 Okay, so I'm glad that all of you are hanging in there just a few hours to go but see your enthusiasm is high as ever, so it's a good sign. So last time so we introduced these Mellon amplitudes and in particular illustrated it for the four-point function and I listed a set of properties, the nice properties. So nice things about Mellon, I'll just sort of very telegraphically list them again and we'll see. So firstly was the fact that they are metamorphic so and then the poles corresponded to the dimensions of operators actually something that's called twist which I'll define, mention the residues are factorized lower point functions so the metamorphic behavior so this in fact there were no branch cuts as I was saying. Then there's channel duality so this is the statement about for instance S and T interchange that you can factorize things in different channels. Then the other nice thing was that for large N CFTs double trace or multi-trace operators automatically included in the extra additional sort of gamma functions that were in the definition, the gamma functions of the definition and finally the flat space limit more uniform in terms of these variables. So these were the various nice features so this in particular these Sij literally went to actual PI.pj so real moment are not the real moment that you label flat space scattering amplitudes by not just fictitious moment that we introduced but of course it's sort of the fact that we were writing Sij in this form is inspired by this fact. Okay so any questions about things we had seen so far? So now let me try to motivate some of these I just listed these as I stated them but actually these follow from the OPE so all these nice features essentially follow from the OPE and let me try to motivate that and the mode I'll motivate it mostly so what I say here is sort of true for general Mellon amplitude for an endpoint function but I'll mostly focus on the four-point function as I said. So if you are looking at let's say the original four-point function and so we let's say take the S channel OPE which means basically you are sort of bringing together things one and two together three and four so this is there's not a Feynman diagram this is just a mnemonic for the order in which you are taking the OPE you're bringing the points one and two together you're bringing the points three and four together it's just to sort of show that so this is what you call an S channel OPE because that's the order in which you are taking things so you could have of course taken one and three together and two and four together that would be that would correspond to the T channel OPE and similarly there's a U channel OPE but let's focus on one channel so so O delta one of X1 O delta two of X2 the OPE so the OPE is of course an expansion an operator identity that holds in any correlator and the nice thing about CFT's is that this expansion is a convergent expansion and so we can expand it in set of operators and so the coefficients so the expansion you would have in general some spin L operator can enter into into the expansion basically even if the external operators are scalars because they are not sort of coincident even you expand it you can have all kinds of spin orbital momentum angular momentum if you wish so so you can have operators with spin L so that's what these indices are so you're bringing X1 close to X2 so so of course if you have some spin L operator you need to have since our left hand side is a scalar you need to have the tensorial contraction can only be with powers of X1 2 with factors of X1 2 carrying these tensor indices so so if we if so the operator O itself can have any dimension so let's say let me actually denote it slightly differently so you can have a primary operator O dimension delta but you can also have descendants which involve scalar del square to the n acting on those primaries sorry slightly bigger so you can have C1 2 is a so called OP coefficient it's labeled by the dimension and this the number of derivatives which is a measure of the descendants remember the momentum acts on a primary and generates descendant operators so you can have del square to the n and then you can the operator can carry some spin and and these are the these are just the positions at the separations X1 X1 2 which you need L factors of these to soak up those indices and then by purely by dimensional analysis we need to have so we need to have some number of powers of X1 2 square and you can see what so C1 2 are dimensionless coefficients they just measure the sort of effectively from the 3-point function of of this operate these three operators as I as you've seen the 3-point function is characterized by a number in a CFT so this is that number and so so the X1 2 square by dimensional analysis comes with a power so the left-hand side has a scaling dimension of delta 1 plus delta 2 under the overall scaling transformation so the right-hand side must have the same as well this operator has dimension delta and then there are additional factors of 2n and finally the X1 2 so so if you just match the the scaling behavior what you find is that you have delta 1 plus delta 2 plus minus delta plus minus 2n plus L so you see this has dimension delta so when I when we say scaling it means so this has this goes like lambda to the power minus delta 1 plus delta 2 so or if you wish the mass dimension is delta 1 plus delta 2 so here you this factor takes into account that but then there are you need X1 2 to the L to compensate for this you need to the 2n to compensate for this and the delta for this so that's where these powers come so so that's the that's the sort of the general form of the expansion and and remember what we discuss the right at the beginning when we introduce the melon transform the melon the nice thing about the melon transform is that it picks out the powers and the powers that in a power law expansion it's precisely the poles of the melon amplitude pick out the different scaling powers so so if you compare with the definition that I gave for M of ST so M of ST remember there was it was something like dt ds u to the s over 2 v to the minus s plus t over 2 and then various gamma functions that was the thing where u was essentially x12 square x34 square divided by some denominator but you see u is the thing that's going to zero in this s channel expansion when x1 is going to x2 so we'll so we see that the powers of u are essentially the powers of x12 square that you so in the the s variable will have poles so you'll get a sum of contributions so when you do the sort of M of ST was was here and this was the the position space amplitude so to to get the the power law behavior of that there was all also this additional factors of x12 square and all other there was some these delta ij factors I'm not going to write them all out but I know so a of uv was after we stripped off all that that's that's the part after the that's the one that depends purely on the cross ratio so that there's no such factor but if you compare this with the with the pole behavior or with the power law behavior over here that would be reproduced if M of ST would would have poles in s which are at these values and so the delta one plus delta two if you go back to your notes from yesterday you'll see that this piece was just a trivial overall piece which was which was there in the amplitude the overall definition of the amplitude so the non-trivial piece which depends on delta and n the particular pieces is this so you so you get pole at each of these values corresponding to operators that are present in the OPE of delta one and delta two and the and the and the residue will be of course I will say something about the residue later but right now it will be some function of t which will generally depend on n so you can sort of write the behavior of M of ST in this way so so each of these terms in the OPE so this is a discrete sum so that's why you get sort of a discrete sum of poles so that's why it's meromorphic so and some of simple poles and in fact you they are all simple poles there you can't have double poles because if you had double poles if in this expansion you would get log terms in so you you can see that if I had a piece which was one over s minus some constant square and then you were to do this u integral I mean the s integral then at the double pole you will pick up terms which involve u to some power times a log piece just from the the usual thing about the double pole when you evaluate the sort of the Cauchy integral so so you get some of simple poles so that's sort of the first thing there are no branch cuts because no branch cuts because the spectrum is discrete operators delta n one second is discrete so in the OPE you don't have a continuum of operators appearing you get a discrete set of operators so so there's no branch cut you would get a branch cut only if there were a continuum in the sum yeah only if the poles are sort of accumulating but here they're always discrete so at least okay I don't know if there's any proof of the statement but I think in any quantum field theory you don't have accumulation point of dimensions of operators no I think in a general so you get of course infinite cities I mean there's of course these which are discreetly spaced by one unit but even these the delta in a CFT in dimension greater than two in two dimensions of course you have continuous dimension spectrum of dimensions but in higher than two dimensions there's typically there's no accumulation point I think of the spectrum is always sort of the sum is infinite there are infinite number of delta but there's no branch cut that's all I'm saying you can have an infinite set of sequence of poles but okay well then maybe meromorphic is not the right word like a gamma function would you call it meromorphic or not I think it's sort of like a gamma function it has a set of poles of course gamma function has some exponential singularities which this doesn't have but yeah it's in the same sense so maybe a holomorphic with infinite sequence of poles it's also true that if you want to I said over here the flat space limit is more uniform in flat space when you take the limit you get branch cuts so flat space asymmetries of course have branch cuts and that you can see when you go to very you have to take a certain scaling limit and then the poles start sort of coalescing and then you form you get branch cuts so in a if you try to take the flat space limit that's a somewhat singular limit in which in which many of these poles can coalesce to form a branch cut but away from that it's sort of just a set of simple poles okay so so it's meromorphic there are no branch cuts because the spectrum is discrete and then okay so the poles so the location of the poles poles at not quite what I wrote dimension that's why I wanted to qualify it by this thing called twist so in the dual theory yeah in a ADS when in a large n large coupling theory which has an ADS gravity limit and you take the flat space limit then yeah that's what I mean flat space limit is more uniform as what I said in this in these variables meaning the this melon amplitude goes over to the flat space S matrix in the by taking an appropriate scaling limit it's more uniform there's a simple transform between the two so so the location of the poles by this argument by just this scaling behavior are at at these values and this delta minus L which is the dimension minus the spin the dimension of the primary minus its spin this is sometimes called the twist this is a old notation I think going back to gross and Tremon but someone some young student asked me today is it anything to do with twist topological twisting so I think it's a matter of how far our field has gone that when you think of twist you think of topological twisting or I don't know twisted boundary conditions or something like that but this twist has nothing to do with that it's just I don't know exactly then the distinct of the origin of the term but I think it was gross and Tremon who first introduced this in the context of deep and elastic scattering in QCD and so and so the light cone if you consider the light cone OPE in a general quantum field theory it's organized in terms of operators of twist higher and higher twist so sorry the gamma functions will have poles so those are the less interesting poles at least for larger and CFTs that's what I was saying over here that those I'll come to that when I come to point number five and the multi-trace operators those will just correspond to multi-trace operators so those poles will be so these are if you wish the poles of the single trace so we'll in this expansion there are these plus multi-trace okay let me make a distinction so these are these are single trace operators and then there can be multi-trace operators multi-trace I mean in the OPE of O delta 1 O delta 2 I will have things like O delta 1 many derivatives O delta 2 that is also going to be present in the OPE because it just has the same quantum numbers and everything so I so this operator it's a it's built out of two of these single trace operators single trace primaries but it's what is called the double trace operators so so let's say in this particular case for a four point function essentially double trace operators like this can also appear but I will come as I said those will be taken into account at least for large and CFTs by by by the additional gamma functions that are there over here so that's why that's the nice thing about this melon amplitude that this piece captures the single trace the contribution of the single trace operators which for a string theory are that correspond to the single particle states so the you don't have to worry about the contamination between single and multi-particle states at least in the perturbative string theory or a large end CFT not I don't think in a general abstract CFT you have a flat space limit it's firstly it has to be a large end CFT but let's suppose that those conditions are met that there's some factorization and so on but I had the flat space limit basically exists if the dual ADS has a radius which is adjustable parameter so that you can consider particles which are much more energetic than the ADS scales so that they are they are insensitive to the ADS curvature so effectively you need a toft coupling which is large so you need a parameter one parameter family of CFTs in which you can take a strong coupling limit I think only for so these are so all so if you wish the properties listed here are sort of more general but here you need large N here you need large N plus large toft coupling in the cases where you have these then you can take in these cases you can take a flat space limit and and then what I said there is true yes the poles of course yeah the value of the poles depend on the on the coupling because the delta in an interacting theory depends on the coupling so we are to go be near flat space you're taking that coupling very large and so yeah there will be a dependence on the coupling and so the location of the poles will will be sensitive to that and that's why as I was answering earlier you can have the phenomenon in the strong coupling limit that many of the poles start coming together and that's when you can form a branch cut because the dimensions are sensitive and in the strong coupling limit you can sort of have a coalescence but that's in a sort of a singular limit or an extreme limit that happens for finite coupling typically discrete okay so so the poles are at these values so it depends on the twist and this is the descendants contribution of the descendants and so there's an infinite sequence of poles for each delta there's a infinite sequence it may of course happen that the residue vanishes or something and in some of the DS computations that happens but typically that requires delta to be an integer or something like that any case so so infinite number of poles so so there are descendant poles and labeled by n and for each primary delta a single trace primary that's what so you have a and in fact the residues I'll say something about the residues soon but so this is about the poles I told you about these two so let me say something about the residues so so I took over here 0102 and I wrote down this expansion if I take 03 and 04 I'll have a similar expansion and then so you can imagine another sequence of operators but then it reduce then this four-point function reduces to a two-point function of the two sets of operators and now you normally choose the basis of your primary such that and the two-point function is diagonal and so so there'll be a non-zero contribution only when you get when the same operators appear on from from the two OPs so so they there will be so the the factor is it so the OPE will sort of OPE factorizes into something like C12 delta n so you get so that I should say the four-point function factorizes factorizes into something like this and then some contribution from the and then roughly speaking it's the O delta n two-point function and so this is just a schematic thing and then there'll be the corresponding x12 square to some power x34 square to some power which is the piece that gives you the cross ratio so so the so the full four-point function will factorize because of the OPE into something like this so that will translate into the statement about so in addition to the poles and we can talk about so the poles come from this power law behavior that's what it picks out but this numerator piece is what will go into the residue and from this we can say something about the that residue which I just state but before that there was a question yeah it's a matter of choice what you call it yeah I've there can be descendants which are just purely derivatives but I've sort of included them here and so you can have del mu 1 del mu 2 of O mu 1 mu 2 mu L minus 2 things like that but I've sort of included them over here but yeah you could consider descendants having spin but but it doesn't affect anything that I've said over here yeah so so then of course you it will depend on what you call us delta so so in this difference they cancel off okay so so because of this OPE so yeah by the way yeah since you people I've had a course in in in the bootstrap let me write it in a way which is probably more familiar to you you've seen this conformal block decomposition so this is another way to to write this so here so this is the these are the conformal blocks which include the sum so here there's a sum only over delta this includes all the sum over n piece so you can split up this delta n sum into a sum over delta the sum over n and essentially because the contribution of the descendants is determined by that of the primary you can sort of pull out the primary contribution and the three point function out and then there is this kinematic piece so this remember is kinematic it just depends on the conformal symmetry so that's another way to to say this so the main point is that you can factorize the OP a four point function using the OPE into sort of three point functions two sets of three point functions and and this translates so this factorization in melons space this translates in melons space to the fact that the M of ST we can write as there will be these numbers c12 delta for for all these and I'll write it particular way so by the way this twist is sometimes denoted by tau so so you can write it in so this part is just the familiar part that is coming from here but these these conformal blocks the G delta L of UV has an expansion in in in terms of in powers of u something like u to the delta by 2 or more generally u to the tau delta by 2 times a power series which comes from the descendants and so so basically this so I talked about the U piece the U piece the picking up the scaling then you gave you these poles but then there's the t integral and that's essentially coming from the functional dependence on B which translates into a certain set and so these these are no remember this is a kinematical thing this is what Slava Ritchkov probably told you about these are known polynomials known functions and you can make an expansion in in terms of a known set of functions like this and this this GN of V if you look at its corresponding melons transform there are a set of polynomials so these are actually some set of orthogonal polynomials which start with t to the j plus lower order terms so it's a polynomial of degree j but of course it's there's a family of them labeled by n so for different n there so when n is equal to 0 the simplest case of the primary they they are what are called Han polynomials any case there are things that mathematicians have studied a bit and so there are a fairly explicit set of polynomials and the nice thing is that they sort of start with t to the spin oh this j is basically the L yeah so it's j is the L of the so this is t to the L so it's the spin of the of the particle of the operator that is being exchanged so this is why so you have a delta and L being exchanged in the 1 2 3 4 in this channel and so you get so this OPE in this S channel in this S channel can be written as a sum like this this is very much like what you would do in momentum space so if you had a momentum space amplitude and like you heard from Nima in the morning when you go sort of near a pole of an intermediate state so the S channel momentum which is p1 plus p2 square approaches some physical value then you get a pole and the residue of the pole factorizes exactly in like in momentum space you see here the factorization and in fact the this polynomial is the analog of the Legendre polynomials pl of t in flat space scattering of flat space scattering amplitudes so there if you had an intermediate particle of spin L you would again get a polynomial in the residue now of degree L in the t variable and in that particular case again it's kinematic and it is just a Legendre polynomial but here it's a more it's a different set of polynomials which again in a appropriate flat space limit go over to the Legendre polynomials and so and so it's a sort of a deformation away from the flat space case but this is to show that the structure is very parallel to what you have in the flat space except that it's in terms of not actual momentum or momentum space but in terms of this S and T variables which are playing the same role okay so any questions yeah you will so this will be at say x2 this will be at x3 is that what you mean no no sorry what no so you you for the the two-point function has to be diagonal in the the two-point function has to be diagonal in the in the operator so it's sort of a basis that you have chosen so such that it is it's it's purely diagonal so that is something the conformal invariance that requires so so any case coming to coming to so this is the statement about the residues factorizing into lower point correlators in this particular case it is just the three-point so these are the three-point functions of course so so it factorizes into three-point functions so this side this side and then there's a sort of a propagator from here which is which is this piece so that's as I said very much like in momentum space more generally if I considered a melon endpoint amplitude just like in momentum space also you have a similar factorization into sort of n minus k or n minus k plus one points and k plus one points like Neemar again drew the same sort of diagram you you have many things that sort of factorizes into the sort of diagram that he was drawing so so you you will have this general factorization for general melon endpoint function okay so so that's about factorization and then channel duality in a sense is quite trivial because it's it's just the S and T exchange when you and so here I did the OPE in this way but the OPE is associative and so whether I expand it in the S channel like this I could have instead done OPE in this channel so that would just correspond to exchanging another set of operators here so the statement about the associativity of the OPE meaning that you can do it in either order and the final answer is the same is the statement that this M of sd you can expand it alternatively you can expand it in terms of poles in the T channel so there'll be again a set of poles those will correspond to the operators that exist in the OPE of Delta 1 with Delta 3 and so that so here you have one three two four and there'll be some other set of maybe Delta prime or an L exchanged over there but the whole amplitude especially when you have supposing the external particle external operators were the same then there will have to be a symmetry under the exchange of T and S and more generally S, T and U so so that is the statement of channel duality which is quite simple okay so so then the additional points which are to do with large NCFTs and I'll just say just a couple of words if you remember if you go back to the expression I wrote down involve so these gamma functions let me just write down some of them in terms of this S and T variable we had Delta 1 plus Delta 2 minus S by 2 we had gamma of Delta 3 plus Delta 4 minus S by 2 and similar ones with T and so in the T variable it was something like Delta 1 plus Delta 3 minus T by 2 gamma of Delta 2 plus Delta 4 minus T by 2 and then there was there were two more gamma functions involving effectively S plus T but what I want to point out is that as someone said that there are poles here as well so these the additional the extra gamma functions have poles at for instance Delta 1 plus Delta 2 minus S by 2 equal to n which is a positive non this equal to minus n with n greater than equal to 0 so in other words S is equal to Delta 1 plus Delta 2 plus 2n with n greater than equal to 0 and so there are poles at this value then the other gamma function are also at S equal to Delta 3 plus Delta 4 plus 2n and then in the T channel you'll get T channel you get T equal to Delta 1 plus Delta 3 plus 2n etc so there are all these additional poles in these extra gamma function factors which were sort of introduced as part of the definition but we see that these are very useful to sort of pull out these these additional poles because firstly they they they they are exactly in sort of agreement with the kind of poles you would expect for these double trace operators because by the same sort of argument they in the OPE they are definitely there and they will have dimensions Delta 1 plus Delta 2 plus 2n you can have again derivatives over here and and the nice thing is so this will be their dimensions in a large n cn in a large n CFT or Delta 1 let's say del square n or Delta 2 have dimensions Delta 1 plus Delta 2 plus 2n plus 1 over n corrections so the so the so that's the statement about the large n theory that these operators the anomalous dimensions are suppressed so they so so we see that this are the right candidates and in fact their OPE coefficients are also determined in terms of they're sort of they are fixed so their OPE coefficients again at large n are also determined in a way kinematically I should say because it's sort of the and we are considering O1 O2 and then this sort of O1 delta the 2n O2 and and and they are just basically determined by so the three point function of this is basically set of two point function so so the OPE so that's what is reflected in the gamma functions in the gamma functions the poles all have the same residue that's one of the things you learn about the gamma function right and the gamma function gamma of x is basically something like 1 over x plus n and greater than or equal to 0 plus maybe whatever so there's a the the poles all have the same residue and that's consistent in the large n with the fact that that basically you will get the same they're all all the residues are simply proportional to each other and so so this gamma functions play a very nice role as I said you can sort of forget about by including them in this definition when you look at MFST you need to only focus on the single trace operators which is very nice when you're trying to make a connection with the dual string theory because then you can try to connect it with the single string scattering amplitudes so okay so I think I'm quite sort of behind my this thing so what I want to do so these were about the flat space limit well it's probably I'll refer you to the paper of Benidonis in 2008 and where he sort of talked about the flat space limit I won't I won't say more about that now but it's I think one of the nice features also of the Mellon amplitude so what I want to do next is to just show you a contrast couple of sample computations in in perturbative CFTs and in ADS and that's sort of similar different in some ways but so I'll do that probably I'll start on that next time but we just make some so so all these were sort of general facts that I told you it would be nice to see how these amplitudes actually look in specific cases so so there are two specific cases I will say I won't be able to describe in detail but I'll in the last lecture I'll just show you some of the explicit results and so the other one is in ADS gravity so these are if you wish the weak coupling and this is sort of the strong coupling CFTs and so so the the amplitudes will look they of course the weak coupling one you would try to analyze by sort of Feynman diagram like techniques and this one the strong coupling you would look at the Feynman diagrams in ADS which are what are called these Witten diagrams and so so both of these limits you can try to explicitly look at a set of diagrams involving the four-point function so in a perturbative CFT we'll look at some tree diagrams like this and so in this particular case and and this one we look at very similar diagrams but in ADS so so this is the difference between Witten diagrams and Feynman diagrams you just draw a circle around one of them that's the basic difference though this is so this is an ADS this is the boundary of ADS and so that's so we look at this sample sort of four point functions and and just see how you compute them and how they illustrate some of the general features that we have seen so far okay so let me just stop