 So, what we saw was essentially the kinematics of CFT amplitudes in position space and the final sort of thing that we saw was we can write the general amplitude for an endpoint function of primary operators, we can write it in this way up to a factor which takes care of the covariance. These UAs are the N into N minus 3 by 2 cross ratios, independent cross ratios and the delta ij obey sum over j. So, you have the fact that the symmetry tells you that things must take this form. So, fine. So, now let's introduce the Mellon representation of the amplitude. So, when you have, so normally in quantum field theory momentum space is very natural because of the basic fact about Fourier transforms that you learn that if you have a superposition of sines and cosines of harmonic modes, the Fourier transform is the natural way to pick out the different harmonic frequencies of the modes. But in a conformal field theory, we have a sort of a power law decomposition. So, when you have things behaving like a power law, then the Mellon transform is the natural one, picks out the different scaling behaviors. So, that's a basic mathematical fact. So, let me remind you what the Mellon transform is. So, supposing you have a function, you can decompose it in terms of power loss x to the minus s times the Mellon transform. So, this if you have a function. So, this is the sort of the Mellon transform of the function. And this s is actually a complex variable which you integrate along a sort of a contour which is for many most purposes something parallel to the imaginary axis. So, this the contour c goes from minus i infinity to plus i infinity. Sometimes, you have to adjust it a little bit to pick up the appropriate poles. So, it's essentially like a Fourier transform, but it's just like the Laplace transform and Fourier transform. So, this is just a more natural way to pick up the appropriate scaling powers. So, it's just in some ways it's a sort of this thing, but usually the power law of this thing is over. So, this written in this way it will be more manifest that you pick up the way. So, they're all decompositions of the same function space, but it's just more natural to write it in this way. So, for instance, so this function if it's sufficiently well behaved, then what you do is you essentially close the contour on some side to pick up the appropriate poles. So, supposing we have for f tilde of s which is like 1 over s minus delta. So, you let's say you have the s plane, you have a pole at some point and then just like you do it for in the Fourier transform. So, this is you close the contour depending on the depending on basically x where the x is greater than 1 or less than 1. So, for instance for with x greater than 1, you close the contour on the right and you so delta is positive let's say. So, this is let's say the pole then this gives you something like 1 over x to the delta. So, the nice thing about this representation is that the different power laws that you see are associated with sort of the poles of poles of f tilde of s and yeah. So, that's what in fact for our amplitudes it will be that this will be a sort of a metamorphic function. So, but yeah in fact and then you have to assume it sort of dies off sufficiently well so that you can close the contour at infinity and pick up the poles. So, in any case this is just mathematical transform. So, just to say that it's the natural it's a natural thing if you want to pick out sort of different power law behavior. So, you can see that this f tilde of s essentially if your f of x has sort of contributions of different scaling dimensions then the f tilde of s would pick up would be sensitive to that it's just like a harmonic analysis for the Fourier transform but it's now in terms of the exponents. So, so this is what it's because of this that Mack defined. So, the Mellon representation of a CFT amplitude as defined by Mack basically uses this property and he defines. So, he defines it as an integral. So, just like over here an integral but over now several variables and I'll define it more precisely. So, what he so we saw that the general amplitude depends has this sort of structure and so to capture that behavior so this to capture that behavior what Mack introduces are essentially the same these s's are the Mellon variables this s i j's there are they are conjugate to the x i j's like over here and this is the Mellon transform and the Mellon amplitude and so this so of course there is this factor of gamma of s i j overall i and j which is sort of put in over here this could have been absorbed over here but in a sense it's a very it's a matter of convention but it's a very nice it will turn out to be very convenient especially for large end CFTs to explicitly take out this factor because the gamma functions has certain poles and those have a natural interpretation in the case of large end CFTs but let me actually say something about this so this s i j's actually are not so you might think there are n into n minus one over two s i j's over here but we know that the amplitude really depends on non-trivialy on just n into n minus three by two cross ratios so actually these s i j's are not all independent they obey and that's what this the square bracket is essentially defining it's saying that you you have the the measure has an additional has an additional piece which enforces certain constraints that the s i j sum over j is equal to delta i this is this is essentially the sort of constraint that takes care of the covariance just like over here so there's a overall piece so the so this so you have n into n minus one by two s i j's if you just consider the antisymmetric combinations here but then minus the n constraints so these are constraints imposed as delta functions so so the measure if you wish it's a ds i j this is just times various delta functions n of these delta functions which which impose this constraints that are n of these constraints and so so there are really only n into n minus three by two independent s i j's which is of course exactly the same number as the cross ratios so in a sense you can though this definition looks a little bit complicated essentially and we'll see this explicitly for the case of the four point function you can view this as just a melon transform of the cross ratios so there are n into n minus three by two cross ratios and you can you can write them in terms of n into n minus three by two independent uh melon variables and and and and essentially the transform is a melon transform of those independent variables and this will this is a sort of just a more fancier way of writing it so this to make it sort of more co-variant or so that you don't have to sort of pick it's sort of more uh uniform way of writing it in terms there's more symmetric way of writing it in terms of all the n different points and this gamma function as I said is is a matter of convention but it's a very useful one so that's what the melon representation is any questions about the definition so I just want to make one comment though Mac himself defined this transform for CFT amplitudes you can imagine doing the same thing for even a scale invariant field theory and all you need to do is is modify you can define it exactly in this way but all you need to do is modify the measure so for a scale invariant but not conformal field theory you can define it in exactly the same way except that this dsi j the constraints uh are are not as many as you had over here uh it's just an overall one which sums over all i and j which we saw is the sort of overall constraint that we have uh and there you can just view it as the s i j uh therefore there are only n into n minus one by two minus one independent s i j which are conjugate variables to the uh the corresponding ratios okay so uh so that's anyway just a side comment um so uh so you can view these things as conjugate to the cross ratios in the general conformal invariant case uh that's a very nice uh so so this is just a definition uh but there's a very nice uh so the um uh there's a suggestive way to implement uh the constraints uh and so this these uh uh so these n constraints uh so there are n of these uh equations there's a nice way to implement them uh by identifying s i j with some kind of Mandelstam like variables so imagine them to be uh some uh dot products or some fictitious momentum uh so we'll just call them momentum but if you wish you can just think of uh given any s i j's you find uh momentum p i which such that the dot product uh is equal to s i j and uh which satisfies um which satisfy uh so these are not any kind of physical momentum but uh uh uh but uh but there are some uh vectors which uh we demand that they satisfy uh um some kind of momentum conservation so um so so if we uh so if we assume uh something like this and define also p i dot p i to be equal to minus delta i so uh the norm uh uh to be equal to to delta i uh then the uh uh so then this condition summation uh the summation s i j equal to delta i this was j not equal to i but you can see that this is just the condition that summation p i uh uh dot so sum over all j is equal to 1 to n because uh so the delta i piece gives you the so that's minus uh of p i dot p i uh uh so uh so this condition is basically the statement that uh that these momenta are conserved so uh so if you dot it with any of the vectors then uh then you uh then you uh you get zero uh so this constraint can be viewed as uh following from uh uh following from some kind of momentum conservation uh uh uh provided these momenta obey some kind of on-shell condition so at the moment these are just sort of words and suggestive terminology but uh uh this is useful because uh firstly we'll see that they behave very much these s i j's will behave very much like uh the corresponding Mandelstamm invariant invariant so if these were actual momenta these would be what are called Mandelstamm invariance generalizing the s d and u that you would uh uh uh that we normally introduce no i don't think so yeah so i don't think there's a there's a there's any nice way to do that in a sense the yeah the the number of degrees of freedom are different this is sort of a d dimensional vector with n n d components and n this thing and then uh so it's difficult to write it in terms of the p's except it depends only on the invariance the p's so it's yeah there may be some nice way but uh which was right right so so there may be some uh some nice way i thought a little bit about this at some point but it didn't make any uh useful headway uh but it seems like an attractive picture to try to uh uh to yeah but uh yeah so yeah maybe it's in d plus one so so at the moment i will actually only be thinking of these invariants and uh and so we can view these conditions uh on these s i j's as just basically as if these moment are conserved and and they are they obey uh some on shell uh quantity and uh there must i mean it's of course on uh so you're using the scaling behavior uh uh here so that part it's some kind of a decomposition in terms of the sort of the eigenvalues of uh the uh so the so the fact that the s i j's come in is the fact that uh it's in terms of eigenvalues but i'm not very sure what's the nicest way to say it in uh in terms of the full s o d comma two because the rest normally you label things by just the spin and the dimension so this is in some sense the dimension it's like behaving uh but uh uh what these are the more uh so so these are some kind of vectors which are sort of these are sort of invariant these s i j's by the way are invariant under all the special conformal transformations so uh maybe there's a way to relate these momentum to some the special conformal transformations but uh i don't know i don't know if uh so the best group theoretic language to state it is i don't have anything very more useful to say about that so so by the way i just want to say one thing again uh for the the conformal invariance and the uh scale invariance if you just had scale invariance where you had only you didn't have all these uh n constraints you can still talk about the s i j so if you just had scale invariance then you relax the the on shell condition k i p i square equal to minus delta i uh and just impose so we just impose one condition which is something like summation p i square is equal to summation in delta i uh so in many ways it seems as a conform at least it will uh will appear later when we look at some explicit computations that the conformal invariance the amplitudes for the cft are sort of on shell amplitudes whereas if you relax that condition so that you are sort of off shell in some ways then that's more like the amplitudes for a scale invariant theory and perhaps that's a useful way to to view this because because many of the so so in the cfts which i will be primarily of interest uh the viewed in terms of these momenta there are sort of on shell momenta but as we know in uh in usual quantum field theory many properties are more manifest when you go off shell so that's why this may be a more useful way to uh to view this so um okay so let me look at the special case of the four point function and uh and because we will be uh mainly working with the four point function uh so let's write down this more explicitly uh in that particular case uh so so in the four point function it's we'll define this s variable in in analogy with our conventional Mandelstam variables as minus p1 plus p2 square so and using the fact that p1 square is delta 1 uh i will talk about that but it will be basically um that there will be exchanged operators exchange say in the four point function there'll be operators which are being exchanged with that dimension uh of that um but i'll i'll i'll be talking about that um uh i just wanted to motivate this here by saying that the melon transform is a nice way to pick out the different scaling behaviors of one over x to the delta and that shows up in the poles so it'll be just a more sophisticated version of that uh that will uh happen this is just a multi-dimensional sort of melon transform so you'll now have poles and different variables and so on uh so we'll illustrate many of these things with the four point function quite explicitly so i'm defining here uh so uh so basically in the four point function so we have n into n minus 3 by 2 is equal to 2 uh for n equal to 4 uh so we had the two cross ratios u and v so um uh so if we look at this melon amplitude uh we have um the number of s i j's over here are six but there are these four constraints so there are only two independent s i j's which correspond to the uh uh two independent cross ratios u and v that we had for the four point function so to isolate just that i want i don't want to work in terms of the s i j's but i want to write it in terms of two independent variables which i'll call s and t which are like the mandelstam in variance for the four point function so in a four point function when you do kinematics uh you you define uh basically these so if you have p1 p2 p3 p4 then this is what you call the s channel with p1 and p2 this is what you call the t channel with p1 and p3 uh and you define uh uh these invariants so uh so i'm just defining these combinations and from the definition p1 square is delta 1 p2 square is delta 2 and p1 dot p2 is s12 so uh so this is what the s variable is and similarly here it's delta 1 plus delta 3 minus 2 s13 so these these are the i will choose these as the two independent uh invariants out of all the six different s i j's so in fact you can uh so so of course it follows from this that s12 is a is nothing but uh half of delta 1 plus delta 2 minus s s13 is half of and um and then there are the four constraint equations uh and these which if you write them out in full uh i'll just do it once so uh this is the case with the with i equal to one and you sum over the other j's similarly here i is equal to two and uh so these are the four constraint equations on the other s's uh you can uh so uh so in principle given s12 and s13 in terms of s and t if i plug it in here you can see that i have enough equations to solve for all the other s i j's so there were six as we remember started out uh but s12 and s13 we've already expressed in terms of little s and little t uh and so then these are the equations basically give you equations to solve for uh solve for uh the other variables i'll just write down the solutions just uh once again just so that it's all explicit um so these are just some linear equations so the solutions are uh very straightforward to write down you see everything is expressed just in terms of little s and little t as it should be uh there are just two independent kinematic variables by the way sometimes you define u which is equal to minus s plus t uh and then things become more symmetric uh um but uh but anyway i want to write in terms of just two independent variables uh so which which one uh okay yeah it yeah it might be useful to define with a shift sometimes yeah different people define things with little shifts here and you you could define it that way i think this is probably the most symmetric definition uh um yeah so any case um uh we can uh so if you're now uh a plug in so now all you have to do is plug this in into the uh into this formula uh uh and we solved we've already solved for these constraints by writing things in terms of the s and t so uh so we can write this integral purely as an integral over the two independent variables that remain little s and little t uh so this measure just becomes a usual one on v s and d t these are linear equations so you just get and the delta functions you've solved for so uh so the in this in this particular case we we can write things in terms of this reduced function of u and v and uh and so when you put this in here all these extra factors of delta three plus delta four by two you see there are all these uh each the form for each of the s i j is that there's some combination of deltas and then and then some s and t variable these combinations of delta you can take outside the integral then they appear over here and those in fact give you uh some some particular exponential factor of delta i j's that's sort of a fixed factor that's the the covariant piece which is not very much of interest so so basically what you find is that this a of u v can be written as as I said an integral over just the s and t variables this the six-dimensional integral once you've solved for the delta functions is just the two-dimensional integral uh on uh on this s and t and uh and and then uh yeah and then the nice thing is that if you look at all the remaining factors so it's just a little bit of algebra but if you're interested I would urge you to look through it if you just substitute for s one two s one three and just write out all the combinations you'll get things involving x one two square so you see that there'll be an x one two square which will come with an s and then an x three four square which will also come with an s or s by two and then in the denominator will come x two three square with an s and x four square uh x uh x one four square with an s so it will all combine to just the combination we defined as u uh earlier and in fact uh just u and v uh so as it should of course so that was the whole idea that you have the you took out the piece that was co-ordered and then you got something which just depended on these cross ratios uh and uh uh and this u and v and then there are all these gamma functions so you can write out all those gamma functions uh writing them out because uh there are these six gamma functions uh now we write them in terms of and so basically I'm substituting these values of s i j uh and the reason to write this out in gory detail is that these gamma functions will play a role and then we can write the melin amplitude as an amplitude which depends only on these two variables s and t so you see as I said uh basically this melin amplitude for the four point function here is is essentially the melin transformed with respect to the cross ratios uh u and v up to various uh sort of conventional factors of gammas which have been sort of pulled out uh explicitly out of this m yeah so there's a specific formula for the delta i g s which you can write out like delta one two is half of delta one plus delta two and so on I won't write it all out but there's a uh there's a particular uh covariant factor but you needn't worry about that very much anymore uh because the information about the cross ratios is is now being encoded so you started with an amplitude function of two cross ratios you're trading it for a function of two melin variables which are suggestively written in terms of these s and t uh variables by the way let me remind you again since it's not up on the board u the conventions for u and v that we are following are so that's why you can see that the the u comes with a power of s by two uh because the x one the s one two has an s and then the one three channel uh you have t so that appears in both this so that's why you get an s plus t over two so this is uh so this is the uh the the kind of form that you have for the four point function okay so uh so this is uh to explicitly illustrate that uh but let me let me now talk about some let me first list some general properties of these melin amplitudes and really illustrate this for this sort of four point function uh at least feels I sort of sketch how it so we've just made a transform a change of variables so how is this useful it's useful because m of s and t has nice properties uh which are not manifest in the original position space amplitude so so m of s t has which firstly they are meromorphic functions of uh so I'll I'll first list them all out and then I'll sort of justify them a little bit more uh so they are meromorphic functions of the s i j or in our case s and t the independent s and t so it's not just in fact it's nicer than in momentum space not just any analytic functions they have uh no branch cuts this is related to the fact that there are no uh the spectrum is discreet uh and uh uh and so there's no branch cuts in normal momentum space come from uh continuum and here also you can imagine that when you did this transform uh if there were a continuum of dimensions that were contributing here you would have gotten something like a branch cut but uh uh uh so they are basically meromorphic functions and in fact even their pole structure will be uh is very nice so the poles uh so as I said no due to discreteness of spectrum of dimensions so the poles in the different channels so when I say different channels I mean it's a in this particular case it's a function of two complex variables s and t you can look at poles in s poles in t uh and these different channels correspond to uh dimensions actually what are more precisely twists which we will see uh of operators exchanged in that channel if you wish intermediate states so um so that's so the poles uh the location of the poles uh have uh uh have uh have a meaning just like uh in momentum space except there there were the masses and so on of the one particle states here uh there are the twists of operators uh and the yeah there was a question so we will see the there will be both primaries and the descendant there'll be poles corresponding in general to the descendants also there'll be a so for each primary there'll be a whole c series of poles uh so the residues so so when you have a pole of course the residue usually carries some interesting information and residues are related to the lower point correlators uh are example three-point function for instance in this particular case uh so this is again the kind of factorization that you expect in momentum space amplitudes in ordinary quantum field theory where when you go near an intermediate particle going on shell the residues are sort of the uh so you have uh you have some supposing you have an intermediate particle uh the uh the residues are related to the sort of uh factorized into the amplitudes for the lower point correlator uh so so these obey the same property so so uh so these are this is about the residues and the poles and the the channel dualities are that is the fact that you can factorize the ope in different channels in s channel or t channel simply manifest as s t exchange etc yeah I mean yeah it's just uh yeah so so in fact the nice thing is about the that if you consider large n cft so where multi trace correlators factorize so so the dimensions uh so example so therefore double trace operators like 0 1 0 2 have dimension delta 1 plus delta so so in the in large n limits where in the large n cfts in which the single trace operators uh essentially you can view it as a sort of free theory of of these single trace operators uh the uh the additional gamma functions which seem like some sort of general junk floating along with it all these extra gamma functions very naturally the additional gamma functions take care of of the uh contributions of of such operators meaning the m of s t is purely built from the single trace it purely captures the single trace operator so you don't have to introduce additional poles so all the poles on intermediate channel are all of single trace operators it's not the poles that are there surely the ope has contributions from the double trace operators but those are poles that are poles of these gamma functions and the residues of those gamma functions uh capture the factorization properties for the double trace operators so in a sense you can sort of in the larger limit kinematically take into account the presence of the double trace operators by some trivial multiplicative factor and focus on just the single particle sector in this m of s t this is very convenient because if you want to for instance look at the dual string scattering uh so if as I began the lectures we want to use this to understand whether are all large n cfts dual to perturbative string theories and in string theory you're looking at single particle scattering at so the large n cft is uh limit is one where you're looking at perturbative scattering let's say at tree level of single particle operators and that's completely captured by this this m of s t you don't have to worry about all the multi-particle all the multi-trace contributions that are so it's a nice way to disentangle the multi-trace contributions in the OP there's there will be a disconnected piece yeah so that's there's a trivial piece at some u by v to the delta or something like that that's a kind of a forward delta function you kind of get yeah that's in fact you can almost yeah you can see that that's just like a delta function so so there's the disconnected piece but this is the piece which gives you the interacting part of the so this so this is what makes it very natural to try to look at the so it's this is so all the all these properties are very reminiscent of what you would expect for a string scattering amplitude as we have been discussing and so the consistency conditions for for these are the bootstrap conditions are like the conditions you would impose for a consistent string scattering amplitude yeah yeah that's what the additional poles will give you yeah you will you will have the ones with different twists as well including the different symmetry so I'll talk about the the states intermediate states with spin but those those will be included as well so finally the last nice property which I think makes this thing nice is that so as I said they are all very reminiscent of string scattering amplitude but in fact in sort of a large radius limit of ADS amplitudes they go over to flat space scattering amplitudes and and in fact the PI dot these do become the SIJ do go over to sort of the Mandelstam in variance in flat space up to a sort of a proportionality factor so so there it's so there you can view these as on shell this becomes indeed an on shell condition and in the language of the flat space scattering and the and this and the M of ST is essentially up to a sort of a convolution it is the flat space scattering amplitude as was shown by well this was something I haven't been giving references very well but after the original paper of Mac, Penedonis was the person who sort of tried to use this for and conjectured a precise form for taking this limit and later I think it was sort of put on a firmer footing by Fitzpatrick and Kaplan so yeah I mean these don't care so much about the signature because these Mandelstam in variance you can view as complex functions which you can rotate from Euclidean signature to to Minkowski signature I mean the the amplitudes are analytic functions of of this of these Mandelstam in variance like the flat space scattering amplitudes are so so it doesn't care so much about whether it's Euclidean or Minkowski yeah so you're generally considering these as generally complex numbers which you can go to a Euclidean or Minkowski and sort of signature and then you can take Sij to obey the appropriate so there'll be some physical region of as it stands is just a complex number Sij and just represent so PIs are some complex momenta if you wish but there'll be some in in Minkowski space there's some triangle or something in the Mandelstam in variance space there's a physical region which is the sort of for physical momenta but but the amplitudes are in variant and you can continue them beyond those to unphysical momenta Euclidean momenta any general complex momenta so all these features how many minutes do I have okay I will just so I'll just okay I'll just say a couple of words and then stop so all these properties essentially follow from the OPE from the OPE it's factorization and so I'll yeah so let me just maybe make so if I have I'll continue later but you have in the four point function we know that we can split it into I mean we can write it in in terms of so now we look at only the single trace operators here and we will have the OPE generally containing terms so this is X1 2 you can have some number of derivatives so and so the general form of the OPE when you have some spin L operator you get some some pieces over in the numerator which are like that and then the denominator comes with a power which is something like delta 1 plus delta 2 minus delta divided plus 2n plus L 2n is the number of these scalar descendants L is the number of these spin so basically I mean I'll I'll elaborate next time more but you can see already that this scaling behavior by our general these things are on the melaintransform picking out the appropriate power loss you can see that it will pick out a pole and in fact this delta 1 plus delta 2 in the S channel is already taken into account in our definition of S12 so the S variable will have a pole at S equal to delta plus 2n minus L and so this is what is the twist plus 2n so the twist is dimension minus the spin so you can already see that there will be poles that will come from so basically the fact that there are meromorphic functions will follow from the fact that this OPE is a convergent series which and then each of the terms will give rise to poles and the poles will have values which will be these powers which correspond to the twist plus and there's a whole series n goes from zero to infinity so there will be a whole series which corresponds to the descendants as well and the residues will be related to the lower point functions as I said over here and so on so I'll elaborate on this a little bit more next time but basically these properties all follow from the fact that there is a convergent OPE in the CFT which has all these properties so it's a so this MFST captures these things in this nice way instead of the more sort of the messy OPE expression this and this is a more efficient way of capturing the same information okay let me stop here thanks