 Well so I'm going to talk about an aspect of an algebraic aspect of perturbative quantum field theory that's probably beneath the radar for many of you because the kind of quantum field theory that this talk is about is the tree level gauge theory amplitudes and the kind of relevant algebraic structures are quite elementary. They're just leap polynomials and cubic trees but it's so nice, the proof in this talk and somehow the subject of this talk which has been explored only really at tree level doesn't really have any nice simple proof in it so I think it's still worth understanding and some of the things in this talk I think are robust enough that they can help point the way to more interesting things at loop level. So the subject of the talk goes back to the KLT relation in the 1980s which related closed string integrals to sums of disc integrals. So this calligraphic A here is a Veneziano amplitude which would normally be written as some sort of disc integral you know z1 less than z2 etc from 0 to 1 of a multi-valued function zij to the minus sij. If you haven't seen string theory before then don't worry but there you go and then immediately after that people realized that a relation of this kind between a closed string amplitude and two to disc integrals like this implied by taking the so-called field theory limit of this string theory. Excuse me, I think no one sees the paper. If you want people to see the paper you're writing on you need to enable the sound with this device so it will be opine. I think we got it sorted sorry I think it's good now. I think I just managed to pin it for everyone sorry for this delay on my part but I understand now everybody can see it clearly. Sorry. I've also changed my microphone and I hope that this Yeah everybody said they can see the paper now so I think we're good sorry for that. Is it fine now? Yes yes it is. Good okay. So all I was saying is that in the 80s string theorists wrote down they said that a relation like this existed amazingly nobody wrote down a formula so this is a relation which suggests that there's an inner product between the disc integrals that gives you this closed string integral which is a very different integral and nobody wrote down a formula for this SAB until 10 years ago and nobody really proved that this was correct until very recently. In any case a relation of this kind at the string theory implies a relation of the following kinds for field theory tree amplitudes. So if Mn here is the gravity n-point tree amplitude then you expect a relation of this form where a Yang-Mills here is say the partial amplitude of Yang-Mills and I'm going to if you don't remember these sorts of physics terms then also don't worry because everything we need I will bring up and do course. So in 2011 various people Byron Ball, Dan Gard, other people worked out a formula for SAB here which is in terms of the which is a function with a certain weight of these variables Sij which are the Mandelstam variables that arise when you compute these abstractions. So the whole point of this talk is to give a very elementary proof of this formula for SAB which occurs in this so-called double copy relation because there's no this is a reasonably important formula but nobody had really written down a direct path to to give you this straight from field theory considerations. So the the plan at the top then is I'm just going to tell you remind you what a partial amplitude is and then remind you of a few very very old you know 50, 60 year old facts about the polynomials and then I will explain the proof. So hopefully I don't labour the point of this slide too much you can compute Yang-Mills using Feynman rules when you do that you write down Feynman graphs and associated to each graph you you have a product of vertex factors and the only reason I wrote this down is to remind you what the vertex factors look like. So you know the trivalent vertex factor is associated to an FABC which is a structure constant of a Lie algebra usually SUM and these lambda a are a basis of the Lie algebra and there's some invariant traits on the Lie algebra and then likewise the quartic vertex in Yang-Mills is a sum of three terms and each term is the product of two structure constants which you can get by taking a trace of a Lie bracketing of some elements in the Lie algebra with some fourth elements of the Lie algebra here and I've drawn the associated trees so I can think of a Lie bracket bracketing like this or an eminomial like this as a tree and so I've drawn where you know each bracket is represented in the tree as a vertex coming together you see okay so the point of this slide is to remind you that if you were going to be you know computing a Yang-Mills amplitude for the first time at an issue what you would actually have to do is write down a lot of terms and each term it corresponds to a cubic tree in some way you first write down the all the Feynman diagrams some of which are not cubic trees because they have quartic vertices but in the ends you get an expression that looks like this where this sum is over cubic trees which I'll say something about the second and another point to emphasize is that for the tree amplitude of almost any gauge theory you're you're going to end up with an expansion of this form so what I'm saying is not specific to Yang-Mills if I start with some other gauge theory then I and I take the Feynman rules and I expand the conventional expression then it reorganizes in this form where C gamma here is a color factor associated to a Lie polynomial in the obvious way so if I have a Lie polynomial one bracket two bracket three like this then I associate a color factor C gamma by taking the Lie polynomial the Lie bracketing of the associated elements of the Lie algebra and tracing it with some fourth fixed element now I can think of this as a sum of the cubic trees because each Lie polynomial corresponds to a cubic tree one subtlety is that you know the Lie polynomial corresponds to the cubic tree or the binary tree only up to sine and so but the sines are important for me because these C gammas you know C of minus gamma is equal to minus C gamma so what I mean by this sum is that for each cubic tree or for each binary tree I choose one of the two Lie monomials gamma or minus gamma and and put it in here now this is not conventionally how physicists starting with the hadronic physics revolution of the 1960s and then the birth of strength there in 1970s this is not how physicists write their amplitudes the way they write their amplitudes is as a sum over permutations so they write them as sum over a where a is a permutation of one through to n minus one and the this coefficient here a a is related to that coefficient there by a sum a gamma C gamma over all cubic trees where this a gamma here is plus or minus one and we'll there's a natural inner product between permutations and and and the monomials as as I'll explain a second but the point of saying all this is that a a here is what is called the partial amplitude and it's those a of a's which which appear in the in the KLT field theory KLT relation so to explain how the properties of these partial amplitudes I'm just going to say a few things about Lie polynomials these are such well-known facts that hopefully we'll have come across them at some point in in their various guises so the the free Lie algebra which I'll write Lie of a I can think of as being a subspace of the free associative algebra if I think of the associative algebra not as an algebra but as a vector space and the the elements of of Lie of a are the primitives in the free associative algebra with respect to the shuffle code product so that that anyways the is probably familiar to most of you in some form or another what this means is that well I still haven't defined this inner product that I'm using but there's a an obvious inner product on on say on the free associative algebra where if I have two words a b in the free monoid on a start on a a some set then the inner product is is one if a equals b or zero otherwise and so what what the statement that what this statement means is that if I have some gamma which is a Lie polynomial then for any two non-empty words a and b this inner product has to be zero so that's that's probably called Ries theorem for many people and what I'm going to do now is restrict to the setting where restrict to the subsets la or subspace la inside Lie which I'll call the multi-linear Lie polynomials and what I mean by a multi-linear Lie polynomial is just it's a Lie polynomial in which for for each it's a Lie polynomial polynomial gamma in which no word has repeated letters so the freely algebra has you know the freely algebra on on some some integers one two three treating the integers as just letters contains things with repeat Lie polynomials with repeated letters like this but that's certainly not what I want because I want to to be using the correspondence with trees where all of the leaves have distinct labels that that's all I that's why I'm just introducing this notation elevate there's there's nothing tricky there and so then in the context in that context Ries theorem if I think of elevators vector space says that elevators the orthogonal space of of the shuffle subspace shuffle inside w where w is the free vector space spanned by the words with no repeating letters and the shuffle subspace is spanned by the expressions a shuffle b where a and b are not the empty word and so the dual statement to to to this is to say that the the dual the vector space dual of elevate is the quotient w by by shuffles and this explains some of the basic facts that are all throughout the physical literature but they're never really prove that sort of their word their sentence sentences which are given to explain some of these things but Ries theorem implies a number of well-known identities that were were conjectured in the 80s so the the partial amplitude of of any any gauge theory as I said before can be written in this form times where the sum here is over some the polynomials gamma these are some coefficients that come from the specific gauge theory that you are studying and it immediately follows from from Ries theorem that for example a the partial amplitude satisfies this identity you see because because if I put a total shuffle here a shuffle b into this formula then by Ries theorem a shuffle b with gamma zero another identity which let's see so this this as a special case has something called the u1 decupting identity where i is a letter and another another conjectured identity is called the class queer identity which which also follows in a in a nice simple way from from Ries theorem and and a statement about bases for the freely algebra so you you might have come across an alternative characterization of the freely algebra as the image of the of the full bracketing map l so l takes any words and and gives you back the limanomial which is a full bracketing a sort of left justified full bracketing of that word but the and and these these monomials are not every limanomial but the dink and stack waiver theorem implies that these monomials span the freely algebra or in my restricted case this this multi-linear component but the the monomials la where a is a word do not form a basis they're not linearly independent because for example the jacobi identity means that you know that that's zero but you can show that a basis for out of a is given by those monomials full bracketing of a where the word a begins with its smallest letter with respect to some ordering some total ordering of a that you can choose and then dual to that is the basis where I just choose the words a where again a begins on on the smallest letter and and it turns out that these are dual bases with respect to the pairing I described earlier and then the the Christquiff relation follows follows from that because there's a nice explicit formula for for the comb operation l of a word if i is the first letter of that word then the full bracketing of that can be written as a sum over bc of b tilde ic where b tilde is minus one and the length of b times the reverse of b so this is the antipode operation on the hop out the hop out group words and then with respect to the pairing the the adjoint of of of this formula tells you that the adjoint of l so the adjoint of l star is explicitly given by a sum that looks like like this and this is tantamount to an identity which was widely used in physics without apparently being proved which is that the partial amplitude b ic is equal to uh minus one to the length of b partial aptitude i uh b b reverse shuffle c so this is some sum of partial aptitudes on the right hand side and you can combine this basis statement with this nice formula for for the full bracketing to to give you this sort of identity um so uh all of that is is is supposed to be just saying uh reminding you of of some classical facts um and explaining that they have an elementary um that they were known by physicists in a very elementary form uh and the reason they the reason that physicists knew about these identities even if they wouldn't know really how to prove them is that uh the physicist is constantly computing with these color factors uh which involve totally practice um and so so now i'm going to uh explain how you can um in a very elementary way just find the the KLT matrix that um uh was was first uh written down at least 10 years ago um um and uh to do that uh uh uh just let um really just a tiny bit more uh notation just so um for for a subset i uh let little s sub i be the mandelstam variable associated to i um and so in uh you know for mandelstam what that meant was that uh uh s sub i was the the rents um norm of the vector uh that you get by summing over the momenta uh case of i inside that subset and so it's clear from that definition that s sub i has an expansion like this um where uh we sum over all pairs ij which which appear in i um and i'm assuming here that the the vectors the momenta ki are uh individually null um and then the kinds of functions that so functions of mandelstams arise in in amplitude computation but the kind of functions that arise are um are the form uh the kinds of functions you expect are in the uh the rom ring with s i uh sub i as the as the variables uh but modulo this relation here um so you don't expect to see functions like one over s12 plus s23 um because s12 plus s23 is not a um uh can't be written as some momentum uh k squared uh that you do expect to see functions uh where you have um one over s123 for example uh and these satisfy uh these relations and in particular the the kinds of functions that we really expect to see are related to to cubic graphs um and so right for for some uh limonomial gamma uh write s gamma for the product of the the s i um where you take each each i which arises as an edge of the corresponding tree so for example um you know one two three is is a is a limon limonomial um um uh actually it's a sort of a too boring one um this is a slightly less boring one uh and the associated tree would be this and you see that there are um there's one two internal edges and i'm going to count the root of the tree also as an internal edge um and the associated subsets to for the for these internal edges are um you know two three for that edge one two three for that edge and then uh one two three four for the root and so associated to this limonomial is the uh is the Mandelstam expression s23 s123 s1234 um and this would be the the product of of of propagator factors that arise if you were computing the the conventional Feynman contribution um from a cubic graph like this uh obviously the so the the propagator contribution would look like one over s gamma and you would add back some i epsilon so so um and then the uh so with that with that notation if the s gamma understood the um klt relation is actually a statement about uh this object here which i'm calling t uh and um which i saw in a talk that Mikhail Kopranov gave uh which he didn't prepare but which he uh he for i don't know how it happened but he stumbled on a room of physicists and had a conversation with a few of them and um and gave a talk on the back of it um and this this object t here is if you like a sort of prototype um of the gravity tree amplitude it might not look like the gravity tree amplitude but the reason it's a prototype is because it has all this it has the same uh pole structure you see because this is a sum over cubic trees again i take each tree each tree i choose either the limonomial gamma or the limonomial minus gamma um and so i i have and these uh one over s gamma factors uh contain all the singularities of the of of a gravity tree amplitude it's just this this numerator uh which doesn't look like the tree amplitude um and so uh because this is some objects then in la tensor la with values in the ring of Mandelstams um but it turns out that there's a um there's some homomorphism from la to uh a space of appropriate functions of of um of Lorentz invariance uh such that n tensor n acts on t to give the gravity amplitude um this is something that is not proved to my satisfaction but is proved to the satisfaction of a number of uh uh so to speak friends of mine and if even even if it's proved or even if they think it's proved i think uh uh everyone agrees that somehow the that map is not very well understood um but this anyway is the prototype of the um uh gravity amplitude and then the prototype of the yang millers amplitude or or the partial amplitudes for any gauge theory um are these t of a here because i can think of t as a map from the uh the dual of the lee lee monomial space elevator uh to elevate um just by taking the inner product with one of these factors uh of t um and so uh yes so so i i obtained this map from h t of a this has the prototype of um a prototype of the partial amplitude uh and the big claim is that t is invertible and the inverse of t is essentially um tantamount to the k l t matrix or in other words tantamount to the k l t relation um and so in order to see that t is invertible uh there's um a really charmingly um simple um uh thing that you can you can show which is that there's a very natural looking lee bracket on the uh dual of the um the dual vector space of the multi-linear the algebra so uh here i'm suppressing that i what i really mean is elevator dual tensor the ring of mandel slums um and there's a lee bracket uh on elevator dual and you define it inductively in the following way so if i have less is i and j then the lee bracket of uh then the the braces of i and j is s i j times i j uh and that might not look symmetric uh but remember that um uh l la dual here is um is is this quotient space by shuffles so in particular i j is equal to minus j i um in in all of a dual so so this is some anti-symmetric bracket uh and you inductively define it uh in this way um and you have a so so here if you have a letter two letters i and j and two words a and b uh then you can expand it like this and there's an analogous relation to b um and so an example would be for a word a uh the curly bracket of of a with i is this sum over the concatenations of the word a where s i c here is the sum of s i j where j is a letter that appears in the word c um and so it's not obvious that this is a lee bracket it's not obvious that this is a bracket you even uh care about yet uh but the the reason you care about it is that it satisfies this very nice property uh which i've put in a box so t of braces a b is the ordinary lee bracket of a and b where t is the is the map i defined on the previous slide uh which goes uh which which is the prototype of of partial gauge theory amplitudes uh in particular i can nest this relation so t of curly one two curly two three sorry three four uh is the lee bracket one two is the limanomial one two three four uh and more generally i can write uh if gamma is a limanomial i can write braces gamma for the uh expression in the uh in in the dual which i obtain by writing gamma as a nested bracketing of commutators and then replacing every commutator by a uh a pair of braces braces instead uh so for example uh going from here to here i can go from there to there by replacing every set of brackets by set of braces and then the claim is that the map which takes a limanomial gamma and returns the uh expression in the dual that you obtain by uh replacing the brackets with braces uh that this turns out to actually define a map from l to its dual uh it's not quite obvious that it defines a map because you don't know that this this this curly lee bracket is a lee bracket and so it's not obvious that um what i've written on the first slide is even well defined um but it turns out that it is the curly braces is a lee bracket and uh s is the inverse of t and in one direction it's easy to see that s is the inverse of t because um uh you know if if i take uh uh t of s of gamma that's just t of the uh gamma with braces in it but just as i wrote on the previous page by nesting this this nice relation which this bracket satisfies um t of that curly gamma gives me back a limanomial gives me back the limanomial i started with but uh you can you can you can go in the in the other direction as well um and then the the um so so this is the main result um and then uh an important mark in fact is that the matrix elements of of this map s are indeed the um so on the first page i i wrote uh the the formula that that had been um uh laboriously arrived at by by this group uh and there there this formula here uh turns out to be a formula for the matrix elements of the map that i've just written down um so to be somewhat precise so uh if i if i take uh the the the full bracketing with the curly braces of of word a uh then that's some expression in the door and so i i can write it in a basis making use of the uh of this particular basis uh where i choose some ordering and then i sum over words b such that they begin with their smallest letter um then these here uh if i write them s a b uh these are the physicists or the the are the sort of uh conjectured formula um that i wrote on the on the previous page and um so so that's the uh that's the first uh thing to say and then the second thing to say about the main result is that uh it implies the the klt relation uh in the following form um which is to say that if i write t again and remember that this is the prototype um gravity amplitude to sum over all cubic trees like this uh then i can write that as a sum over over over ordering to a and b again um restricting to orderings that begin with their smallest letter um uh times times these coefficients s a b um and so and these are the putative uh yang-mills amplitudes um and so this expression is the klt relation and i should say that it's actually slightly stronger than the klt relation as the physicists say that um because these involve words of different lengths so it's um uh a slightly different statement that it implies the physicists statement um and eric stoner's video on which means that i will stop sorry heli i i think you you finished greatly on time thank you thanks heli for the talk um so everybody's invited to ask questions um i i had one which i mean first of all i thank you for this very clear presentation it cleared up a lot of confusion i had from listening to what physicists say sometimes about these things um but i was wondering all the structure that that you that you exposed here um in this tree level case i mean when i think of loop level i still have all this lee algebra as the color structures flying around and there must be a lot of structure in there so i'm just curious uh when you go to loop level what how much of these kind of structures do you think extend in some form or another uh yes i think the the the key thing is um uh the key insight here is that it's interesting to to study these um uh well i i should say first that the the the the algebra uh the the lima lee polynomial story immediately goes away at at uh at further orders in the perturbation series you don't you don't have um the the the color structures don't um uh don't correspond to lee polynomials anymore um but it i think the the story you'll notice doesn't depend on the specifically algebra i'm choosing because i'm i'm just trying to use the uh the nice word combinatorics properties of lee polynomials and there there isn't you could do analogous things at loop level so the main main structure in this story was uh if you like t uh ab i haven't written that in this formula but these are the other if you like sum over all gamma which are compatible with both orderings a and b times one over s gamma and you can consider an analogous object at at all orders in the in the gate theory um uh perturbation series except that there what you're doing is um uh and it takes a while to work through you know it's it's the old story you draw quark graphs with um uh you know ribbon graphs and uh associated to ribbon graphs uh associated fat graphs or surfaces and associated the surfaces uh color structures and so on uh but when you do that you get uh t's labeled by marked surfaces um and these two marked surfaces can have different apologies uh and i i strongly suspect that if you study these t's enough uh you you should be able to formulate nobody's really formulated what this should be uh at all orders so i think the question is how do you formulate it and you can study these which i think is proxy yeah thank you edley um i don't see further questions right at the moment so let's thank edley again