functions have one over pi square plus delta i. So this is behaving like an external propagator. So it's as if the conformal invariant amplitude is the one in which you replace the external legs by just sort of one, and the contact interaction is also a number. But if you were to relax the marginality condition, then in Mellon space, it's as if you're going off-shell and putting factors instead of propagators, you're putting these gamma functions up playing the role of propagators on the external legs. So this is a sort of off-shell leg factor. So it's a little bit suggestive of the fact that somehow the conformal invariant amplitudes are on-shell, whereas just scale invariant amplitudes are more like off-shell amplitudes. And you have some kind of external leg propagators, external leg factors associated with them. Any case? So this is the case when, of course, since you don't have the delta functions, as I said in the first lecture, the Sij is that are n into n minus 1 minus n into n minus 1 by 2 minus 1, independent ones. There's just one constraint. That's why there's a minus 1, which is coming from the scale invariance. So any case, that's if you go off-shell, but yeah. But you can, yeah, it's a sort of in the sense that what I'm thinking of over here is that supposing I were to compute the four-point function in the interacting theory, so I have the free answer, and then I compute by pulling down some factor. So it's, if you wish, one in which I have a large n factorization because I pull down one factor also of 1, 0, 2, 0, 3, 0, 4, and I'm sort of big contracting these. Yeah, so if you, in the free, if it was just a free, right. Well, I mean, I think in a large n, but theories also, you would have the similar sort of generalized free field theories. So yeah, but I just kept it a little general so that, but the most you can think of it as just a free field expansion. So you can do something similar with this exchange diagram. So you have, once again, delta 1, delta 2, some delta. And so now what you're writing in position space is an integral over two of these interaction points, u1, u2, and then something like 1 over x1 minus u1 to the 2 delta 1, 1 over x2 minus u1 to the 2 delta 2, x3 minus u2 to the 2 delta 3. So just free field propagators. And then one exchanged operator, u1 minus u2 to the power 2 delta. And so you have now a more complicated integral. I won't go through how you do it. You sort of adopt similar steps for each of these. You introduce a Schringer parameter, sigma 1, sigma 2, sigma 3, sigma 4, and one for the internal leg. So you basically exponentiate all these. And then once again, the u1 and u2 integrals are Gaussian integrals. You can do them. You have to do some various rescaling and so on. So you have to do some rescaling. And you get some more complicated version of this. So now let me just write the steps. You introduce Schringer parameters, all the legs, all the edges, and then do the Gaussian integrals over u1 and u2. Both are Gaussian. You can do it in some successive steps. And the end result will be an integral over the Schringer parameters, which will depend only on the various separations just like over here, which you convert to a Mellon integral using the same identity. When you do the Gaussian integral, you'll again get some Gaussians. And you use the same identity to write it as a Mellon integral, and then you do the Schringer parameter integral just like over here. So those are the sort of steps. By the way, again, there's some sort of condition for the conformality that delta 1 plus delta 2 plus little delta should be equal to d delta 3 plus delta 4 plus little delta should be also equal to d. So the integrals simplify when you impose these conformality conditions. And the final answer for the Mellon amplitude is very simple. So what you find here, of course, you had M of s t, which was just one. What you find here is the next most non-trivial behavior. M of s t for this diagram actually depends only on s. It's independent of t, because we are looking at it where we, so the reason is that we are, this is a sort of an s channel exchange diagram. And so it depends only on s. And the final answer is just a beta function actually. So let me just write it. So it depends only on s. And it's just a beta function. So remember beta of AB is gamma of A, gamma of B. And there's some proportionality factors which are constants independent. So there's only an s dependence. And this again ties up with what we saw in the morning as the general. So remember s was what in terms of the Mellon variables, we call minus p1 plus p2, the whole square, which is delta1 plus delta2 plus s1 minus 2s12. So that's the same s we have had. And so this has the properties that, this is of course a little bit more non-trivial now than what we had before, which was just a constant. And you see that it is meromorphic. In fact, you have a beta function. So it has just the poles of these gamma functions. So there are no branch cuts, et cetera. And the poles are at, so in fact, this is just a constant. It just depends on the delta, which is that intermediate operator. And so this part of the gamma function is trivial. So it's just coming from here. So gamma poles are from delta minus s. And they happen when half delta minus s is a negative integer. So in other words, s is equal to delta plus 2n. So this is exactly what you expect. So these were by, I didn't explicitly say it, but these were this here. I was assuming you have a scalar exchange. And so the spin is 0. The dimension is delta. And the pole is at delta plus all the descendants. So delta plus 2n. And that's what this beta function shows. And moreover, the residue is just a constant proportional to the three-point function. And in fact, that's why there is no real T dependence, because the exchange particle here was a scalar. So remember, we had the general form. I had T to the L. So this is just T to the 0 here. So that's why a constant. So there was a polynomial in general in T as the residue. And the degree of the polynomial was the spin. And here, the spin is 0. So that's why it's a constant. In a sense, that's why it's independent of T. So if you just consider a diagram like this, it is an exchange like this then that has only an S dependence. Of course, you would have to add in explicitly, like we do in Feynman diagrams. We add in all the different channels. Then you would get an amplitude, which is, so this would be sort of S to T and S to U, or whatever. If you added in all of them, you would get something more symmetric, more crossing invariant. And so it's already, I mean, interesting, because it's sort of reminiscent. I mean, it's a sort of a baby version of a Veneziano amplitude, except that it's only beta function of not S and T, but this beta function of S. So already a free theory gives you an interesting, I mean, a very trivial extension of a free theory gives you an interesting malin amplitude. So the first non-trivial kind of malin amplitude you can imagine. So you can actually do more with tree diagrams. I won't describe in detail, but you can consider more general trees, arbitrary tree. And what you find is that the malin amplitude for the whole thing is just a product of beta functions for each of the external internal legs. And the arguments are basically the S variable, the analog of the S variable, the momentum flow, the sum of all the momentum flowing into this. And that has the same properties that it factorizes on all these intermediate channels. So at least for arbitrary tree diagrams, you can write something in terms of just these beta functions. For if you want to generalize this to loops, it seems to be more difficult. And maybe there's a general way to try to derive Feynman rules for the malin amplitude. You would like to sort of ideally, having seen that a contact interaction, the malin amplitude is just one. For exchange interaction, it's a beta function. And more generally, for a tree diagram, you see that it's a product of each vertex, gives you a constant. At least the scalar vertices give you a constant. And the edges give you beta functions. So you might think that there are some kind of more. You would like to see some more general Feynman rules for an arbitrary perturbative graph in which you assign similar beta functions to internal legs and then integrate over some kind of malin loop momenta. I think that's potentially possible, but has not been done yet. So anyway, this was just to illustrate some tree-level computations. Yeah, we can do the same things now in ADS. And that's what, in fact, pentadones and collaborators started to try to apply this malin technology there. I'll just show you that they're very similar to what you get for the perturbative field theory. Though this is, if you wish, a strong coupling. So in theories with an ADS dual, you can try to compute again with a diagram corresponding to a similar interaction. Now in ADS, an interaction in ADS, a quartic interaction, a contact interaction of four scalar fields. And so this written diagram in position space by the usual technology of written diagrams is given by now this is an internal point in ADS. And these are boundary points which are labeled as before by x1, x2, x3, x4. So the position space amplitude by the usual rules of written diagrams is given by an integral over this, integral over all of ADS. So that's why there's a d plus 1 dimensional integral over this internal point y times a product of these bulk-to-boundary propagators. So that's what I'm denoting here, bulk-to-boundary propagators from each of these xi's to y, and the corresponding to fields of dimension delta i. So this is the sort of, remember in this case you had a similar integral, but there it was just the d-dimensional propagator. Now we are considering the d plus 1 dimensional ADS propagator, but very similar kind of integral. And you can also do it by similar techniques using, in fact, what is convenient is something called the embedding formalism. So you can use the embedding space description for each of these bulk-to-boundary propagators. They look much simpler in this embedding space, which is this lift to d, 2 space in which the conformal or the ADS isometries act very simply. So in any case, you can evaluate this. And what you find, I'll just be brief. So you find that melan amplitude, in this case, is also just a constant. I mean, just the interaction. So it's very much the same. So even though it's sort of very different looking, but you get melan amplitude, which is independent of S&T. So that looks quite boring. But you can look at the next most interesting case with, once again, some delta 1, delta 2, delta 3, delta 4, and some exchanged scalar of dimension delta, just like over here. So the position space amplitude is, now there are sort of two cubic couplings. And then there's an integral over two internal points, y1 and y2 in ADS. And you get bulk-to-boundary propagators from x1, y1. Similarly, x2, y1, and these are the ones on this side. And then, similarly, from x3 to y2, x4 to y2. And then the new thing is that there's a bulk-to-bulk propagator, y1 to y2. So it's sort of, again, a generalization of this one here. By the way, I should have said something over here. This might look very trivial, but this amplitude is 1. But this integral in position space is something that people, in fact, haven't done. They define this integral. They don't know how to write it in terms of other simple or elementary functions. In fact, this is basically defined as a D function of delta. It's a special function defined to be basically this integral. So in position space, it looks very complicated. And in fact, in the early days of ADS-CFT, people were developing a lot of machinery in position space to do these sort of integrals. They define, finally, the sort of D functions and try to express all other things, many other things in terms of D functions. But even something like this is very complicated if you look at the expressions that are there for the position space. Even in terms of these D functions, it's some horrible mess of an answer. So the simplicity is, in a sense, very misleading. Misleading in the sense that it looks almost trivial. But it's, in a sense, I think capturing the fact that most of that complexity in position space is really irrelevant. The main amplitude, basically, it's a contact interaction. It's just giving you a contact term. There's no real physics in this thing. So that's, I think, part of the utility of the Mellon representation, that it's sort of a lot of the complexity is sort of stripped off when you take this Mellon transform. So similarly, over here, so we have this big messy integral. But once again, you can do sort of similar steps, introduce Schwinger parameters for all the edges. Now in this embedding formalism in ADS, there'll be, again, Gaussian integrals over instead of U1, U2. Y1, Y2, ADS integrals. You get the integral over Schwinger parameters can be written in terms of a Mellon integral, and you can do the Schwinger parameter integral. So you can actually do exactly the same steps for the written diagrams. And I'll just, again, write down the answer for you. You can look at the papers of Paulos and then Penedonis and many collaborators. I think Fitzpatrick, Kaplan, David Simmons, Duffin, Subrat Raju, Bolt-Wandri's, yeah, a big collaboration. So they worked out many of these written diagrams explicitly in, I mean, the Mellon transform of these. So the Mellon amplitude for this kind of diagram is a little bit more complicated. But it also has the same feature that we found here, that it's only a function of s. It's independent of t. And instead of being a beta function, it takes the following form. It's just in terms of a slightly more complicated version of a beta function, which is a hypogeometric function. I'll just write out the final answer just to show that it's very explicit. And it's sort of a generalization of, so this 3f2 has five parameters. It's a hypogeometric function evaluated at z equal to 1, the argument. But you see the, so these are some just parameters involving the external dimensions, delta 1, delta 2, and little delta. But these are very much like what you have over here. You have a numerator, gamma function, and a denominator, gamma function. That's what these two arguments in the hypogeometric function give you. And then this piece is also a little bit like this one over there. So it's just a little bit more complicated version. And it's nothing compared to all these d functions that appear over here. But the main thing is that you can again expand it. Of course, it's not as simple as a gamma function whose poles are just basically the same. But the coefficients here are more complicated. In fact, the coefficients do depend on n, quite non-trivially, unlike the gamma functions, which have all essentially a constant residue. So this hypogeometric function essentially has the, so what you see is that you have the same set of poles. So you have meromorphics of the function. This hypogeometric function is meromorphic as a function of its s variable. And the poles are basically at the same location, corresponding to the fact that the exchanged operator is a single trace operator here, which is delta. And it's descendants. And what you can also show is that these coefficients correspond to the three-point functions for each of these descendant operators. And so it's just a little bit more complicated version of the free field answer. So it suggests that somehow weak coupling, you have something which is a beta function. Strong coupling, you have something which is a hypogeometric function. They sort of have the same kind of poles. And there must be some function which interpolates between the two in some nice way and exhibits all these properties. So I think one might be able to use analyticity on these properties to somehow even, perhaps, guess the behavior as a function of lambda. Any case, I'm nearly out of time. The last comment I wanted to make was that Paulos and also Benedonis and collaborators generalized this to more general written diagrams, just like you could generalize here to arbitrary tree diagrams. And they were built from these beta functions. Here they could similarly generalize. And they found you can draw an arbitrary written diagram. They have some many crazy cases, a 12-point function or something, which you couldn't even imagine doing in a position space. But in this melon space, it becomes more complicated, but still in terms of some generalized hypogeometric functions, you can. And they have some sort of rules. Once again, there are poles in the poles of the sky. And then there's a factorization of the numerator into the lower point function. So there's a very explicit set of Feynman rules, which especially Paulos has given in a clear fashion, which enable you to write down the melon amplitude now for some more general endpoint function and showing that you can sort of basically sew them together, sew these amplitudes together from the lower point ones. So that way it seems, again, very similar to what you have over here. So this was just to give you some taste of what the explicit expressions for these melon amplitudes look like in cases where you can actually do some computations. So once again, I think the power of holomorphy as we have seen in holomorphy is a very powerful thing, as we have seen in supersymmetric field theories in this BCFW thing that Nima talked about today. And I think there should be a similar way to exploit the power of analytic behavior to say more about string theories, dual to CFTs. And as I said, perhaps also to make progress in the bootstrap, these ideas will probably, I think, help. But it's a very underdeveloped subject. So those who are interested, I would suddenly urge you to think more about it. It deserves more exposure, which is the reason why I talked about this. So OK, let me stop here, and thank you.