 Okay, I'll begin. So I'll begin with some reminders from last time. So we had G, a connected Feynman graph with N, G edges, HG loops and kinematics, masses and momenta Q, M. And we had graph polynomials psi G, which is the sum of the spanning trees and then the product of the Schringer parameters not in each spanning tree. And then I'll directly define this other polynomial psi, which depends on Q and M. And it's the sum of all spanning two trees. We take the product of the edges not in the spanning two tree. Then we take the total momentum through one of those trees, doesn't matter which all squared. Then some factor that involves all the masses and psi G. And from this we concocted the Feynman integral, which was IGQM, which was some trivial factor which I'll drop. And integral over sigma of omega GQM, where omega GQM was 1 over psi G to the dimension of space time over 2, psi G over psi G NG minus HG D over 2 times omega G. And sigma was contained in projective NG minus 1 space and it was the coordinate simplex, which is where all the Schringer parameters are positive. Okay, then from this we defined... Oh, alright then. It's a reminder. I don't want to redo the entire course. Are there new people? Yeah. Oh, wow. Okay. Goodness. Okay, in that case, certainly, omega G is I equals 1 to NG minus 1 to the I alpha I, we omit. Okay. So this integral can converge or diverge. We'll see exactly when in a minute. So from this we define hypersurfaces. Yeah. Yeah. So we had hypersurfaces XG, which is what I call X psi G, union X psi G, where this is just the vanishing locus of the first polynomial, which is homogeneous. And this is the vanishing locus of the second polynomial. It's a family of hypersurfaces, but we're going to think of it for the time being as just fiber by fiber, so for a particular choice of masses and momenta. So this is contained in P and G minus 1. Now there's an important point here that in the case where G has no masses or momenta, then I'm going to put psi G equals psi psi G, so that's very important. There's another sort of stupid case that if you, and this quantity here is, if NG is strictly bigger than HGD over 2, then this factor here is going to cancel with this factor here and we're not going to get it in the denominator. And in fact we can consider a simpler geometry in which we just have this hypersurfaces and not the other one. I'm not going to consider that case because it will just mess up the notations. But one can do slightly better in that case. V of F means the locus where M vanishes as equals. V, the vanishing locus, yeah. Absolutely, it's a standard terminology. Vanishing locus. Okay, so an example just to, I'm sorry I've given so many examples in this course, but I think it's a quick way to upload the main ideas. To your heads. So massive lines are depicted with a double edge, a thickened line. So the graph polynomial, let's just look at this one. It's the most interesting one. Equals, okay, and so we had a notion last time of motic subgraphs. So these are the ones where something bad happens in the geometry. So the strict motic subgraphs in this case are the, so motic subgraphs are subgraphs such that either, when you cut an edge, either the loop number goes down or when you cut an edge, it no longer connects to all the momenta and contains all the masses. So here, in fact, there are just two strict motic subgraphs in this case. This one, because if you cut any edge to, for example, it will no longer connect to this external momentum. And if you cut one, then it will no longer contain the massive line. One. And likewise, this is also motic. And these motic subgraphs tell us that something happens when certain coordinates go to zero. This tells us that when alpha one equals alpha two equals naught, the polynomial vanishes. And likewise, for this one. Sorry. You consider the remaining coordinates intercom. Alpha one equals alpha two. You have omitted alpha three. But you say now alpha one and alpha two. So a motic subgraph defines a subset of edges, subset of variables. And when it's motic, it tells us that there's something bad in the geometry happening when those variables go to zero. What means motic? Motic. Yeah, I defined it last time. It's quite a technical definition. It's some type of subgraph that depends on the kinematics and the masses. Which carries a seeminality. Yeah. It's a slightly subtle definition. Yeah, I'm afraid we won't need it this time very much. OK, so here's a picture in projective space. So we have L one. So last time I called the vanishing of the hyperplanes was called L. And this is the graph hypersurface x psi g. We also mustn't forget the other graph hypersurface. It's going to lie somewhere. It's going to go around, but it's a hyperplane. So I'll just draw one part of it here. And so there's something, a non-normal crossing situation going on in precisely the motic loci where those variables vanish. This is the domain of integration here. By the way, when all the masses, if all masses were non-zero, I would have only u v problem. If all masses are non-zero. Yes. Yes. Well, then the only motic subgraph would be the whole thing itself. So it's only when all alphas are zero. Yes. Well, all alphas zero. That's not a point of the projective space. You can't take all alphas zero. But it's nowhere. You can't take all alphas zero in projective space. Yes. Yeah. So you're only interested in strict motic. In that case, the only motic subgraph would be the whole subgraph, the whole graph. But we are only interested in strict motic subgraphs. And another way to say it is if all the alphas go to zero, that defines a point that's not in projective space. But still, this is where you want to renormalize the thing. No. No, I don't. Because, OK, you can have an overall. So we've already taken that into account. There was a gamma factor in the Feynman integral in the first lecture. And there's an overall divergence that we've already taken care of. So that's when you pass the projective coordinates, you're already regularizing the overall divergence of a graph. In this case, there won't be one. Yeah, so then last time we did some blow-ups along all the motic subspaces. And in this case, it looked like this. We blow up the red points. And we get something like this. So this has boundary divisors d1, d2, d3, and d1, 3, d2, 3. So this is what I call the Feynman polytope. It was defined in a canonical way, sigma g tilde. And in this blown-up space, the graph hyper surface, as we'll show in a minute, looks like this. And it was called y. And now I'll be the other one out here, y. That's the strict transform of the hyper surfaces downstairs. So that's the picture we had. Oh, this works now. Great. That's where the denominators are of the integrand. All the possible singularities of the integrand are red. And the boundary of the domain of integration is white. So in general, we had xg contained in PNG minus 1. And we also had these coordinate hyperplanes, le. And we blew up this precise prescription with these motic subgraphs to get a new space, pg. And in it, we have the strict transforms of the hyper surfaces. And in the case when there are no masses or momenta, we only have, we ignore this second hyper surface. So these are the strict transforms of the respective x's. And here we had a device I called d, which has devices of two types, those which are just facets that came from coordinate linear hyperplanes downstairs. And then there are new ones which come from blowing stuff up, gamma motic. So for today, we could say motic is precisely that which needs to be blown up to get a minimal compactification uniformly for all Feynman graphs. And last time, the main result was that we got a, so this is strict normal crossing. And last time, we got a recursive product structure on everything, on the graph hyper surfaces, on these devices, and on the space itself. So let me write that down. So first of all, for the boundary devices, so dE is a point cross dG slash E. You contract the edge in the graph, sorry, that's not right, pG slash E, that's what I meant. And for the things we've blown up, there's a product structure, d gamma is p gamma cross pG mod gamma, that's the quotient graph where you contract all the connected components of gamma down to a point, down to a vertex. And these graph hyper surfaces have this remarkable structure that when you intersect with the boundary, they likewise decompose as a kind of product. And yG intersection d gamma equals, yeah, y gamma cross pG mod gamma. dN has two components, p gamma cross yG mod gamma. And these identities were proved by proving some asymptotic factorization formulae for the graph polynomials. In the limit as some of the alpha variables go to zero, we saw that these psi's and psi polynomials miraculously factor into products of polynomials of the same type. To lowest degree, yeah, to lowest degree, which is the key to many things. Okay, so now I want to interpret this integral as a period of cohomology, as I explained in the second lecture. So we need to think about the integration domain. This is the Betty homology class. So to make sense of this, we need to define a good space, good parameter space for the kinematics. So I will define, for want of a better word, the generalized Euclidean region. I show my ignorance, perhaps there's a better and another name already in the physics literature, but this seems fairly reasonable. So that will be a set U, which will be in some space of kinematics, the space of possible q's and m's, by which I'll be deliberately slightly vague for now, where we demand that the real parts of the squares of the masses are positive for all edges which are thickened, which carry non-zero masses. So I think in the first lecture, I called the edges which had zero mass, the vanishing mass, Vm, the mass vanishing edges. And we'll also impose that the real part of the sum of the partial squares of the momenta, so the squares of the partial sums of the momenta have positive real part for all subsets of external vertices, external half edges, which carry momentum. And this is contained in a space where in this space, which I didn't write down last time, I perhaps should have done. So this is what I vaguely referred to last time as the space of generic momenta. So there was this assumption, all these blow-up conditions were conditional on being in this region of generic momenta, and now I'm going to restrict to a strict subspace of that for the following thing. This was in the first lecture, this was the set of internal edges which don't carry mass, and this is the set of external edges which don't carry momenta. So maybe I shouldn't have used this notation. This means that for all edges which carry mass, which are thickened lines, if the mass is non-zero, it's strictly positive in a sense. If the momenta are non-zero, then that's strictly positive. Because we have some legs which carry zero external momenta, and there's no condition on those, of course. But of course you could introduce a more standard way of working. I'll come later, yeah. Yeah, sure you could. So a theorem for all masses and momenta in this space, the Feynman-Pottitope, which I defined last time, the pullback of the closer for the analytic topology of the pullback of the interior of the coordinate simplex does not meet the strict transform of the graph hyper surface. So in that picture it means that the red line does not meet the polytope over which we want to integrate. Did I forget something? No. So proof. So the proof is by induction. This polytope is stratified by its faces. So in this picture, the stratification it has, I don't really know how to draw it, but we have the big open cell in the middle, and then we have the open, could I mention, one faces and so on and so forth. So the big open stratum, let's call it sigma tilde g o for the interior. This is contained in pg minus d. By definition of the blow-up, this is isomorphic to png minus one minus the union of the coordinate hyperplanes. And this contains the open coordinate simplex, where alpha i is strictly positive. And this isomorphism sends this open stratum isomorphically onto, homomorphically onto this thing, sigma naught. So here's a point that I said was an important point last time that seemed very trivial, and I sort of glossed over it. But that's that the graph polynomial is not identically zero. So I said that carefully last time, and this is where it's important, and the second, this psi polynomial is also not identically zero if in the case when g has any non-trivial kinematics at all. So this was crucial in the definition of MOTIC, that this property- Can you please look at the q and m of u? No. Well, it has kinematics, but no, but the case q and m is zero are incorporated because we only look at- No, but- It would be a different graph. I don't. I said I don't have a denominator of psi if I have no external kinematics. Because it's a sum of an empty set, so you get one or something like this. When you get a product of an empty set. Yeah. Yeah, if there's no kinematics, we shouldn't have psi in that. It's just a power of psi that you integrate. If there's no kinematics. Yeah, it's just a power of psi you integrate. Like in the examples I gave in the first lecture. Yeah, exactly. Where was I? Okay, so now the point is that the coefficients in the graph polynomial of psi g, that's why I wrote it up, are all plus one, are all one or zero. But it's not identically zero, so that implies that the graph polynomial is strictly positive on the locus where all the alpha i's are strictly positive. And similarly, the coefficients in psi g are all things like squares of momenta and me squared, stuff like this. And that implies that psi g, because it's non-zero, is positive, I'll put the qm back in, on sigma cross u. Where u is this domain here. So that means that yg intersect sigma g tilde open is xg intersect sigma interior, and that's empty. So we've shown that the graph hyper surface doesn't meet the interior of this polytope. And now by induction, we just need to check all the faces. So proceed by induction on the faces, but the faces have the good measure of being of exactly the same type, their graphs, their Feynman polytopes of quotient, Mottic and Sub and quotient graphs. So this is a formula we showed last time. And so by induction, the components of the graph hyper surface on the faces are again graph hyper surfaces, and we've already shown that they don't meet the corresponding polytopes. So that's done. Okay, fine. So we can define now the graph motive. Again, it's an abuse of the word motive. It's really a mixed order structure. Mott g, which will be the cohomology ng minus 1 pg minus yg relative to d minus d intersect yg. And we think of this as an object in some category. I'll be more precise about this a little bit later on. We think of this as a Durham cohomology group and a Betty cohomology group and a comparison isomorphism. So that was in lecture two. I'll be a bit more precise about that in a little bit later. And so the point of this previous theorem is that we can take the homology class of the domain of integration of this Feynman polytope and it gives a well-defined class, which is in fact universal. It doesn't really depend on the construction. It's the same for all graphs, which is nice. Sigma g, which to save on notation will mean the homology, the relative homology class of this Feynman polytope in the Betty homology of this thing. And this is the Betty realisation of this relative dual. So d refers to the boundary of the integration domain? d is the divisors, these divisors here, and the integration domain. So the domain... Absolutely. So we're looking at... We haven't done the differential form yet, but the boundary of this... First of all, this theorem says that a priori, sigma g tilde, is contained in p g r, right? So it's a homology cycle in p g relative to d. But because it doesn't meet y at all, we can restrict to p g minus y g with impunity. And the boundary of sigma tilde g is contained in d. And in fact, it's contained in d minus d intersection y g because of the theorem. And so that's precisely saying that it defines a relative homology class in this group. This bound is... Yeah. And that was the whole issue that we had to go through with z to two example in lecture two. So the issue was getting the domain of integration well-defined homology class. So a mark is that mod g agrees it's the same definition with the so-called graph motive defined by block-in-on-crimer in a special case where there are no masses or momenta and possibly some further restrictions. Okay, so now we... So we have a domain of integration. We have a homology class. Now we need something to integrate. So we need the diram co-homology class. Okay, so we had... Recall that pi g is the map from this blown-up space is the blow-down to projective space. And we want to write our integral, our Feynman integral, q m. We want to put it back to the space upstairs. So we're going to integrate over the Feynman polytope and we're going to integrate the pullback of the differential form, omega g, q, m. And we're going to take q and m in, of course, in this space, a subspace of generalized kinematics. Sorry, the generalized Euclidean region. Otherwise, so this is also always conditional. So I should... Oh, it's written here. I said very clearly that it's... This depends on the fact that we're in the generalized Euclidean region. But still, this integral can still diverge and that's the whole problem of renormalization. So now I claim... It's not very difficult to see, but it's sort of a general fact that our domain of integration... They're two key remarks. First of all, the domain of integration upstairs is a compact polytope. And secondly, that the real part of the thing we want to integrate is strictly positive on... So we're integrating essentially a positive function on a compact... No, it's... It's dull because you raise to some powers for actual things and real parts can be no longer positive. Are raised to powers? Yeah, because if you form, you make a ratio of two polynomial. Yeah, these are raised to integer powers? No, but if now the flow is not strictly real, not real, but complex, and after raising to power, we don't get to the positive real part. Yeah. Yeah, I think it's made of a small domain, if you're... Yeah. Yeah, it's still on real points, yeah, but it couldn't draw near real points. No, but I'm on the real... Give yourself real points. Yeah. Okay, small detail. Okay, thank you for that. Yeah, that slipped my attention. So, well... Okay, maybe on some slightly smaller domain. The integral is absolutely convergent on some smaller domain, let's call it V. If and only if pi upper star g omega gq has no poles along the gamma, where gamma is motic. Okay, yeah, so perhaps we need to shrink the domain a little bit. And so, to get some, or some sufficient conditions for convergence, we just need to compute the order of the pole along each divisor d. So, yeah, okay, so definition. The superficial degree of divergence, SDG equals the number of loops times dimension over 2 minus ng. And the following lemma, which is rather easy to prove, that the order of the pole of pi upper star omega gqd along a boundary divisor d gamma is either 1 plus SD gamma if the subgraph is not mm. So, mm meant mass momentum spanning. It meant that the subgraph connected to every external momentum, which was non-trivial, and contained all the masses, essentially. And in case when it is mm, so let's correspond to an infrared sub-divergence, we get this formula of gamma is mm. And the proof is use the factorization theorem as I stated last time. Plus mm means, it was a definition I gave last time, mass momentum spanning. So, it meant that me, if there's a massive line, then the edge is in the edge of our subgraph. And it means that some connected component of gamma meets, if you like, or contains all external legs external half edges, which carry momentum. Oh, sorry, yeah, sorry. I feel like I've been lecturing continuously for six hours. Okay, oh yeah, so use the factorization theorems. This is a simple calculation. You just use the factorization theorems, which I gave last time, and then use explicit coordinates for blow-up. In fact, there was a formula last time where we computed the order of vanishing of a graph polynomial along a divisor using these factorizations. And the blow-up coordinates were simple changes of variables like b to 1 equals alpha 1 over alpha i and so on. You just plug in these variables. So, this is very straightforward. It's called power counting in the physics literature. And so, we get conditions for a finite integral to converge, which may or may not be satisfied. So, let's call this condition here that it has no poles, the convergence condition, that it has no poles along the divisor's D. So, if star-conv holds, we can define the Dirac class, pi upper star omega g in the Dirac group associated to the motive of g, which is h and g minus 1 Dirac of this whole thing, pg minus yg relative to d minus d into section yg. So, of course, when it doesn't hold, then we have a problem, and that's the subject of renormalization. So, it can happen that when you pull back your integrand upstairs, you find you still have poles along the divisor's D. And in the ultraviolet case, we know how to deal with that. You renormalize. It's a different topic of what I'm talking about, but it's just an inclusion-exclusion on this picture. Essentially, you have a, when you pull back your differential form, you find that it has poles along some of these divisors. But because of this recursive structure, the residue will be, if it's a simple pole, the residue will be a product of Feynman integrands of sub-unquotient graphs. And then you need a recipe, which comes from physics, to find a different form which has the same residue along that divisor, and you just subtract it. And you do inclusion-exclusion. You subtract these singularities along this polytope by induction on the dimension, and that gives you exactly the prescription in physics to renormalize. And you get the BPHZ prescription, and the forest formula comes for free out of the geometry, and the Canon-Somanski equation pops out as well, again from this sort of factorization structure. So it's the geometry which determines everything. That's this ultraviolet, yeah. For infrared, I don't know what to do. I think. Change of diver. Yeah. So on the nine point of view, it's the same interval in which you're changing. Absolutely. So if it's diver, it's still diver. Absolutely. Absolutely. So I'm just saying that I'm not considering the case of diver. And so we could put this in some more work. Yeah, so one thing to do is to incorporate the case of the theory of renormalization into this business. So with duck crime, we defined the corresponding motive for divergent processes which have just a single scale. It's a different geometry. So here we concentrate on the non-divergent. We're just looking at it. I'm just looking at integrals which converge. Which means UV infrared safe. Exactly. UV infrared safe, absolutely. So three. Funny, yeah. Motivic Feynman amplitudes and periods. Okay. So let G be a connected Feynman graph as above. QM in U. And let's assume that our integral is convergent in this geometric sense. And then we can define the Motivic Feynman amplitude to be, so I'll be more precise about this in a minute. But it'll subscript M. I, Motivic GQM. And it's an element of a ring of Motivic periods of some category, some Tanakin category H about which I will say some more in a minute. So I'll say something about that later. So that's the following the construction in the second lecture of thinking about, a Motivic period is a, is a, an element of the affine ring of the torsor of isomorphisms between, from a diram to a betty fibre functor in a certain category of realizations. And the gain is that we get a Galois action on it. And the period, of course. So these things had a period map, period homomorphism, and it goes without saying that the period of the Motivic amplitude is, of course, the amplitude we started off with. So it had better converge otherwise, you know, that's why we certainly need to restrict ourselves to the convergent case for the time being for the first step. So any, any. Okay, so what's, oh sorry, I forgot to define it, my apologies. Motivic final amplitude is, what is it? Is the triple, the matrix coefficient, the Motiv of G, then framed by the pullback of the Feynman differential form, upstairs, and the homology class given by the Feynman polytope. And this, this class is indeed a class in the Dirac homology up here, and this is a class on the dual of the Betty realization of this thing. So that's a perfectly well-defined Motivic period. Now the next definition is slightly confusing, but I don't, couldn't think of a very good terminology. Now instead of looking at, so we can change this, we don't have to look at this differential form, we can look at a different differential form that converges and integrate that instead. So we can actually look at any Dirac class, and I'm just going to call it omega, and I hope it's not confusing. No, I'm going to call it omega qm, because it'll depend on q and m. So omega subscript of G will be the canonical one that comes from the physics, the actual scalar Feynman integral we're looking at. This will be a generalized integrand. So we can allow ourselves to look at these more general integrals, and recall this, the Motivic Feynman, not amplitude, but period. So maybe this is a bad terminology, but I want to use the word amplitude to mean the original integral we started off with, and period for a generalization where we consider any more general integrand. And so Img omega, I'll drop the q and m, will be omega sigma G. So again, a Motivic period in some category that I haven't quite defined yet. And we'll denote the period by iG, iG omega. So what is this? Let's look at some examples. So an example of such a generalized Feynman amplitude or Feynman period would be something like this. In the case where we have kinematics, we could take any numerator and we could raise, we could take the denominators to arbitrary powers. So that would be an example of a Feynman period. It's not necessarily a Feynman amplitude. So here A, B, we could put polynomials in the kinematics. And in fact, you'd put elements of a Clifford algebra if you were considering a gauge theory. Yeah, of course, thank you. So I'm just going to write that. A, B and Z and P as rational coefficients. So it's any polynomial in the Schringer parameters, such that the whole thing is homogeneous. So P in particular is homogeneous. And the degree has to match up. So such that the entire differential form defines a form on projective space. It needs to be homogeneous to a degree 0. You should try to form along the rest of P. That's the next condition. Thank you, degree 0. And it has no poles along d gamma, gamma modic. So by a simple power counting argument, you can write some explicit inequalities that this polynomial has to satisfy. You can write sufficient conditions. So an example of such thing are all gauge theories. Well, I believe certainly most gauge theories. When you write, when you take a gauge theory and you write it in a parametric representation and you do the Schringer trick and so on, you get an integrand of this form. And the numerators will contain stuff in a Clifford algebra, for example, that contains all the data of your gauge theory. So Kremer has been working on this recently and has an efficient way to produce integrands. Can you hear me? Sir, it means all integral sets are convergent. All integral sets are convergent. So an example of a Feynman period is an integral of this type that is convergent. Ah, convergent sets conditional integral and gauge. Oh, so you're asking about gauge theories? Yeah. So what was the question then? No, actually it's a parametric gauge series. And gauge theory won't automatically converge, no. It will still die hard. Again, you have to renormalize. This is an offhand remark to say that when you consider amplitudes in a gauge theory, you normally write them in momentum space or something, or then if you write them in parametric form, it will naturally produce integrands of this type with higher powers of the graph polynomials occurring in the denominator and some typically extremely complicated numerators involving traces of gamma matrices and all sorts of stuff. But the general shape is of this. So in particular, the amplitudes which are convergent that you get from gauge theories will always be linear combinations of these, will be periods of the same motive. So if you know all the periods of your motive, you know all possible amplitudes for all possible theories with that topology. So you're saying it's like the theory of Avalian integrals of the first and second cancel that you can express all Avalian integrals in combination? Absolutely. So I was going to say this later on, but let me be set now. In some sense, the keyword for physics would be the phrase master integrals. So there are lots of huge amount of work trying to reduce amplitudes to a small class of master amplitudes out of which you can express everything. And the mathematical way to think about this is just to write down a basis for the DRAMc homology. So I feel like saying that if you write down, if you can compute this homology, and we'll get this at the end, and write down the differential forms and calculate the periods, then you're done. You've done all possible, you've computed all possible amplitudes. And then whichever theory you look at will pick out which particular linear combinations of amplitudes you need. So one should only compute Feynman amplitudes once and once and for all, and then never again. And the particular choice of theory will select which amplitudes, well, sorry, which periods are of relevance in that theory. OK, so these are examples of what I call Feynman periods, and I hope the term is not confusing. OK, so now finally the cosmic Galois group. So let graph Qm, the set of connected Feynman graphs with Q external momenta. So we're going to fix the number of external momenta because this is what we observe. Some particles coming in and some particles coming out. And in a theory with m possible masses. These are the possible masses that particles are allowed to take. And we can define a space of kinematics, somewhat crudely to be AD. So this is the possible momenta we could have, and these are the possible masses we could have. And the field of fractions of this affine space we call K. And it's just the field in which we join the momenta and the masses. And of course this is a slightly dumb thing to do because as physicists know extremely well, we only insted in the invariance under the orthogonal group. So it would be better to look at just the dot products of momenta and so on. In other words, use model sum variables. Sorry, square of masses, yeah. Because the amplitude only depends on some subset of the full data of masses momenta. So we can do better, but physicists know this extremely well. There's no need to go into details. Yes. Oh, sorry, yeah. Oh, thank you. Thank you. So yeah, I must say these are called model sum variables. But we don't really, we can just work with this for now. So HQM will be a category whose objects are, so apologies for being slightly technical. It'll be something betty, something diram, with a weight filtration and a hodge filtration and a comparison isomorphism. And these will be on some, as a risky open subset of the space of kinematics. So we fix the number of masses and the number of external variables we're allowed. I mean the masses can occur with repeats of course. It's just the number of possible particle masses that are allowed. Here the category consists of triples, or rather pairs, where this is a local system of finite dimensional Q vector spaces on the complex points of sum open in the space of kinematics. V diram is an algebraic vector bundle over Q, the finite type with a flat connection and regular singularities at infinity on SQ, sorry on SQ. And there's a whole bunch of axioms that I don't want to go into. So this is, so think of this, we had this co-homology before, we had in lecture two we had betty homology, we had a betty homology of something. So we had mod G, okay so fiber-wise we had mod G had its betty incarnation which was PG minus YGC relative to D minus D intersect YGC and it also had the diram which is differential forms. Okay so now I was thinking of YG as static, we fixed Q and M, now I let Q and M move. So this is a vector space that is moved, this is a finite dimensional Q vector space that moves. So that defines a locally constant sheaf, it's called the local system. And on such a thing you have a monodromy in action of a fundamental group which encodes the notion of monodromy. On the other hand differential forms, now again we can move the differential forms because they have parameters in them and we get a vector bundle. And we can also differentiate with respect to masses or differentiate with respect to any parameter and that gives us the Gauss-Mannien connection which is this connection here. And on each fiber we can compare betty and diram after tensioning with C and that's what I explained in the second lecture. And if you package all everything one would like about this possible structures we have some category in which it contains all the data we might want to use at some point. But it contains this, for first this this is the Picard-Fuchs equations it contains all the information about differentiation with respect to masses and momenta. This thing contains information about monodromy. I'm sorry it's slightly technical but yes. C is comparison so remember that these after tensioning with C that's integration that tells us how to take a differential form and pair it with an integration cycle. So this satisfies a bunch of axioms that's slightly long but that's why I postponed this to the last lecture. We have a fiber functor omega diram so actually the simplest is to view this category over this big field. We have a fiber functor to vector vector spaces over this big field of fractions which is the fiber at the generic point of the vector bundle V diram. And so, oh sorry H, so HQM is a Tanakian category over the field KQM. And if you don't like all this stuff then we can just take the case Q and M to be zero no masses no momenta and then it's just Q and this category H naught naught is exactly the category I called H in the second lecture. Okay so from this we can define the cosmic galore group at last. So, oh yeah so we have, as I explained in the second lecture we have the automorphisms of this fiber functor is an affine group scheme and it acts on the diram of the cohomology on the differential forms if you like for all graphs of the type we're interested in. I beg your pardon? And I can take the model because this field can take over rational numbers. Yeah, I'm nervous about that. You want to say that you want to do some descent. So the force of local system. Yeah, I'm nervous about that. I'm nervous about that and the reason is because the periods will involve rational functions of masses and momenta. So if we had wanted to have a tenacian category of a Q then the periods of weight zero would, so the galore events would be rational numbers and we'd only get rational numbers out. But clearly when you can write down an integral that gives a rational function in Q and M. Still this category of... It's the define of a Q. I know it's kind of define of a Q and I'm nervous about this because these theorems in, the claim is you can construct... The periods of rational functions are incidental functions of Q and M. No, no, no. But the coefficients of the periods will have to be rational functions of Q and M. Yeah, so you can write, if you write down the first final integral we compute will be you can write down an integral that's M1 squared or something. So this is a period of a trivial representation and means you have to have coefficients in M. Yeah, you can apply it by rational... And you can write down, you know, the integrals will be of this type. So it's crucial to work over this bigger field. And if you want to, if you could find a point that's... Or you could then specialize and fix the massive momentum and go to Q. But I don't want to do that. No ramification depending on the family just doing M. That's what you say. No, no, there could be ramification everywhere. One of the things about this is that you could... These final integrals could have singularities over a subspace of kinematic space. It becomes dense when you look at all Feynman diagrams. So you could have ramifications as a dense subset of the space and that's what makes me nervous when you want to do a Witt rotation. So there are results in the physics literature that claim that you can continue on the physical sheet or something. I don't understand them and I don't want to go into that. So there's something that needs to be seriously looked at. Okay, so we have a Tanica group that acts on all these things and it has a huge subgroup which is trivial. So let Triv QM be the subgroup acting trivially on all cosmologies of Feynman diagrams. Yeah, sorry. And then finally we can define the cosmic Galois group. And long last, G cosmic QM is this huge Tanica group. Modulo, the bit we're not interested in which is the one that acts trivially on all Feynman diagrams. So this is an affine group scheme over this field. And so what we get is this group acts on all notific Feynman periods. PMHQM for all G in graph QM. Note that it can happen that your graph, the Feynman amplitude you're initially interested in doesn't converge but it has other periods by putting numerators or so on for which it does converge. So we need to consider all the graphs, not just the convergent ones. Some of the genes are for Galois and some are for graph. Oh, yeah. That's why it's got a cause here. CUS. Okay, so some consequences and then we'll have a break. So, yeah, the first remark is then what are the Galois conjugates of a, let's just look at the principle, sort of the Feynman amplitude, the original one we understood then. The Galois conjugates of a convergent Feynman amplitude are the Feynman periods, are the motivic Feynman periods. So if you're interested in the structure of the periods, you're forced to consider these more general types of integrals, I, M, G, omega. And there's a formula for the corresponding co-action, which is, I'll remind you from the second lecture. So we have a co-action nabler from the ring of motivic periods to the affine ring of cos QM. So this encodes the action of the Galois group on the ring of periods. And this is, did I say nabler? I meant delta. Thank you, sorry. So the formula is this. You take this matrix coefficient, let's apply it to a general Feynman period, and it maps to omega EI Chech. And the sum is over where EI, a basis of the Dirac homology, which is something we can compute. And that EI Chech is the dual basis. So this is a completely computable formula. That's why you're forced to get out of there. That's why you're forced. So the Galois, you start off with an amplitude here and you're forced to consider, the Galois conjugates are these guys. So that's your force to consider more general integrals. But still the same Galois. Still the same. But geometry is the same. Geometry is the same, yeah. Can you explain please what it means without break? Or should I join the break? Do you want to break? Yeah, maybe we should have a break now and then I'll give the consequences afterwards. Okay, I'll just give a list of consequences and I'll come back to Tipo's question in a minute. So what's the upshot of this machinery? It's that every motivic Feynman period, in particular Feynman amplitude, I mg omega now defines, in fact it generates a representation, a finite dimensional representation of this huge group. But we shouldn't take the group so seriously. Something like absolute Galois group of Q or something, it's huge but what we're interested in is these finite dimensional representations and there's something that we can compute in cases using this kind of formula. So in particular every amplitude now defines a representation and it has a dimension for example. So we get a new invariant of a Feynman graph which is the dimension of the corresponding representation. We also gain a weight filtration on Feynman amplitudes and motivic Feynman periods. Another remark is that the period... Can we also get connection? Yeah, we get all sorts of stuff. So I'll give a small list of consequences but there'll be lots and lots of applications although I don't have time to even scratch the surface off, as you say. A remark is that the period of the actual Feynman amplitude, because of the geometry we did, actually it's immediate to show that it's a single-valued function on the Euclidean region. So as Maxim pointed out, we also get free, we get pick-out-fuchs equations, we get a connection. So we can differentiate with respect to masses and momenta and we can apply all the theorems from algebraic geometry and we have monodromy and all these structures are compatible with this Galois action on this co-action, with the motivic co-action. So let me say something about pick-out-fuchs equations. It's clear what happens when you start off with an integrand that depends on masses and momenta and you differentiate under the integral sign, so we differentiate omega, then it's the same as differentiating omega here and omega doesn't appear here, so you see immediately that there's an equation that the action of Nabla factors through the left-hand side of this formula. So there are lots of things one can do in that direction. We can also then rigorously speak of mixtate, mixtatic periods. So those are the mixtatic Feynman amplitudes where the mixtate structure is equivalent to one which is mixtate. And these ones don't just have a weight filtration, they have a weight grading, so we can actually speak about the, we can assign a degree to each of these things and they typically give polylogarithms by general theorems about variations of such things. And from this we can define symbols. So I mention this because it's become a very big industry in physics at the moment. So the symbol, there's a considerable confusion about the symbol, but the symbol it would seem to me would be what you get from this material period where you forget almost everything. You forget the domain of integration and you just retain the differential form and this connection. And then from that you can construct some symbol from it and people have been using this a lot. But in some sense the symbol is a bad thing to do because you've lost the period, you've lost the connection with numbers. Once you throw away the domain of integration you are not allowed to integrate anymore and so it's much better just to work with the material period. So I just gave a list of applications of this theory. So for example now we have something like the anomalous magnetic moment of the electron which is a sequence of numbers and now we can replace it with a sequence of representations. So numbers can be lifted to finite dimensional representations of some group and what that means for physics I have no idea but I think it may have some impact on questions of resumability and so on. Resumability. So when we want to resum you're adding, we're naively adding numbers up together and we hope that it's convergent or Borel resumable but these numbers aren't just numbers, they're representations. So perhaps we should sum more intelligently and take into account the types of representations. There we go. So can we understand this automorphically in one example? What is this action on this set of variables? So these are the things I gave in the very first lecture. I gave, oh which you missed, I gave some examples. I gave examples like, I gave something like log 2 and the co-action, I mean the Galois conjugate is just 1. So the representation associated is is q log 2 sorry, some q. So it's a two-dimensional, I call this v I think, it's a two-dimensional representation and the, how does the group act? Well given a g in the appropriate Galois group, gq, it sends, it acts on log 2 via, it scales it by some rational multiple and it adds some other rational number. So this is in q star and this is in q and so the action of this group factors through, and so the image of the Galois group in this case is an additive group, semi-direct and multiplicative group. It's fixed because it's a rational, it's the invariance under this material. So I gave lots of examples like this in my first lecture and... In a way it allows to change the branch of the logarithm. In some sense, yeah, in some sense. In a theoretical way it changes the branch of the logarithm. Absolutely. So when we have, so I wrote down the motive cross-pointing to this I think in the second lecture and we computed that it was two-dimensional, we computed a basis for the cohomology and just from the definitions we worked out that this was the way that the Galois group had to act. And so you have numbers and you know how they are transformed under the native Galois group. So this one here has weight. No, because I'm looking at the geram, geram Galois group. So let me just finish this chain of thought. So we have numbers and we know how they transform like multiplicative values and we have five minitacles and I'm going to say in a minute something about how they transform and when we put the two together we will get some very strong constraints about the possible numbers that can occur in physics. So to come back to Maxime's question about geram the reason I look at the geram action is because this Betty class is given to us. The last thing I want to do is move it around. It's absolutely canonical and it's really simple. The differential form on the other hand is something that changes for every graph. It's got all this complicated psi and psi in it. So moving that, it's already complicated. It doesn't matter moving around but I don't want to mess with the Betty class. Is every finite dimensional representation of this time, I mean, of this group the cosmic Galois group by definition every finite dimensional representation of this cosmic Galois group is a, yeah. Yeah. So one way to define that the Hopf algebra of this group is by the exactly these symbols and with the co-product, well let me write it down. It's the Hopf algebra generated by symbols mod G V, F, D, R. There's an equivalence relation hidden in here that's very subtle, don't forget. And then the co-product is the same formula exactly as I wrote in the second lecture. Now V and F are of the same family. Yeah, so V and F. So the Dram here means in the notation of the second lecture omega Dram, omega Dram. So V is in mod G Dram and F is in mod G Dram dual. So if you like, this is a more concrete way to do it. You look at every Feynman graph or even just a fixed Feynman graph. We're not interested in all of them. And we have this equivalence class of triples and there's this explicit formula for the co-product that defines a Hopf algebra and its spectrum is the quotient of this huge group which acts on this particular graph. So the huge group is artificial. It's like a projective limit of all these all possible or the Hopf algebra is the limit of all possible such Hopf algebras. It's a bit of multiplication corresponding to the cosmic Galois group. Yeah, so this corresponds to multiplication of the cosmic Galois group, yeah. And but the problem, yeah. It's always in Galois field. I mean you have a huge Galois group and you control my quotient. Yeah, so we shouldn't take the cosmic Galois group so seriously. It's just a a projective limit of all all possible quotients that we'd ever be interested in, okay. It's a projective limit of finite dimension of the group. Absolutely, absolutely. It's a pro algebraic group. Okay, so there are a lot of consequences of this set up, but in some sense what we've done so far is nothing because there's no theorem yet. And it's just a definition. It's just a formalism. And to actually make this do something useful for us we need a theorem. So that's what's coming next. So we need to use this very special geometry. So I call these face maps and we're going to define relations between motivic periods for different graphs. So we recall that we had d e contained in p g and d gamma, these facets and we have the inclusion, there's a morphism which is inclusion of facets on more generally faces but let's look at facets. So the complement of the graph hyper surface in each facet can be embedded into the big space at the bottom of the boundary. And likewise p gamma minus y gamma cross p g mod gamma minus y g mod gamma embeds. And so I claim that these induce morphisms on cohomology on relative cohomology both Betty and Ram. So this is a sort of standard fact but let me try to motivate it with a very simplistic example. So when we do relative cohomology as I mentioned in the second lecture there's a long exact cohomology sequence when you have a sub space z contained in x, I think I wrote this down in the second lecture. And so the long exact sequence gives you a map from something in the boundary into the relative cohomology and it changes the degree. So we're interested in this map where you the boundary goes into we get some information from the boundary. So theorem for every facet f so it'll be of the form d e it's one of these devices d e or d gamma a blown up one exceptional divisor. There is a canonical map I f and according to the two different situations we're in it will send the motive of the quotient graph in which we've contracted an edge into the motive of the full graph and the tensor product of the motives of the sub graph and the quotient graph into mod g in the case where f equals d gamma. And what does it do on the Betty so this does something terrible on differential forms it's quite hard to compute what this does on differential forms but on the Betty things it's very nice and that's one reason why I don't want to change the Betty classes ever. So it'll give a map the other way and what it does to our polytope is very nice it takes essentially the final polytope in the big space and you just take the corresponding face the corresponding facet in this case Is it not sigma tiled up or into what? Yeah why I did a slight because I was lazy with notations I called sigma g was the homology class of the tilde I wanted to simplify notations because I didn't want to keep writing Now by geometry you have to blow up something Yeah we've already done the blow up so we're just it's just the restriction map the map when you intersect with the face and on the level of of homology classes it does this but geometrically it's just restricting intersecting with a taking that part of the boundary so this implies face relations between matific final periods and what are they so by the definition of a matrix coefficient amorphism gives us a relation so what are the let me write them out so the matrix coefficient mod g e omega sigma g e is equivalent to i f to the r of omega and then sigma g and we get another relation so omega 1 is equal to sigma g so what is this, this is an equation that takes a generalized Feynman period for big graph and says it's actually equal to certain generalized Feynman periods on the big graph is actually equal to a period of a smaller graph, a graph in which you contract an edge and likewise here it's saying that certain periods for big graphs are actually products of periods for sub and quotient graphs so these are relations between periods and this is the key to the whole story and likewise we can either iterate this or look at faces of higher code I mentioned, I only looked at facets just for simplicity and the remark and so I'm running behind so I'll say the remark in words is that curiosity here is that if we started off with a graph with general kinematics it's always the case that either the sub or the quotient has no kinematics at all and that means the amplitudes of diagrams with no kinematics plays a very special role so if you're only interested in Feynman diagrams with lots of external particles when you study the periods they will always keep dipping into the space of periods of graphs with no kinematics at all and physicists often say to me why are you looking at five to four theory you know it's not very relevant theory it's got no masses, got no momenta, we're not interested in that there are lots of good answers to that criticism but this is probably the best because it says even if you're interested in graphs with many momenta and many masses you will end up finding these periods in the massless and momenta case coming in whether you like it or not okay so now I need to say something so then the key point now is that every every Feynman period of small weight will actually come from a sub or quotient graph so so now I need to talk about weights and then state the main theorem so this ring of periods he has an increasing weight filtration and it was defined by the weight filtration on the DRAM realisation on the weight of the differential form in the second lecture so perhaps I should remind you that a motivic period m omega sigma is of weight less than n if omega was in weight n m DRAM some behaviour at infinity according to the linear the weight is defined by going to some compactification oh we'll see how to think about the weight it's kind of subtle so a general fact about the weights we had the motive is in DRAM is this relative thing and it's a general fact we know that it has the weight minus 1 part of this is 0 and the weight 2n thing is the entire space so that means it's weights are contained in 0 2n in some sense it's a filtration of course but you can write that the weights are contained between 0 and 2n so let's think about just as a baby example to look at weights we had this relative comology sequence and then it goes into n is number of edges minus 1 yeah thank you and then this maps to hn pg minus yg and so on and so forth so let's look at the sequence and think about the weights so this thing has weights anywhere between 0 and 2n and this thing by similar argument has weights anywhere between 0 and 2n minus 2 because the comological degree is n minus 1 and by by deline this has weights in the interval n 2n since this is smooth no no smooth is I don't know it's not affine because we've blown up it's not affine definitely not because we've got all these exceptional devices so so this is sort of a simple case to motivate what I want to do next so since we know that the a deep fact that the weight filtration is a strict filtration so that means that the graded weight is an exact functor and that means that if we had any class in der anker homology which is of weight and less than or equal to n minus 1 so that means that omega is weight n minus 1 what g'd around then it would go to 0 here because the weights are between n and 2n it would land in w n minus 1 of this which is 0 and therefore it has to come from the boundary it has to come from this thing so it actually lies in the image of this which is w in fact it comes from w n minus 1 of h n minus 1 dr d so the slogan is that the differential forms of low weight must necessarily come from the boundary it's not obvious that the weight is preserved in these mappings these are morphisms that makes hot structures so they preserve the weight, absolutely contrary to the dimension the degree of the differential form which is not preserved the degree of the differential form is not preserved but the weights are preserved absolutely so yeah so the diram classes of low weight come from the boundary ok so what's the general theorem let me write fi for the face of code I mentioned i so as I code I mentioned mod i so i is some indexing set telling me which devices d I want to intersect so i could be this point it's d1 intersect d13 I'm slightly behind so I won't write all the messy notations and each face defines a motive in the same way as we saw in this in the previous paragraph so we have a corresponding graph motive for every face and the theorem then which uses all this specific structure of the graph motive that we sum over all facets of a certain code I mentioned i equals ng minus k minus 1 and we look at the weight k part of the diram of the facet and we have all the face maps from the previous so these are i fi diram and it maps into mod g diram and the theorem is that this map is subjective so it means that every differential form on the big Feynman graph which is of low weight must come from part of the boundary so the weight zero periods come from the zero dimensional boundary stuff comes from the one dimensional boundary and so on and so forth so we have a total control on the low weight of this motive so the proof is this geometric product structure we have plus a box standard spectral sequence and the theory of weights so at last we can get to the main theorem the main theorem so g connected Feynman graph and let's look at its some metallic Feynman period so the main theorem which is a sort of Galois closure type of theorem and it says that every Galois conjugate g i m g omega so g and g cos of the appropriate let's put hqm so every Galois conjugate which is of low weight is something that we already know about it comes from a small graph so if it's of weight less than k then it is a linear combination of products where gamma i are quotients of motic subgraphs so quotient means contracting edges and such that the number of edges e gamma r is at most k plus 1 and the proof is the previous theorem plus the face relations well in particular every Galois yeah oh yeah it doesn't change much okay so what is this what is this theorem saying it's saying that that not every Galois conjugate is a period of a smaller graph but every Galois conjugate small weight is a period of a smaller graph so we want to think of this as a statement of the kind Feynman periods are closed are closed under the action of this Galois group with in this sense with this single restriction so that's a very strong constraint on on aptitudes and I'll explain explain how to make this work so this is the principle of small graphs so I hope this is starting to tie up with what I said in the first lecture and the point is that there are very few there are very few Feynman graphs with a bounded number of edges and so the main theorem so we shall write them down compute their periods and the main theorem will then give us a constraint on all amplitudes to all loop orders the main theorem will give constraints on the possible amplitudes all loop orders no I don't need you I can just say that I'll give some more examples in the spirit of the first lecture so what's the program then instead of computing amplitudes as numbers what we should be doing is computing the motifs compute for very small graphs and these are pathetically small I'll show you in a minute you get non-trivial statements by doing some very simple calculations and as I said before the cohomology if you compute this vector space so once you've done this up to a certain point you know everything about all possible master integrals up to that point and you know something about all possible periods up to that weight for any quantum field theory so what could have happened so if we were thinking about periods have a weight filtration you'd say okay let's look at the periods of weight one or two like the logarithm log two and you look at graphs of one loop two loops and you compute them and you get some numbers and a priority as you keep going up and up with higher and higher loop orders you keep getting more and more periods of low weight and it become an infinite space that's out of control but the main theorem tells you that that can't happen that it saturates if you've not seen a period at k plus one edges then you'll never see it ever again so let's do this well let's give some examples I'll maybe I'll give myself an extra ten minutes if that's okay since it is the last lecture so let's start off with let's start off with graphs with masses and momenta and do some really some pathetic calculations but the point is to illustrate the power of this theorem so the one inch graphs just two children I'm not going to bother with that two edges so what we want to do is write down graphs with two edges so here's a graph with two edges so two massive lines and momentum coming in going out q and the only interesting graph polynomial here is the xi which is this and so this stuff the hyperplane when p1 it's pretty simple this just defines a point in p1 so the motive is just h1 of p1 minus a point and there's nothing to blow up here the boundary the coordinate simplex is just 2.0 infinity and so this is just a q of nought so the only generalized periods you could ever get from this graph are rational numbers one of not interesting we only get rational numbers or rational functions of q and m and then there's another slightly more interesting one loop graph two edge graph which is this one and now the graph polynomial or the xi polynomial is a bit more interesting defines quadratic function of alpha 1 alpha 2 and so we can compute the motive there's also psi which is alpha 1 plus alpha 2 and the motive is just h1 p1 minus 3 points relative to 0 infinity and that we can we know what that is it's just direct sum of two kumar extensions and so the material periods are only log a 1 material logarithm of some quantity x1 and material logarithm of some other quantity x2 which you can compute it's just some function of q and m and I think it's called the Chilean function in physics and then there are other more trivial examples you know which are kind of trivial so that's it we've understood all we've classified all possible material periods that can come from two edges and so x1 is some some some function of q and q squared and the mi squared where given by the intersections of these hypersurfaces with the various coordinate hyperplanes so there's some rational functions it's some horrible expression square root of q squared minus m1 squared plus m2 squared minus 4m1 m2 so it's just given by Newton's formula I didn't want to write it down but in the case q equals 0 one of them is just m1 over m2 this is one of the one of the quantities you can get out of this if you put q squared equals 0 it's simpler and if you put m1 equals 0 you get I think something like log q squared plus m1 squared over m2 squared but the general things got some some square root some complicated expression I don't want to write it down you just solve this equation yeah you just solve this equation equals 0 or something you just solve these two equations in projective space and you get some quadratic thing I didn't want to write it down because it's ugly but I think it's called the Chilean function I'm not entirely sure okay so we've classified all possible metallic Feynman periods at two edges that's kind of trivial and they're all of the form logarithm of some rational function so I'm sorry some algebraic function of the masses and momenta which I didn't write down and this is a special case algebraic they can have a square root because of Newton's formula so now we we've understood that so now we move on to three edges and let's look at the triangle graph with masses everywhere one two three and now we look at the motive it's just h2 of p2 minus a quadratic maybe quadratic union hyperplane actually sorry relative to in the generic case there'll be nothing to blow up it'll be relative to the coordinate hyperplanes in any case it's clear to see that this is mixed state and the weight graded pieces of type q0 q-1 q-2 you can do that without any calculation really and so we know by general facts about variations of mixed state or structures that the periods are linear combinations of dialogithms okay so the motivic dialogithm has its co-action is given by so it's reduced co-action is given by this formula this is a logarithm and what we know from the main theorem is that the right-hand side of the co-action maybe needs to be slightly modified in this case but we know we know that the right-hand side of the co-action on the amplitude is necessarily of the form of something we've already computed so lc of the Feynman periods of the quotient graphs with two edges the two edge quotients of this graph so what you do is you contract an edge here and you get back this graph and we've already computed its amplitude and that tells you what the right-hand side of the corresponding dialog term is in the full expression so the reason I mention this is because there are many some recent results in the physics literature where people have done exactly this they've computed some small amplitudes like this as analytic functions very complicated expressions with dialog rhythms of crazy arguments and then they've written down the symbol which is not unrelated to this they compute the co-product on the symbol and then they observe and conjecture that it can be re-expressed in terms of Feynman diagrams and the main theorem says that we know this is always true that the right-hand side will always be expressible in terms of Feynman diagrams so this explains a lot of a lot of phenomena in the physics literature so we can keep playing that game but I'm going to stop that for now and switch to switch to cases with no masses and no momenta like in the first lecture so now let's look at some more exciting examples from massless fight the fourth theory so I did a lot of these in lecture one so the only way, let's think about how we could get a period a period a material period of weight up to two so something like a logarithm by the the theorem is from an H1 or an H2 so we only need to look at two or three edge graphs so the most interesting case is the three edge graphs so what we do on the back of an envelope we write down three edge graphs with no masses and momenta so there's this one there's this one I think that's it and now because we're looking in the massless momenta most case we're only interested in the first graph polynomial so here it's alpha one plus alpha two plus alpha three so we have these four examples and in each case we want to write down the motive and we can do this but in fact just by stirring at it it's obvious it's very easy to see that it's mixed state it's just a bubble it's three tadpoles glued together yeah it still has six lines yeah it still has six lines but that's not a problem because we need to consider when we look at the Galois conjugates of an amplitude it will involve the quotient graphs in which we contract edges and even if you start off with a four valent theory we may be forced to contract this line and it will give us a six point function so the Galois group will necessarily go outside graphs of a fixed degree of vertices but it doesn't matter there are very few cases we check that they're all mixed state and unrammified so unrammified means that when you reduce modulo any prime something funny doesn't happen and this is obvious because all the coefficients in these polynomials are one so nothing no monomial will disappear for example when you reduce modulo two or three so without doing any thought in fact without any calculation we know that there are no non-trivial periods coming from three edge graphs so there's nothing and in particular that means there are no periods of up to weight two at all occurring ever in fight the four theory as Galois conjugates so as a very special case we certainly get no log p for p a prime and in particular we get no log two so this so it never occurs as a Galois conjugate at any loop order in the entire perturbative expansion sorry yeah why don't you get two pi i because the period is real the domain of integration is Frobenius invariant so the Frobenius fix the Betty path so I know it's Frobenius invariant it's a real number so I can't get two i pi right so yeah so we get a corollary so this is a theorem a statement that's true to all orders in perspiration theory that's completely rigorously proved and then make it even more special let me take a graph in fight the four theory which is primitive log divergent this statement to this very special class of graphs where the motive is mixed eight and ramified it two I mean this is not necessary condition then then the corresponding amplitude never has logged two as a Galois conjugate so that's a statement about infinitely many fine graphs and fight the four theory has a good portion of giving examples which expect to satisfy this condition so for example we can look at there are some nine and ten loop examples in fight the four theory which lie in weights 12 14 and 15 which have been computed by Panzer and Schnetz so just a few years ago this would have taken hundreds of years of computing time to even come close to this sort of order of magnitude but these guys have actually computed such examples and they give Euler sums if you like this is an experiment that's going to test this theorem so Euler sums are things like minus one to the some power m1 to the k1 so I wrote this down in the first lecture I think so Euler sums are certainly periods of this of motives of this type and so you can as I explained in the first lecture you write down these these Euler sums you place them with their matipic versions so contractually that's well defined and you test this you see whether the Galois conjugates involve a log 2 and they don't so a priori such a period sits in a 400 dimensional vector space or something but knowing that it doesn't have log 2 as a conjugate immediately reduces you down to a vector space about half the size that's exactly the sort of thing that I was talking about in the first lecture but this is obtained by a completely trivial computation of small graphs we get a very very strong and highly non-trivial constraint at very high loop order and I'm trying to finish up so again next we look at we go up we look at weight we want to understand all periods of weight at most 4 dialogues and things like this we look at 5 edge graphs and that kind of easy you know so on and so forth and without doing any work whatsoever it's completely obvious that the geometry of these hypersurfaces never produces a 6th route of unity so a dialogue evaluated a 6th route of unity certainly never occurs as a period of such a graph I don't have to work at all to do that it's completely clear just and so corollary another corollary look at graphs and fight the 4th 3 is above such that the motive is mixed to 8 over q a join 6th route of unity unrampified if you like and certainly this very small sub family of graphs by the main theorem never has you can never produce a lead 2 of z to 6 as a conjugate and conveniently our good friends panzer and schnitz have found examples in fight the 4th 3 of weight 11 and 8 33 of weight 13 and they evaluate mz v's at 6th route of unity so they sit in the space of periods of mixtape motives at 6th route of unity some absolutely gigantic space at this weight but we have this constraint that we know that the galore conjugates can never be of this type so immediately by pure thought we know that the amplitude these very difficult graphs has to line some much smaller subspace of possible amplitudes and so on and so forth so it's exactly what I promised in the first lecture this is numerical so as I explained you compute something no sorry this is a theorem so it's not numerical it's a theorem that it's analytically proven to be exactly in mz v's at 6th route of unity and then you lift that number to the matific version and you compute the galore action and you find that there's this this strong constraint that it never has this so if we keep going the expectation is that this the main theorem plus the small graphs principle will eventually completely explain all the all the structure of the amplitudes we saw in the first lecture should explain so I won't prove this co-action conjecture that I mentioned in the first lecture due to Oliver Schnitz but it will, if you like, explain all the symptoms all the consequence of this conjecture so the first thing to do I'm afraid I didn't do this yet because I didn't get around to it but one needs to check that there's no no z to 2 and work your way up and this will explain why the amplitudes have the structure they have so the conclusion is that the amplitudes and quantum field theory are in some sense full of holes so we're used to the sort of plum pudding model of the atom this is the Swiss cheese model of amplitudes so this is a piece of Swiss cheese so if you like this is a picture of all periods and sort of the the cheese is going to be so it's going to have holes in it so this is the amplitudes in coming from Feynman integrals and as you've shown this huge group this cosmic Galois group is in this sense going to move these periods around and what happens is you take a period here like at 11 or 13 loops or something some crazy period that we have a lot of trouble understanding but the Galois group may send it to a period that in very low weight we have already eliminated that means it can't occur so that means there must be a hole up here all the things that map under the action of this group into a place where there was no period cannot have occurred so that's a sort of intuitive picture of all the amplitudes in quantum field theory and what remains to be done it's a final comment what remains to be done questions is to connect with to understand gauge theories we know that the amplitudes have some very special structures that cancellations and so forth that aren't true in this very general picture to understand the connection with renormalization and there's a whole list of other questions but I think I'll stop here 20 years ahead of us how do you use to grasp this small number of pages so so we have some amplitude so we have some differential form in some vector space and the group moves it so it has some conjugate in the same space and if it's a small weight we know that it comes from this face map comes from the boundary by general theorems about and weights and then the face maps because of this special geometry the faces are themselves happen to be so if we looked at general periods if you wrote down a general period you'd have some complicated singularity of the integrand and its galois conjugates would be arbitrarily complicated because when you go into the boundary and blow up and take the limit you'll find more and more complicated polynomials appearing and the whole situation is out of control but the thing that's constraining physics is that that doesn't happen graph polynomials of smaller graphs and so that's what constrains constrains them I don't know if that answered your question so it's a combination of general theorems about weights and the fact that the the geometry is very special it's exactly this equation here but we never go outside the class of of graph hypersurfaces and this explains why the Galois group is in this sense closed on sends Feynman aptitudes to Feynman aptitudes in this sense but still in spite of the holds as the weight increases one gets more more transcendental as the weight increases so this is the weight going no, but one has to ask the right question so it's not possible to keep just computing amplitudes to higher and higher loop order we're looking, as I said in the first lecture we look for theorems that are true for all amplitudes that are true to all orders in post-patient theory and here is a structure that is true to all orders so it's a negative constraint but this is not really the point this is just to give examples of very powerful predictive theorems that one can say about amplitudes so physicists are computing amplitudes they'll do something numerical to try and fit it in some vector space and this structure collapses the size of that vector space down to something small so it has a practical application but I don't think it's the main point I think the main point is the existence of all this mathematics that's the point the theorem is not restricted to convergence yes it's restricted to convergence yes so if you want to look at renormalized amplitudes okay so if you want to renormalize then you can if you renormalize in parametric space what happens is that you you can cook up counter terms of this sort of type with a log of some numerators which are very very close to graph polynomials and so you have to control the geometry of these guys and it's very close to the original geometry so I don't think that the renormalized amplitudes which give all these counter terms will change much and experience shows that the renormalized amplitudes of graphs and sub-divergences are actually simpler than the ones which are convergent so yeah that's a an important task is to um fit this with the theory of renormalization but the geometry is all there it's all there in the blow ups and the structure of the this debate question of what are the possible periods what are the possible periods in physics we don't know we have no clue so in the olden days there were these conjectures in the case of fight the fourth theory that they would be all multi-positive values for example and that's so that's been a lot of work due to myself and collaborators showing that that's completely false and so then if that's true if physics is much more complicated than we expected then what theorems are left to say about amplitudes and this structure nevertheless survives all these negative results which of the type Feynman amplitudes are much more complicated than we thought so they're very very complicated but they still have extremely special class of numbers or functions in the class of all possible and the constraint you found do you think they are all the constraints no ok so in the first lecture I stated Schnetz's co-action conjecture which stated that the periods of fight the four are themselves closed a very very strong conjecture and I'm skeptical about it experimentally so they checked hundreds of examples that gives all possible constraints if you know the class of numbers there are multiple zeta values say and you impose this co-action conjecture then in fact that successfully predicts exactly the vector space of all the amplitudes you get gives exactly the right size so that's at some point when the weight increased you have new numbers coming in inside of control well we know that we get modular forms so the theorem of myself and Oliver is to prove at eight loops we get graphs whose motive for modular form and so we get a new type of number that doesn't have a name and we expect it to get arbitrarily complicated so as the so the program of trying to describe explicitly the types of numbers coming in to quantum filter is hopeless I think because we have lots of examples where things at every loop order historically every loop order there's been a conjecture or several conjectures that have died a horrible death that what we expected to be true was actually completely false and this has gone on over the last few decades so I think one can not be too pessimistic about the nature of amplitudes how do we get the modular form so that you need an intimate sequence of its coefficients so you have a graph of an eight loop graph P18 dimensional space P18 or something it's a hyper surface of degree nine and it has a million terms in it and the claim is that its co-homology has a sub quotient which is the motive of a modular form and you prove that by fibring it successively and doing some operations on the graph hyper surface to decrease the dimension down and down and down and eventually you get a K3 surface and the K3 surface a singular K3 surface you look it up in the classification and it's known that it's modular in that case using modularity lifting terms and so you know that the motive is a symmetric square of an elliptic curve and that rules out that means it can't possibly be a period of a mixtape motive it can't be a multiple zeta it can't be any of these nice numbers we know and so we have lots of examples like this of wild nature where the co-homology looks as if it's going to be arbitrarily complicated so the original program that I explained, Cathy's original conjecture was given that we expected all these amplitudes to be MZVs MZVs have a grouping acting on them there should be a group acting on the amplitudes but the original plan of trying to control all Feynman amplitudes is just not possible because we have these exotic counter examples but the group exists and that structure still works and it's all the more surprising because the amplitudes go in a completely orthogonal direction to multiple zeta values or what we would have liked so I think the program of trying to write down amplitudes explicitly is beyond what's reasonable for the foreseeable future but instead we have these ideas coming from this country we have new invariance of amplitudes we have representations instead of just looking at numbers and we have weights we have differential equations we can use all that machinery I think that's the way to go