 Okay I'll start so thank you for coming and I appreciate that as expected there's a mixed audience of mathematicians and physicists so this first lecture will just consist of an extended introduction with lots of examples. I'll explain for mathematicians what a Feynman integral is and why it's of interest to number theorists and I will try to in the second half illustrate for the benefit of physicists what this Galois cosmic Galois group does and why it's a very very powerful tool in understanding aptitudes. I also appreciate that you might not want to listen to me droning on for eight hours so at least in this lecture you'll see everything that's that's going to happen in the course. So this is just introduction and examples so the goal is to define and study a Galois theory of Feynman amplitudes in a completely rigorous manner there will be no nothing conjectural about this at all and I should really mention my intellectual debt so that the inspiration the story behind this is quite a long one but I believe it started off with calculations due to Broadhurst and Kramer in the 90s in particular 95 who found multiple zeta values which I'll define later as amplitudes in massless fight the fourth theory which is a certain physical theory which I'll come to you later. Okay so multiple zeta values are certain numbers and a few years earlier the linear Ihara and Drenfeld independently and at exactly the same time 89 had developed the theory of the of the motivic fundamental group P1 minus 3 points and the sort of philosophy that emerges from this is that these numbers multiple zeta values are intimately related to a certain group called the motivic Galois group of a certain category called mixtape motives of a Z which at the time didn't exist and all that was conjectural at that time. So what happened next is that Cartier in around 1998 said well we have amplitudes in physics related to numbers in mathematics that are related to some motivic Galois group well could there be and he coined the term cosmic so that could there be a cosmic Galois group that that somehow acts on these amplitudes and corresponds to this this group here which we which was conjectured to be underlying the structure of multiple zeta values. So he made some vague statements but he invented the word cosmic Galois group. Another important contribution was due to Konsevich in 1998 who suggested counting points of graph hypersurfaces over finite fields by which I won't say much but it's proved to be a very useful tool in in trying to understand the structure of amplitudes. I should certainly mention work of Alacorn and Mathilde Marcoli who in 2004 and 2005 wrote a paper in which they constructed what they called and I'll put it in his latest commas because it's different from what I'm going to define. So they constructed a cosmic Galois group and it is related to the renormalization group so I'm not going to say anything about this in this course at all it's just so that you're aware that there is a phrase out there cosmic Galois group in the literature I don't know of any connection with what I'm going to do and I won't say anything about this. Then an important contribution due to Belkali and Brosnan in 2003 which started to cost some doubt on whether amplitudes in this theory were in fact multiple zeta values at all. Yes so they found that graph hypersurfaces are of general type in other words when you take you look at the the count points over finite fields of hypersurfaces defined from Feynman graphs you get pretty much anything possible but their counter examples as you mentioned were physically completely unrealistic so they corresponded to graphs which would never occur in any quantum field theory. So when you say they found multiple zeta values does it mean they found them in a few cases or for an infinite series of all the graphs? No a few cases that so that no until very recently there were no infinite families known. They just began they well I'll come to their results later in the second half they computed numerically many examples I can't tell you how many but a convincing number and they found that they agreed up to very high precision with multiple zeta values. So they did a numerical fit. No they didn't prove that they didn't prove the except in some harmful of cases that the amplitude was actually a multiple zeta value it was just verified numerically to a high accuracy. So I'll come to that later this is being sketchy here but I want to mention everything that that came before. Another important contribution is a paper by Bloch, Inno and Kramer in 2006 who defined what they called the motive of a certain family of graphs in this theory called primitive graphs in this particular theory five four and four dimensions and there's been a huge amount of recent work that I'm not going to say very much about at all I'm afraid because time is shorter than I than I expected. I will mention the names of my collaborators and Dmitri Dorin, Eric Panzer, Oliver Schnetz and Karen Yates who have enormously influenced the way I think about about this but the the bottom line of quite a large body of work is that we now know that there exist amplitudes in fight the fourth theory in this particular theory which are in fact not or expected not to be multiple zeta values so this initial sort of correspondence sort of doesn't really work and in fact this the story is even more complicated than that because not all multiple zeta values actually occur as MZVs so it would seem that this whole project this and so yeah I think that statement's false thank you as I'm finished thank you for that I said in in in fight for so the initial analogy that it inspired Cartier to coin this cosmic gala group has sort of fallen apart because amplitudes are much much more complicated than multiple zeta values and in fact the story is much more intricate than anybody imagined but the point of this course is to say that in fact despite this complexity this idea of a cosmic gala group holds in some sense and that's what I want to explain so in this course we will define an affine group scheme this is slightly inaccurate it will really be several affine group schemes but for the introduction this will do an affine group scheme over Q Gc cosmic gala group and associate to a to certain families of fine and amplitudes it'll be extremely general by the way the the the certain will be very small restriction so these amplitudes will now in fact will be very general they will depend on arbitrary masses and arbitrary momenta so to such families of fine and amplitudes we would associate a motivic period that I will write I subscript Mg Mq and we will be able to speak of the Galois conjugates G of I am masses momenta for any element in this group so from an opportunity you get something you get something new so remark I use the word motivic in homage to to gotten deep because all of these ideas really come from his work but motors play no role in this and the theory will be entirely rigorous and there'll be no conjectures except today there'll be some conjectures but in the rest of the world so as part of a very special case of this theory here I'm speaking to physicists we will retrieve the notion of the symbol of an amplitude so this is become an industry huge industry recently in high energy physics the symbols only defined for a very particular class of amplitudes which happen to be poly logarithms in fact the motivic period is it will be hold for everything and the symbol is a much weaker notion than the motivic period but we will retrieve this and be able to state general themes about them out of this theory as a side effect will also pop out the renormalization group and in particular the concrime hopf algebra of graphs of renormalization in the third lecture I'll explain that so this will come out for free although the precise connection between the the actual renormalization group and the cosmic Galer group remains to be worked out and a third consequence of this theory which is what I want to talk about today is that it implies a very subtle recursive structure on amplitudes and it gives phenomenal constraints on what you expect to find as amplitudes so it in particular it has a practical application to the study of formula I don't know so from what I've understood these are amplitudes in n equals 4 super Yang-Mills my understanding is there's been a huge progress in understanding the integrands organizing the integrand structure for n equals 4 super Yang-Mills for which it wasn't a prior clear how to write that down but the actual integration where you actually integrate over some domain is still completely open from that point of view it seems so when you say what is play no although so we are supposed to understand that the cosmic Galer group is not like the Tanakian group of some particular you're jumping ahead yes no it okay for just for over it will be a quotient of what I'll do is I'll define a representation of the Tanaka group of a category of realizations I will work with the category Tanakian category of realizations and that has a group but every time I give myself something that comes from algebraic geometry the group action on it will be the correct one well is expected to be the correct one if there was an Abelian category of mixed motives it should give exactly the same answer it's a way to get around all the conjectures no no I'm not constructing category of motives I will explain all of this is this broken was it a strength test okay I'll give up so it is a strength test of strength so you will have both invented and destroyed the cosmic Galer group okay so now Feynman graphs so I will consider scalar a scalar quantum field theory in in Euclidean space so R to the D oops D where D is an even number so even number of spacetime dimensions so a Feynman graph will be a connected graph later on I might make them not connected but for today they're all connected so a graph G is a set of vertices a set of internal edges and a set of external edges so V is set of vertices e.g. are internal edges so these are pairs of vertices not necessarily distinct and very often these edges will in fact be labeled and yes absolutely and we can have tadpoles as well absolutely and then we will have external half edges which are called legs by physicists so we have a graph some topological data and there's kinematic data one a particle mass me so it'll be a real number for every e and e.g. and to a an incoming momentum qi which is a vector in the R to the D where D is the number of spacetime dimensions for every external half edge subject to momentum conservation which is that the sum of all the incoming momenta is zero and and sometimes I will need to specify which masses are going to be zero and which aren't likewise which momenta we're allowed to take to be zero and which aren't and so to do this it's convenient to draw massive lines massive edges that is to say those with me not equal to zero as thickened lines okay so here's an example of Feynman graph maybe I should do like this that's better so we'll have q2 q3 and q1 and this means that we have masses m1 and m2 which will be non-zero but the mass of the third particle here is zero and momentum conservation tells us that q1 plus q2 q3 equals zero it doesn't matter because in the formulae we'll always take the square of the math oh yeah and I will always replace if we have a graph with several incoming momenta q1 q2 up to qn we can always replace it with a single external leg which carries the sum of the momenta so if you like this defines an equivalence relation on these graphs okay so I'm gonna say something slightly tedious but it will be important for later on I won't spend too long on it we can specify the set of edges with vanishing the set of vanishing masses and momenta in some sense I want to think of each each graph comes with the data of which momenta and masses are non-zero and each such family will define a motivic period so what that means is that we will view the Feynman amplitude which I'm about to define I g which is a function of these masses and the momenta as a function so here we we should say what we mean by function it can be multivalued in general that means it's a function on some choice of universal covering of some space it may be it may even be well ill-defined so it may be infinite everywhere so when I write to find an integral sometimes I will not assume that it converges it will just be a formal expression so it will be a function on the complex points of an affine affine variety m g index v which is I'll just write the set of masses and momenta where the momenta the masses are an a1 minus 0 for all e in not in vm and the momenta will be an affine d space non-zero for I not in vq okay and oh I've got something subject of course subject to to the momentum conservation condition so this is a sort of space on which we want to view the Feynman amplitude if it's if it converges it will have singularities all over the place I'm lost sorry here yeah so this is we specify some set of edges for which the masses vanish some set of edges for which the masses don't vanish likewise for the momenta what I don't understand how to do is is take a limit as a mass goes to zero whereas a momentum goes to zero that's very tricky so that's a question that I won't address at all in these lectures so I want to specify who's zero and who is non-zero and for mathematicians that the Euclidean region so what it means to be in Euclidean as opposed to Minkowski space is just a set of real means we restrict to the set of real points of this variety okay okay so now to define the Feynman amplitude so I'm going to to jump in and define the Feynman amplitude directly in parametric space which is very old-fashioned and younger physicists are maybe not so familiar with it but it's in all the derivation from the momentum space representation is in all the textbooks so you trust me that it it works in it it always works so this is a sort of 1960s presentation of Feynman amplitudes and it involves graph polynomials so a spanning k tree written t equals t1 tk is a subgraph with k connected components which are the ti's and each ti is a tree that has no no loops in it such that the vertices of these trees cover all the vertices of the graph so the vertices of g is the disjoint union of the vertex set of each t of the union of the vertices of the t's and then some terminology to each internal edge of the graph we associate what first is called a Schrodinger parameter alpha e and I'm from this we define a couple of polynomials kickoff polynomial graph polynomial is often called the first semanzic polynomial it is written psi g equals the sum over all spanning one trees in the graph and then for every such tree you take the product over edges not in that tree of alpha e and so this is a polynomial in the string of parameters and it has coefficients in z a spanning one tree t t is a spanning one tree so maybe I should write t equals t1 so this is span spanning one tree according to the definition at the top and then now oh yeah so the second semanzic that I'm going to denote by phi gq it's now going to depend on the momenta it's the sum over now spanning two trees I'll write it more neatly this time spanning two trees oh gee and we take the product of the edges not in the union of these trees alpha e times qt1 squared where qt1 also equal to minus qt2 is the total momentum that is incoming that comes into t1 and the square I should nearly forgot to say if when d dimensions so qi has components q1 up to qd let's say then the square is Euclidean norm so an interesting fact which which actually gives rise to all this arithmetic coming out of quantum field theory is the fact that these polynomials here and here have integer coefficients and that's the fact that's almost never used in physics sorry they have inches the fact that this has integer coefficients is very important because it gives all the arithmetic and it's almost never used so let's do an example so let's do this graph again q3 so the spanning one trees are one two three two and one three and so the first semantic polynomial or kickoff polynomial is the sum of the complement the products of the complements of the edges in each spanning tree so here's just alpha 3 here it's just alpha 1 and here's alpha 2 and the spanning two trees are edge 1 and this isolated vertex here this vertex here and edge 3 and then this vertex in edge 2 and so 5g of q is equal to so we take the momentum the total momentum going into into one of these trees so let's say this vertex that's that's just q1 and then we take the product of all the edges not in the spanning tree so that's alpha 2 alpha 3 of course if I took the momentum and into the other tree you'd get 2q plus q3 but because of momentum conservation that's the same thing as q1 squared here we get q3 squared alpha 1 alpha 2 and the last graph we get q2 squared alpha 1 alpha 3 okay so we get some very concrete polynomials coming from graphs and some remarks which is that psi g is always homogeneous and its degree is hg which is the the Betty number first Betty number of the graph in the standard definition and this is and I will often call this the loop number number of loops of a graph it's just the dimension of h1 and then 5g as a function of the alphas is homogeneous as well but of degree one more okay and finally we let define a third polynomial that I will call psi I don't think this is standard terminology to call this psi but it suited me psi m comma g will be the second semantic polynomial plus the sum over all internal edges mass squared of that edge times alpha e multiplied by psi g so in this example we get I'll write it out an alpha 1 plus alpha 2 plus alpha 3 in the alphas yeah I said this and in yes in the alpha is absolutely okay so now for the final integral which will be constructed out of these polynomials number 3 so this is the final integral in parametric form yeah so let me write ng for the number of edges of internal edges in the graph for the time being and then the final integral i g of m q and it also depends on the dimension but I really gonna really gonna fix the dimension for most most of the time it's gamma do to integral sigma omega g d where omega g m q d is one over the first semantic polynomial to the d over two times so m q to the power of n g minus h g d over 2 this is why we want to take the dimension to be even to avoid having square roots everywhere times omega g so I have sigma I'm coming to that first omega g so omega g is sum i equals 1 to n g minus 1 to the i alpha i you emit a mid alpha d alpha i and g and sigma is certain locus in projective space of dimension n g minus 1 it's real points so it's the coordinate simplex it's the real coordinate simplex see maybe I'll write it here put this board to some use so sigma is the in projective coordinates it's the region where the alpha n g's so alpha i is in real and non-negative okay some remarks oh so of course I should say this this may be an ill defined integral it may make diverge and most of the time it will diverge badly okay some remarks is that the inter ground omega g m q d is homogeneous of degree zero so it's a small calculation we know what the degrees of psi g and phi g I told you here and you plug it into the formula and one has to check that it's homogeneous of degree zero in the alphas and once you've made that remark then indeed the integral does make sense as a projective integral if you don't like that you can always restrict an affine chart by setting one of the alphas to one for example and you get a standard integral over over r r to the n minus one or something another remark is that the amplitudes in a general a not necessarily scalar quantum field theory are much more complicated but they can be expressed in parametric form using similar integrals but with numerators so numerators will be some sort of polynomials in the alphas with coefficients in some some Clifford algebra or something the point is that the the geometry of the Feynman integral will not depend on the the numerators so I expect and I hope that everything that I say can be extended to more general quantum field theories but for now it's not much of a restriction to consider the scalar case for this reason yeah not sure I mean not sure I see what you mean that so when you derive this doesn't there's an exponential and that produces this gamma term here but this is not regularized in any way so if you take the if you take the the momentum space definition of a Feynman integral and you do the Schringer trick and you do the momentum integral is using a Gaussian formula for a Gaussian integral you get a formula very close to this yeah so that okay so that that's how you part so you have an e to the minus something and then that's that's where you pass to okay yeah so if you work in affine space you get the integral like you say with an e to the minus something and then you can do one more integral integrate out you get e to the minus a graph polynomial times lambda because it's homogeneous you integrate out lambda which will spit out this gamma function and then you get a projective integral so there's one one stage further than the normal affine maybe you're thinking of the the parametric integral with a delta with a delta function yeah okay and the remark that maybe not all mathematicians are aware of is that almost all the predictions for collider experiments are obtained from computing such quantities and that's why the calculation of Feynman is an enormous enormous industry so perhaps before having a break I'll just give some some first examples of Feynman amplitudes and try to convince you that there that you get interesting quantities from them so here's some sort of random selection of examples from the literature so at one loop what are the sort of things you can get the one-loop graph would be this triangle graph that we looked at earlier with some choice of masses and momenta doesn't really matter which other examples would be polygons like this maybe with many momenta this is in D equals four dimensions then all these families of Feynman integrals it turns out is always expressible in terms of two functions the logarithm which I will write in this way le one of x equals minus log one minus x and the dialogue with them le two of x equals sum x to the k over k squared so this is I don't know if he was the first to observe this but it was there's a beautiful bit by Davide chef and Del Boog or what this is explained so you only need essentially these two functions to describe all the Feynman amplitudes at one loop and what are the arguments of these dialogues and logarithms while they will be some complicated the arguments will be some complicated algebraic functions perhaps with a square root thrown in there of the masses and the momenta I should say this mass squares and momenta no sorry I won't say that so that's really it for one loops one loop rather two loops it gets more interesting one example that's been massively studied in recent years and has quite a long history is the sunrise diagram again with you've got three possible masses and one momentum coming in so let me give you the graph polynomials just for fun if I do you q q squared after one of two over three and this is a very long history but the upshot is that this these families of Feynman integrals give elliptic dialogue with them and the most recent work on this is due to Adams Wagner and Vite seal so presumably in their references there's the full history of this family of integrals but the remark is that the general two loop diagram general two loop amplitude is I believe not known so what does that mean not known it means that it's some function which doesn't have a description in terms of familiar mathematical objects there is a vast array of examples which can be expressed in terms of poly logarithms multiple poly logarithms but the literature physics literature is full of such calculations and there are the interesting examples for number theorists like myself at least such as these family of graphs bn so here we need equals two dimensions and so and edges so these were talked about recently by David Broadhurst in this very room not long ago so here you take q equals naught so q because not means I may admit to to draw the external momentum or just put it a little small line like that to illustrate that it's zero and all the masses are equal to one but perhaps I should thicken these lines a bigger one say oh n can be anything yeah that was to that was the end of two loops and then to the two loop story stops because we don't we're stuck and now now this is just a different family of examples so n bigger than or equal to two and so here that the polynomial psi doesn't depend doesn't depend on anything now it's some alpha i times psi g and again for fun I'll give you psi g is the sum of one of the alpha i's all the alpha i's so that the zero locus of these polynomials define interesting hypersurfaces that have been studied fair amount in mathematics in some cases and here here are some examples of the corresponding aptitudes the i2 is is the integral find an aptitude of i b2 is one i b3 think this is these all due to David Bordhurst is three times the Dirichlet L function for the unique Dirichlet character mod 3 which is which is non-trivial so this is three times sum chi n n squared and we start to get more interesting numbers i b4 equals seven times z to three even z to value and I think beyond that they're not known so these are certainly numbers which are very interesting to a number theorist and in fact there are variations on this graph variants on this integral which are relevant to the study of such Feynman integrals which experimentally mainly in some case it's been proved yield a whole array of special values of l functions of modular forms so where f is f is some modular form for the for SL2 some congruent subgroup of SL2 so before continuing with the main example I should maybe have a 10 minute break coffee so this is this class of graphs is called log divergent precisely because the gamma factor will produce a pole but by abusive notation let us write i g to be modulo this gamma factor what remains of this integral and it is omega g of psi g squared and so this is a number if it converges and it converges if and only if for every sub graph the number of loops that's why the number of edges is strictly bigger than twice the number of loops every strict sub graph so this is called a this means that the big graph is primitive has no sub divergences so an example an example of a graph which is log divergent is this one so now because there are no masses and no external momenta they play no role I can drop them from the pictures so this is certainly log divergent it has four edges and two loops but it is not primitive because it has the sub graph three four contained in it which has two edges and one loop and two times one is two so it violates this inequality why these a comment for the physicist why these quantities are relevant so in this case we get I should say that we get a real number you don't get functions anymore we just get numbers so why these quantities relevant for physics because they give renormalization scheme independent contributions to the beta function of this theory oh I should say also the one I write 5 4 what that means in graph theoretic terms is simply that that the graph has no vertices of degree bigger than or equal to 5 so the valency of each vertex is at most four okay so here examples so these are the calculations originally due to brought us in crime oh which which started off this whole business so one loop there's one loop there's this graph and its amplitude is one just the number one at three loops there's a single example of a primitive log divergent graph which is the wheel with three spokes and its amplitude is six times six each of three at four loops there's a single example which is the wheel with four spokes and it gives 20 zeta of five at five loops there's several examples let me look at one of the most interesting ones without there are a couple of others and this gives six zeta of three squared that's the square of this Feynman amplitude and we understand why that's the case at six loops there are more examples but the most interesting one is this graph which again was computed experimentally way back when but only rather recently has been proved rigorously so and then many others and here the amplitude is something complicated it's 27 over 5 zeta 5 comma 3 plus 45 over 4 zeta 5 times zeta 3 minus 261 over 20 zeta 8 and so these calculations were first due to brought us crime numerically up to a certain degree but there's been spectacular progress in recent years going to much higher loop order and proving the actual quantities rigorously by panzer and schnitz for two they're two different yeah so already for the wheel with three spokes you're right this is a it has six edges so it's a six-dimensional five-dimensional integral the graph has 16 times you can't compute it that way it's terrible so you use different techniques you use momentum space so you use something called the Gagan-Bauer x-space technique and you expand in terms of certain Gagan-Bauer polynomials and then accelerate the convergence there's a whole industry of there's a huge literature on how to compute numerically but that's all been superseded because they're now algorithms that do this using the parametric representation in the case of Eric who's sitting in the audience and all of schnitz is a different approach using single-valued multiple poly logarithms so now many of these can be done much more efficiently and exactly so these all theorems these are nothing is yeah these are these all two theorems and they're examples now up to up to much hard to put up to even 11 loops so there's been a spectacular progress in in recent years and this but I'm for the illustrations of today I'll just look at these examples okay so the first observation is that there are sorry so originally there were no infinite families known but now we know that there are infinitely so we know some infinite families now some which are explicit and some of which are just proved by general theorems which are multiple zeta values so multiple zeta value is this nested some so these were first defined by Euler in the 18th century but now we know this extremely long story about which I regretfully will say nothing that we no longer expect that at some time one did expect all these amplitudes to be multiple zetas but that is no longer the case and we do not expect multiple zeta values in general so the question then is what so to the now there are known examples due to to Panzer and schnitz where you have an evaluation of a Feynman integral as something like an Euler sum well a variant a variation of this definition where you put a root of unity and standard transcendence conjectures predicts that it's not possible to write it in this form but more drastically there are examples due to myself and Oliver schnitz where we proved that the we proved that the graph hyper surface is modular so there's a piece of the co-homology is the motive of a modular form and so that's that's that's an eight-loop that's a much more sort of catastrophic yeah but it's in the same it's still mixed hate yeah if you put a root of unity but a modular form changes changes the the the type of number completely so the question is then what on earth can we say in general what is there left to say in complete generality about amplitudes at all given the vagaries of the examples and the first thing one might toy with is some sort of invariant some weaker invariant of multiple zeta values like the weight so the weight of a multiple zeta value is the sum of the of the arguments and so let's have a look at the weights on on these examples this is short of trying to compute the integral let's see if we can understand the weights so let me put a column here the weight and put another column here twice the loop number minus three the weight here is is zero and here it's kind of a trivial example so we'll ignore that here the weight of zeta three according to this definition is three and twice two times three minus three is three I'm here the weight of zeta five is five and this quantities is two times four minus three which is five here the weight is six and this quantity is seven and this integral here has weight this multiple zeta value has weight eight but 2h minus three is nine so already we see that even looking at a very crude invariant like the weight we have some sort of upper bound which is twice a loop number minus three but in these examples here the bound is not attained and we say that these examples have weight drop so so we see that even understanding the weights is a tricky business but I I should say that we actually understand the weights fairly well at least in this this setting so that the weight is the first hint of something motivic going on so the prototype for and the inspiration for this entire subject is the following experiment where we will try to compute the Galois action on fight the fourth or the amplitudes which we know in fight the fourth theory and in particular those which are MZVs or close to MZVs so in the second lecture next week I will define a ring of an inverted commas motivic periods it's an abuse of the word motivic but I'll keep it anyway so we have a ring that's called PM sub H shall explain next time and it comes with the following structures it comes with a period homomorphism from these elements of this ring to actual complex numbers then it comes with an action of an affine group scheme of a queue that I'll just call G so this group will act on this ring and it has other structures in particular an increasing weight filtration so in general the weight is only a filtration but in in the examples that we're going to look at like more positive values it will happen that the weight is a grading and we can speak about the weight so it's a small abuse of terminology but for what I'm about to say we land in a subspace where the weight is a grading and so I will use the word weight as it were grading but in general one should remember that it is not so what is H I will define that next time it will be a tenacian category of of realizations something yeah but that that will I want to explain what a Galois theory of amplitudes means and and I'm now going to explain how it will explain a lot of the structure that we see so I think these these examples are very striking and what the motivation for constructing this so examples there are things called motivic multiple zeta values which I defined a few years ago again they actually they actually live in a sub ring of another ring of periods which injects into this one but I'm going to identify them with that image in this so I don't think that's very drastic so their objects corresponding to multiple zeta values and the period is the multiple zeta value okay so then what can we do with this group well we can define the Galois conjugates of an element in this ring so all this will be constructed next week the Galois conjugates will be the linear combinations of the g psi where g is in the rational points of this group and therefore every in fact every element will generate a finite dimensional a finite dimensional representation that will call v sub psi of this group and we will get a representation row from this abstract group about which we know very little but to a very concrete group of matrices so the idea is that we want to replace actual numbers with representations of a group so I hope that this idea should resonate with physicists where the idea of replacing objects with representations of groups is familiar so it turns out that we know how to compute a lot about these representations in the case of multiple zeta values so let me give give examples for this in the case of some simple multiple zeta values so the first example is even zeta values so the Galois conjugates of an even zeta value are just this are just itself so v in this case will be q zeta-matific 2n and so how does the group G act on this matific zeta value well it just multiplies it by some number to 2n so what that means that there exists there exists a homomorphism from G to GM which called lambda and every even zeta value gets scaled by lambda to its weight so this is kind of a bit uninteresting to get something more interesting we should look at odd zeta values so now the the the Galois conjugates form the vector space spanned by one and the odd zeta values so we get now two-dimensional representation and G of an odd zeta value is lambda G it's the same lambda as before so the weight plus I'll put it here G so this is and q so the this Galois group matific Galois group if you like takes an odd zeta value scales it but it can also add a rational number to it so that's how it transforms and so it means that we get a representation which in in this basis so we get G to ought V and G and in this basis will map to the matrix 1 s 2n plus 1 lambda to the 2n plus 1 of G so we should think of an odd and even zeta value is a one-dimensional representation and an odd zeta value is a two-dimensional representation sorry SS is a it's a it's a function from G to Q it's a function from well G to f I line if you like it's a function from s 2n plus 1 is a function from G Q to Q and and this thing is a homomorphism let me it's a homomorphism from the additive group semi-direct the multiplicative group so these are functions on G and and this forms a group of me into this group of matrices my apologies I forgot the zero here so to M1 it's zero yeah Zeta material 1 is it well you can define it yeah I prefer yeah you could that's useful it's very useful in many contexts to consider it to be a parameter actually here I prefer it to be zero sorry so then we can take we can take tensor products of representations as we know so the first two examples tells us how to what the Galois conjugates of a product of an odd zeta value and an even zeta value are so what are the Galois conjugates of this it's from the two previous formulae it's 2n plus 2k plus 1 the same thing plus what I call it s 2n plus 1g Zeta M2k that just follows from multiplying the two previous examples together so in particular the Galois conjugates of I know let's just do an example zeta 3 zeta 2 are elements of the vector space zeta 2 and zeta in other words you can take zeta 3 times zeta 2 and there's an element of the group which sends it to zeta 2 okay for the first really interesting example is zeta 3 5 and you have to trust me on this there's a there's a way to compute this this the Galois conjugates it's a it's a big subject would take very long time to explain and we won't need everything but the the upshot is that the Galois conjugates are the number itself zeta 3 and 1 absolutely absolutely yeah the reason it's non-trivial actually follows from the fact that the zeta value is non-zero yeah it follows from that fact because there's a period homomorphism and you know so now we get a representation on this vector space so G to ought V and in this basis the in this basis we can write it as a matrix 1 0 0 lambda cubed lambda to the 8 this is s 3 this is minus 5 times s 5 so there's some computation to be done there to obtain this minus 5 and there's some new map from g synthetic map from g q to q which I call s 3 5 such that this thing forms a is a is a homomorphism of groups so one goes to one yeah so yes one goes to one do you think I should write what I this is dram maybe I'm not torsie bad at getting my left and right mixed up but yeah we can discuss it we can discuss the convention okay yeah yeah maybe you're right okay and then in the I want to keep that so here so in the third lecture so in the second lecture I will try to set this up so that all this has a rigorous meaning and works and the third lecture I will define a motivic period again abuse of the word motivic I am G in particular for example for any G of the type we're looking at here so primitive overall log divergent and fight the four but it will be much more general than that but we're just looking at this case now so we will get an analogous construction I am G in this same ring and such that its period is the Feynman aptitude we want which by these assumptions converged as I said earlier and then the key observation which is the whole purpose of this series of talks is that the subspace spanned by the motoric Feynman amplitudes is is in brackets nearly it's it's in some precise sense but the story is slightly more complicated than it first seems it's actually closed under this gala action so the goal is to make this precise so in some sense Calty's original dream was that the Feynman that there should be a group acting on Feynman amplitudes and it should be the same group as the one acting on multiple situates but since we have all these counter examples and this complications that can't be true but the point is then there is such a group acting on amplitudes but now it's not at all clear that it actually preserves the space of amplitudes there's no reason why if you take an amplitude and take its galore conjugate it should still be an amplitude and that's is nearly the case and is an absolutely astonishing fact and I will now try to explain to you why it's such an astonishing fact on these examples so this this fact has extraordinary predictive power for the amplitudes so the purpose of this talk was really to try to motivate this entire construction with with the examples I'm going to give now so as a to test out the validity of these ideas let's take these amplitudes which are computed as what positive values and just put a little m put a little m everywhere that's reasonable because there's the standard transcendence conjecture for multiple zeta values suggest that the the period map on multiple on motivic multiple zeta values is injective and everyone believes this so that's not that's not a big price to play so let us assume the transcendence conjecture for mzv's and then do the following experiments so then so all of what Oliver Schnetz did is he took this the vast data of amplitudes that were now known in for fight the fourth theory and he computed the galore conjugates of all of them and he made the following conjecture based on that numerical evidence which is that the periods I'm sorry that the vector space spanned by the metallic amplitudes for G as above primitive overall log divergent in fight the fourth theory is closed is closed under the galore action action of G so this has not appeared yet it will be written up in a paper with joint with Eric Pancer I believe who's made many contributions so let's assume this conjecture is true and as a thought experiment let's go through the consequences of this conjecture so for now on let's just assume that the conjecture is true and draw consequences from that and let's assume that in all the examples that we're going to look at that we know that the amplitude is a multiple zeta value so in fact there are theorems that will guarantee this in even infinite families of examples so that's that's not much to assume and let's assume in these examples that we can predict the weight and that's also the case I have a paper with couch nets in which we explain how to predict the weight at least in low degrees okay so I'm out of space okay so we need so let's assume that these amplitudes assume that we know that these amplitudes of multiple zeta values and we know their weights that's not much to assume so here's a table here's a table I run out of board space okay I'll try and squeeze it on him here is a table of multiple zeta values up to weight 8 so the weight to so these are the numbers which we expect to find at least low degrees coming out as amplitudes in fight the fourth theory and here I will write a basis for the space of motivic mz v's in that weight so in weight zero there's just the number one in weight one there's nothing in weight two there's zeta motivic two nothing else in weight three there's zeta motivic three nothing else in weight four zeta motivic four and nothing else in weight five the zeta motivic five and zeta motivic three times zeta motivic two in weight six there's zeta motivic six and zeta motivic three squared in weight seven zeta motivic seven zeta motivic five zeta motivic two zeta motivic three zeta motivic four and in weight eight we get zeta motivic five three this interesting before I would three five but you can take five three it's the same zeta motivic five zeta motivic three sorry let me put zeta motivic eight at the top then zeta motivic three squared zeta motivic two zeta motivic five zeta motivic three and the first motivic multiple zeta value that is not a product of things which came before five three okay so here we go let's we have these data of numbers and we have this correction conjecture and we have these graphs so let's let's put the pieces together let's assume this conjecture okay so first of all there is no graph whose amplitude is a multiple of zeta two there can't be because because we know something about the weights it would correspond to a graph with two and a half loops or it could correspond to a three-loop graph which has a drop in the weight but there is only one three-loop graph it's there so there cannot be a graph whose period is zeta two so then the the correction conjecture says that no I'm saying I just assume I know the weights so that I know that this graph has weight three so and there are no graphs in between the one-loop graph and three-loop graph there's so there's no space to have a zeta two it cannot be a zeta two in in this theory and that implies that there are no numbers whose conjugates are zeta two that can ever occur so this implies that the number zeta any zeta odd times zeta two can never occur to any new order so we take our table of mz v's and we strike out having struck out zeta two the correction conjecture tells us that this number can't occur this number can't occur and this number can't occur and so on add-in for an item so let's do an example so example in weight five let's compute the five so the the four-loop amplitude by pure thought we know we have a graph here and we know that its weight is five so a priori the amplitude since we're assuming it's a we know it's a multiple zeta value should be in this vector space of weight five so it's a rational multiple of zeta five plus a rational multiple of zeta three zeta two so what would its conjugates be by the examples I worked out laboriously beforehand the conjugates would be the number one beta zeta motivic of two and the amplitude itself but we know the conjecture implies that the conjugates must correspond to graphs but there is no graph corresponding to zeta two so that forces beta equal to zero so it tells us that the amplitude cannot be an arbitrary linear combination it has to be a multiple zeta five okay keep going so the weight is is tricky so we can understand when graphs don't have a weight drop and we in some infinite families of cases and we can guarantee that there's one weight drop so the combinatorial criteria where you can just look at this graph and see immediately that it has to have one weight drop but predicting two or three or more weight drops is very tricky and there's no there's no general recipe where you give me a graph and I can write down the weight there's an upper bound so there's not the other bound is it's always 2h minus three and and sometimes that bound goes no no low bound good point yeah so I'm assuming I'm just said I assume you know the weights but there are yeah you can't get a low bound using I will come to that in the fourth lecture so in in Hodge theory there's an upper and lower bound on weights so they're two different things one is can we predict when their weight drops that that's the question I just answered but in Hodge theory you know that the the co-mology hn of a of a of a smooth variety has weights between n and 2n so there's a there's a lower bound as well you have to you have to apply that so in some sense there is a lower bound I'll explain that in the fourth lecture if you're sorry consequence of Hodge theory but in some sense that gives a lower bound which which means that beyond a certain point beyond a certain point the only way you could get as e to 2 is you have you have to do the whole theory you just want to make sure that there is a lower bound which grows with a loop number so we have this upper bound with two times loop number and more of the many connections from the data we have also there is a lower bound growing linearly with numbers okay so there are examples where triple weight drops or as that is always number each loop also the upper bound grows by two and the lower bound experimentally seems to grow by one yeah I don't know how I can answer this satisfactorily without referring forwards but yeah the weight is something extremely subtle and we're very far from understanding it there's no no recipe that predicts that there's some some theorems that tell you from the common talks of the graph that there's a one weight drop but the two weight drops is out of reach now I don't know who's question I'm answering okay so now continue so likewise by staring at the examples there is no no graph corresponds to zeta 4 zeta 6 because there's just not room and so now let's look at six loops so let's look at this example here so most six loop graphs have weight nine and they're they're going to be not very interesting amplitude because of this this correction structure this galois structure but let's look at a weight drop case is going to be weight 8 and is going to give something interesting so this graph here again so a priority amplitude in this basis is going to be alpha zeta 3 zeta 5 plus beta zeta 3 5 plus gamma z to 8 plus delta z to 3 squared z to 2 and the correction conjecture is also we've already established that there should be no z to 2 as a conjugate of this graph so that goes out I should update this so this should be out and this should be out but we are allowed to have the conjugates of any possible linear combination from the calculations I gave earlier can be amongst the numbers 1 z to 3 z to 5 and the amplitude itself but that's okay because these these numbers do occur as previous amplitudes they have one way to answer this this weight conjecture weight weight thing is that the co-action should it should respect the respect the number of edges so if you haven't seen a z to 2 so far you're not going to see it you're not going to see it now finally okay so these do these do occur as amplitudes and let's do one more one final example at weight 11 so this would correspond to a seven-loop graph which until a few few years ago was totally beyond what was computable or even forcibly computable so at seven-leaves of multiple z to graph which involve the following periods so here's a basis of the motivic multiple z to values at weight 11 this is wrong this should be cubed 9 plus 2 is 11 z to the 9 okay so sorry for that so this is a basis of of motivic mz v's in weight 11 and it is nine dimensional and the correction conjecture tells us we're not allowed we don't see z to 2s and z to 4s and z to 6s as previous graphs so these numbers should not be showing up as amplitudes and so what we do now is we take a seven-loop graph so I should say these are extraordinarily difficult to compute these amplitudes but I pick a random seven-loop graph which is called p7 8 and you compute its amplitude and you find after a huge amount of work that it can be written in this basis of multiple z to values and you get some coefficients out and you indeed you notice that these the coefficients of all these four quantities vanish in accordance with the correction conjecture but then the correction conjecture goes beyond that it says that if one computes the correction on these numbers you could certainly find something of weight 8 and so it would have to land in the space spanned by the periods of amplitude and fight the fourth theory and if you work out what the correction conjecture tells you all these galore action conjecture sorry galore action conjecture implies that some constraints on these numbers in fact it implies precisely that 3024 over 5 ratio should be exactly equal to the ratio of z to 235 to z to 8 over there so 27 over 5 over 261 over 20 and you can check that that's indeed correct so I hope that this illustrates that this correction conjecture gives some extraordinary subtle constraints on the possible amplitudes which which up to now has not been considered at all yeah so the correction you take this quantity and you compute the co-action so z to 11 is primitive it's not going to give anything so it's the only things so we're looking for things that will be conjugates of the form z to 5 3 or z to 8 and the the second one in this basis was chosen in such a way that it's galore conjugate with z to 5 times z to 3 because it's and then so the only ones that can occur are the third one and the fourth one and so the the co-action sends so the galore action sends the third one and the fourth one to z to 5 3 and z to 8 respectively and the z to 5 3 and z to 8 have to occur in a combination that has previously occurred in the quantum field theory because this is this weird combination of numbers here the only way it comes in into fight the fourth theory is that z to 5 3 and z to 8 always occur with with this constant of proportionality and so you have a you do your six-loop calculation once and that gives you some information at a completely different loop order by by this conjecture so I find this absolutely astonishing that that knowing something in low degrees gives you constraints to all orders in perturbation theory so it was these calculations were motivated this story so they're about 250 examples that have been checked by by Oliver Schnetz and Panzer of this type the conjecture has been up to 11 loops the constraints as I hope I explained that this correction puts on the impossible amplitudes actually gets stronger as you go higher as the loop order increases so normally it's the opposite in physics you can just do things at very low loop orders here we see something the way it's getting more and more powerful as the order increases and indeed the dimension of the space of amplitudes is exactly four which sits inside the mention space of mz v's weight 11 equals 9 so the physics is choosing a very special subclass of numbers and that class of numbers is entirely predicted by this this Galois action had a five-dimensional but but the the point was that here we had a three-dimensional space of possible amplitudes z to five three z to five to eight there's another graph that gives you to five z to three and at six loops you only get a two-dimensional space out of the full possible three dimensions I forgot to say that my apologies I should have said made that more clear so the fact that that explains yeah so I should have said that more clearly here that you have you sorry I admitted a line here I should have said that at six loops you only get a two out of three dimensional space and that will propagate up for all higher loop orders and you see it occurring here that constraint and feeds none whatsoever because the in general we don't get multiple z to values so and for such numbers this conjecture will be even stronger because it will predict you'll get a much bigger space of possible periods and you'll be in a in a very small subspace so there's a very striking example due to schnets and pans are where you have an Euler sum so you have minus one to some power here and the space of Euler sums at that dimension is hundreds I believe and the the conjecture tells you that it has to sit in a vector space of dimension which is a tiny fraction of that I don't know what the precise numbers are so it goes from four hundred down to five or something so if the numbers are not more positive values this becomes a stronger and stronger prediction because you want a bigger box of possible periods and so the purpose of the remaining lectures will be to prove a version a weaker variant of this conjecture so it'll be much weaker but but it will be valid for arbitrary masses and moment as well yes so the the point the problem will be that when you the galore conjugate of a graph in fight the fourth theory the way if you do the mathematics you expect graphs with higher but in fight the five fifth theory or five to six three so you can get it involves contracting edges in the graph which will increase the degrees of the vertices so it could just be that this conjecture is true because because the amplitudes in hierarchy vertices are no more complicated than the ones with four vertices at this loop order up to this far on the picture so it could be that this conjecture goes wrong in in higher loops but yeah that's exactly the precisely that the statement of we get so it's not clear what whether the conjecture is true and that's one reason why my name is not a conjecture because I'm nervous about it I'm Oliver is a braver man than I am and the eminence is is is phenomenal but we don't know yet whether such a strong sort of statement can be and so very briefly the what is the plan for the for the sequel in the second lecture I will be just pure mathematics I will talk about motivate periods I'll try to do everything in that lecture it will be completely independent from this one the third lecture I will talk about graph motives be some algebraic geometry relating to graph hypersurfaces and it will be totally independent from the previous lecture and in the fourth lecture I'll put everything together and find cosmic aloe group and small