1/4 Motivic periods and the cosmic Galois group
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Transcript: English(auto-generated)
00:31
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
00:43
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 cosmic Galois
01:02
group does and why it's a very, very powerful tool in understanding amplitudes. 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 going to happen in the course. So this is just an introduction and examples. So the goal is to define and study a Galois theory
01:43
of Feynman amplitudes in a completely rigorous manner. There will be nothing conjectural about this at all. And I should really mention my intellectual debt, so that the inspiration,
02:04
the story behind this is quite a long one. But I believe it started off with calculations due to Broadhurst and Kreimer in the 90s, in particular in 95, who found multiple
02:23
zeta values, which I'll define later, as amplitudes in a massless fight the fourth physical theory, which is a certain physical theory, which I'll come to later. Okay, so multiple zeta values are certain numbers. And a few years earlier,
02:44
Deligny, Ihara, and Drenfeld, independently and at exactly the same time, in 89, had developed the theory 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,
03:06
are intimately related to a certain group called the motivic Galois group of a certain category called mixed tape motives of a z, which at the time didn't exist. And all that was
03:20
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 could there be a
03:44
cosmic Galois group that somehow acts on these amplitudes and corresponds to this group here, 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
04:06
contribution was due to Koncevich in 1998, who suggested counting points of graph
04:21
hypersurfaces over finite fields, by which I won't say much, but it's proved to be a very useful tool in trying to understand the structure of amplitudes. I should certainly mention work of Anna Kohn and Mathilde Marcoli, who in 2004, and five, wrote a paper in which
04:46
they constructed what they called, and I'll put it in inverted commas, because it's from what I'm going to define. So they constructed a cosmic Galois group. And it is related to the
05:04
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
05:22
about this. Then, an important contribution due to Belkali and Brosnan in 2003, which started to cast some doubt on whether amplitudes in this theory were in fact, multiple zeta values
05:42
at all. Yes. So they found that graph hypersurfaces are of general type. In other words, when you take, you look at the count points of a finite fields of hypersurfaces defined from
06:04
Feynman graphs, you get pretty much anything possible. But the counter examples, as you mentioned, were physically completely unrealistic. So they corresponded to graphs, which would never occur in any quantum field theory.
06:23
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. No, until very recently, there were no infinite families known. As you said, the beginning of asymptotic is what? They just began. Well, I'll come to their results later
06:43
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, except in some handful of cases, that the amplitude was actually
07:05
a multiple zeta value. It was just verified numerically to a high accuracy. I'll come to that later. This is being sketchy here, but I want to mention everything that came before.
07:21
Another important contribution is a paper by Bloch, Ennore, and Kreimer in 2006, who defined what they called the motive of a certain family of graphs in this theory
07:47
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
08:02
about at all, I'm afraid, because time is shorter than I expected. I will mention many, many, many, many, many, many, many, many, many, many, who have enormously influenced
08:23
the way I think about this. But the bottom line of quite a large body of work is that we now know that there exists amplitudes in the fourth theory, in this particular
08:44
theory, which are in fact not, or expected not to be, multiple zeta values. So this initial correspondence doesn't really work. And in fact, the story is even more complicated than that, because not all multiple zeta values actually occur as MZVs. So it would
09:16
seem that this whole project, so the initial analogy that inspired Cartier to coin this
09:43
cosmic Galo 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,
10:03
this idea of a cosmic Galo 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
10:26
inaccurate, it'll already be several affine group schemes, but for the introduction this will do. An affine group scheme over Q, do you see, cosmic Galo group, and associate
10:46
to certain families of Feynman amplitudes, it'll be extremely general by the way,
11:04
the certain will be very small restriction. So these amplitudes will now in fact, will be very general, they'll depend on arbitrary masses and arbitrary momenta. So two such families of Feynman amplitudes, we will associate a motivic period that I will write
11:29
I subscript M G M Q. And we will be able to speak of the Galois conjugates G of
11:55
I M masses momenta for any element in this group. So from an amplitude you get something
12:09
you get something new. So Mark, I use the word motivic in homage to to gotten it because all of these ideas really come from his work. But motors play no role in this. And
12:26
the theory will be entirely rigorous, and there'll be no conjectures, except today, there'll be some conjectures, but in the rest of them. So as part of a very special case of this theory, and I'm speaking to physicists, we will retrieve the notion
12:50
of the symbol of an amplitude. So this has become an industry huge industry recently
13:01
and high energy physics. The symbols only defined for very particular class of amplitudes which happened to be polylogarithms. In fact, the motivic period 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
13:25
a side effect, we'll also pop out the renormalization group, and in particular, the Kahn-Kramer-Hopf algebra of graphs of renormalization. In the third lecture,
13:41
I'll explain that. So this will come out for free, although the precise connection between the actual renormalization group and the cosmic Galois 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
14:14
it gives phenomenal constraints on what you expect to find as amplitudes. So in particular,
14:21
it has a practical application to the study of formula. Is there any connection of this with the recent work of Nima and Gunturo of Paulson? I don't know. So from what I've understood, these amplitudes in n equals 4 super Yang-Mills, my understanding is there's been a huge progress in understanding the integrands, organizing
14:45
the integrand structure for n equals 4 super Yang-Mills, for which it wasn't a priori 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.
15:01
So when you say what is play, oh, so we are supposed to understand that the cosmic algorithm is not like the Tanakian group of some particular group. You're jumping ahead. Yes, no, OK. Just for OFA, 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.
15:25
I will work with the 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, it is expected to be the correct one. If there was an abelian category of mixed motives, it should give exactly the same answer.
15:45
So it's a way to get around all the conjectures. No, no, I'm not constructing category motives. I will explain all of this. Is this broken, or is it a strength
16:04
test? OK, I'll give up. So it is a test of strength. So you will have both invented
16:36
and destroyed the cosmic algorithm. OK, so now Feynman graphs. So I will consider
16:48
a scalar quantum field theory in Euclidean space, so r to the d, where d
17:06
is an even number, so even number of space-time dimensions. So a Feynman graph will be a connected graph. Later on, I might make them not connected, but for
17:29
today, they're all connected. So a graph G is a set of vertices, a set of
17:41
internal edges, and a set of external edges. So v is a set of vertices. E, G are internal edges, so these are pairs of vertices, not necessarily distinct.
18:03
And very often, these edges will, in fact, be labeled. Can there be more somewhere? Yes, absolutely. And we can have tadpoles as well. Absolutely.
18:22
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.
18:46
m1, a particle mass, me, so it'll be a real number for every e and eg,
19:05
and 2, an incoming momentum, qi, which is a vector in the r to the d,
19:22
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
19:52
momenta is 0. And sometimes, I will need to specify which masses are going
20:09
to be 0 and which aren't. Likewise, which momenta we're allowed to take to be 0 and which aren't. And so to do this, it's convenient to draw massive lines,
20:21
massive edges, that is to say those with me not equal to 0, as thickened lines. Okay, so here's an example of a Feynman graph.
20:51
Oops, maybe I should do it like this. That's better. So we'll have q2, q3, and q1. And this means that we have masses m1 and m2,
21:09
which will be non-zero, but the mass of the third particle here is 0. And momentum conservation tells us that q1 plus q2 plus q3 equals 0.
21:48
It doesn't matter, because in the formulae, we will always take the square of the mass.
22:04
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,
22:26
which carries the sum of the momenta. So if you like, this defines an equivalence relation on these graphs. Okay, so I'm going to say something slightly tedious, but it will be important for later on.
22:40
I won't spend too long on it. We can specify the set of edges with vanishing, the set of vanishing masses and momenta.
23:01
In some sense, I want to think of 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,
23:26
i, g, which is a function of these masses, and the momenta as a function. So here we should say what we mean by a function. It can be multivalued in general.
23:43
That means it's a function on some choice of universal covering of some space. It may even be well-ill-defined, so it may be infinite everywhere. So when I write the Feynman integral, sometimes I will not assume that it converges. It will just be a formal expression.
24:05
So it will be a function on the complex points of an affine variety, m, g, index v,
24:21
which is, I'll just write the set of masses and momenta, where the masses are in a1 minus 0 for all e not in v, m.
24:44
And the momenta will be in affine d-space, non-zero for i not in v, q.
25:02
Oh, I forgot something. Subject, of course, subject to the momentum conservation condition. So this is the sort of space on which we want to view the Feynman amplitude, if it converges. We'll have singularities all over the place.
25:27
I'm lost, sorry? Here. 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.
25:41
What I don't understand how to do is take a limit as a mass goes to 0, or as a momentum goes to 0. 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 0 and who is non-zero. And for mathematicians, the Euclidean region,
26:01
so what it means to be in Euclidean as opposed to Minkowski space, is just the set of real, means we restrict to the set of real points of this variety. Okay. Oh, dear. Okay, so now to define the Feynman amplitude.
26:34
So I'm going to jump in and define the Feynman amplitude directly in parametric space, which is very old-fashioned,
26:41
and the younger physicists are maybe not so familiar with it. The derivation from the momentum space representation is in all the textbooks. So you trust me that it works, and it always works. So this is a sort of 1960s presentation of Feynman amplitudes,
27:04
and it involves graph polynomials. So a spanning K tree written T equals T1 union TK
27:24
is a subgraph with K-connected components, which are the TIs,
27:41
and each TI is a tree, so it has 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,
28:01
of the union of the vertices of the Ts. And then some terminology to each internal edge of the graph, we associate what first is called a Schwinger parameter, alpha E.
28:28
And then from this we define a couple of polynomials. Kirchhoff polynomial, graph polynomial,
28:41
is often called the first semantic polynomial, and it is written psi G equals the sum over all spanning one trees in the graph,
29:06
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 Schwinger parameters,
29:21
and it has coefficients in Z. A spanning one tree.
29:41
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.
30:00
And then now, oh yeah, so the second semantic that I'm going to denote by phi G Q, it's now going to depend on the momenta. It's the sum over now spanning two trees.
30:25
I'll write it more neatly this time. Spanning two trees of G. And we take the product of the edges not in the union of these trees, alpha E times QT1 squared,
30:51
where QT1, also equal to minus QT2, is the total momentum that is incoming, that comes into T1.
31:09
And the square, I've nearly forgot to say, if when D dimensions, so QI has components Q1 up to QD,
31:27
let's say, then the square is Euclidean norm.
31:53
An interesting fact, 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.
32:04
And that's the fact that's almost never used in physics. Sorry, the fact that this has integer coefficients is very important because it gives all the arithmetic and it's almost never used.
32:21
So let's do an example. So let's do this graph again.
32:41
So the spanning one trees are 1, 2, 3, 2, and 1, 3. And so the first semantic polynomial, or kickoff polynomial,
33:02
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's just alpha 1, and here's alpha 2. And the spanning two trees are
33:24
edge 1 and this isolated vertex here, this vertex here in edge 3, and then this vertex in edge 2. And so phi g of q is equal to,
33:43
so we take the total momentum going into one of these trees, so let's say this vertex, that's just q1. And then we take the product of all the edges not in the spanning trees, so that's alpha 2, alpha 3.
34:01
Of course, if I took the momentum 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,
34:21
and the last graph we get q2 squared, alpha 1, alpha 3. So we get some very concrete polynomials coming from graphs. Some remarks, which is that psi g is always homogeneous,
34:46
and its degree is hg, which is the betty number, the first betty number of the graph, in the standard definition.
35:02
And I will often call this the loop number, the number of loops of a graph. It's just the dimension of h1. And then phi g, as a function of the alphas, is homogeneous as well,
35:23
but of degree one more. And finally, we let define a third polynomial that I will call psi.
35:43
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,
36:04
the mass squared of that edge, times alpha e, multiplied by psi g. So in this example, we get, I'll write it out,
36:42
an alpha 1 plus alpha 2 plus alpha 3. In the alphas, yeah, I said this, in, yes, in the alphas, absolutely.
37:02
Good.
37:25
Okay, so now for the Feynman integral, which will be constructed out of these polynomials, number three. So this is the Feynman integral in parametric form.
37:51
Yeah, so let me write n g for the number of edges, of internal edges in the graph,
38:01
the time being. And then the Feynman integral, i g of m q, and it also depends on the dimension, but I'm really going to fix the dimension for most of the time.
38:23
It's gamma d over 2 integral sigma omega g m q d,
38:43
where omega g m q d is 1 over the first semantic polynomial
39:01
to the d over 2, times m q to the power of n g
39:21
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. Sigma, I'm coming to that.
39:43
First, omega g. So omega g is sum i equals 1 to n g minus 1 to the i alpha i emit alpha d alpha i
40:03
n g. And sigma is a certain locus in projective space of dimension n g minus 1. It's real points. So it's the coordinate simplex.
40:24
It's the real coordinate simplex. 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 gs,
40:43
so alpha i is in real and non-negative. OK, some remarks. Oh, so of course I should say this may be an ill-defined integral. It may diverge.
41:01
Most of the time it will diverge badly.
41:33
OK, some remarks. Is that the integrand omega g
41:43
m q d is homogeneous of degree 0. So that's a small calculation. We know what the degrees of psi g and phi g are, I told you here,
42:01
and you plug it into the formula, and one has to check that it's homogeneous of degree 0 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,
42:20
you can always restrict to an affine chart by setting one of the alphas to 1, for example, and you get a standard integral over r, r to the n minus 1 or something. Another remark is that the amplitudes
42:43
in a general, i.e. not necessarily scalar, quantum field theory are much more complicated, but they can be expressed in parametric form
43:04
using similar integrals, but with numerators. So numerators will be
43:21
some sort of polynomials in the alphas, with coefficients in some Clifford algebra or something. The point is that the geometry of the Feynman integral will not depend on the numerators.
43:42
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. Why did you include the exponential factor? The exponential factor? Which exponential factor?
44:01
Which regularized in the infrared or large alphas? I'm not sure I see what you mean. So when you derive this, there's an exponential and that produces this gamma term here. But this is not regularized in any way.
44:21
So if you take the momentum space definition of a Feynman integral and you do the Schrodinger trick and you do the momentum integrals using a formula for a Gaussian integral, you get a formula very close to this. But there is still the exponential of the method?
44:42
So you have an e to the minus something and then that's where you pass to... e to the minus something and then you can do one more integral, integrate out e to the minus, a graph polynomial times lambda
45:00
because it's homogeneous, you integrate out lambda which will spit out this gamma function and then you get a projective integral. So this is one stage further than the normal affine. Maybe you're thinking of the parametric integral with the delta function.
45:22
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.
45:45
And that's why the calculation of Feynman integrals is an enormous industry. So perhaps before having a break I'll just give some first examples of Feynman amplitudes
46:01
and try to convince you that you get interesting quantities from them.
46:21
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
46:41
triangle graph that we looked at earlier with some choice of masses and momenta, it doesn't really matter which. Other examples would be polygons like this maybe with many momenta.
47:03
This is in d equals 4 dimensions. Then all these Feynman integrals it turns out is always expressible in terms of two functions.
47:25
The logarithm which I will write in this way, lee1 of x equals minus log one minus x. And the dialogue with them, lee2 of x
47:40
equals some x to the k over k squared. So this is I don't know who was the first to observe this but there's a beautiful paper by Davidicheff and Del Buge where this is explained.
48:07
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? Well they will be some complicated the arguments will be some complicated
48:22
algebraic functions perhaps with a square root thrown in there of the masses and the momenta. I should say the mass squares and momenta.
48:41
Sorry I won't say that. So that's really it for one loop. One loop rather.
49:07
Two loops gets more interesting. One example that's been massively studied in recent years and has quite a long history is the sunrise diagram.
49:30
Again you've got three possible masses and one momentum coming in. So let me give you the graph polynomials just for fun.
49:48
If I do Q Q squared alpha one alpha two, alpha three. And this has a very long history
50:01
but the upshot is that these Feynman integrals give elliptic dialogues and the most recent work on this is due to Adams, Wagner and Weitzel.
50:21
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.
50:44
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
51:06
which can be expressed in terms of polylogarithms, multiple polylogarithms. In fact the literature, physics literature is full of such calculations.
51:22
There are other interesting examples for number theorists like myself at least. Such as these family of graphs bn, so here we need
51:42
t equals two dimensions and so n edges. So these were talked about recently by David Bortest in this very room not long ago. So here you take q equals naught so
52:01
q equals naught means I may admit to draw the external momentum I'll just put a little small line like that to illustrate that it's zero and all the masses are equal to one perhaps I should thicken these lines
52:26
I beg your pardon? Oh, n can be anything. Yeah, that was the end of two loops and then the two loop story stops because we're stuck. And now this is just a different family of examples.
52:42
So n bigger than or equal to two and so here the polynomial psi doesn't depend on anything now. It's sum alpha i
53:00
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
53:20
interesting hypersurfaces that have been studied a fair amount in mathematics in some cases. And here are some examples of the corresponding amplitudes. The I2 is the integral Feynman amplitude of Ib2 is one Ib3 I think
53:40
this is all due to David Brodhurst is three times the Dirichlet L function for the unique Dirichlet character mod three which is non-trivial. So this is three times sum chi n n squared.
54:03
And we start to get more interesting numbers. Ib4 equals seven times zeta three Reimann zeta value and I think beyond that they're not known. So these are certainly numbers which are very interesting to a number
54:21
of theorists. 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
54:41
mainly in some cases been proved yield a whole array of special values of L functions of modular forms. So f is some
55:07
modular form for SL2 some congruent subgroup of SL2. So before continuing with the main example we should maybe have a ten minute break coffee.
55:22
So this class of graphs is called log divergent precisely because the gamma factor will produce a pole. But by abusive notation let us write ig to be modulo this gamma factor what remains
55:41
of this integral and it is omega g over psi g squared. And so this is a number if it converges and it converges
56:03
if and only if for every subgraph the number of loops sorry the number of edges is actually bigger than twice the number of loops for every strict subgraph.
56:21
So this is called this means that the big graph is primitive it has no subdivergences. So an example an example of a graph which is log divergent is this one so now because there are no masses
56:40
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 subgraph three four contained
57:00
in it which has two edges and one loop and two times one is two. So it violates this inequality. Why are these a comment for the physicists why are these quantities relevant
57:20
so in this case we get I should say we get a real number we don't get functions anymore we just get numbers. So why are these quantities relevant for physics because they give renormalization scheme
57:43
independent contributions to the beta function of this theory. Oh I should say also the when I write five four what that means in graph theoretic terms is simply
58:01
that the graph has no vertices of degree bigger than or equal to five. So the valency of each vertex is at most four.
58:26
So here are examples.
58:45
So these are the calculations originally due to Broadhurst and Kreimer which started off this whole business. So one loop there's one loop there's this graph and its amplitude
59:01
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 zeta of three.
59:22
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
59:41
let me look at one of the most interesting ones there are a couple of others and this gives six
01:00:01
zeta of 3 squared. That's the square of this Feynman amplitude, and we understand why that's the case. At six loops, there's more examples, but the most interesting one is this graph, which again was
01:00:21
computed experimentally way back when, but only rather recently has been proved rigorously. And there are many others. And here, the amplitude is something complicated.
01:00:41
It's 27 over 5 zeta 5 comma 3 plus 45 over 4 zeta 5 times zeta 3 minus 261 over 20 zeta 8.
01:01:04
And so these calculations were first used to broadcast Krymer numerically up to a certain degree, but there's been spectacular progress in recent years going to a much higher loop order and proving the actual quantities
01:01:20
rigorously by Panzer and Schnetz. They're two different. So what do you say, numerically, in the sense that they're computed 12-dimensional integral with very high precision? Yeah. So already for the Wu-3 spokes, you're right. It has six edges, so it's a five-dimensional integral. The graph polynomial has 16 terms. You can't compute it that way.
01:01:42
It's terrible. So you use different techniques. You use momentum space. So you use something called the Gegenbauer X-space technique. And you expand in terms of certain Gegenbauer polynomials and then accelerate the convergence. There's a whole industry of, there's a huge literature on
01:02:02
how to compute numerically. But that's all been superseded because there are now algorithms that do this using the parametric representation, in the case of Eric, who's sitting in the audience. And Oliver Schnetz has a different approach using single-valued multiple-poly logarithms. So now many of these can be done much more efficiently
01:02:22
and exactly. So these are all theorems. Nothing is. You don't want the one where it's a full theorem, yeah? Yeah, these are all true theorems. And they're examples now up to a much harder report, up to even 11 loops. So there's been a spectacular progress in recent years in this. But for the illustrations of today, I'll just look at
01:02:45
these examples. OK, so the first observation is that there are, oh, sorry. So originally, there were no infinite families known.
01:03:02
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.
01:03:25
So multiple zeta value is this nested sum.
01:03:41
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, 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
01:04:03
multiple zeta values in general. So the question then is, so now there are known examples due to Panzer and Schnetz, where you have an evaluation
01:04:22
of a Feynman integral as something like an Euler sum or 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
01:04:40
and Oliver Schnetz, where we proved that the graph hypersurface is modular. So a piece of the cohomology is the motive of a modular form. And so that's an eight loop. That's much more sort of catastrophic. But putting a root of unity is a combination of all of this.
01:05:03
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 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
01:05:24
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
01:05:44
values, like the weight. So the weight of a multiple zeta value is the sum of the arguments. And so let's have a look at the weights on these examples. So this is short of trying to compute the integral.
01:06:01
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 3. The weight here is 0. And here, it's kind of a trivial example, so
01:06:22
we'll ignore that. Here, the weight of zeta 3 according to this definition is 3, and twice 2 times 3 minus 3 is 3. Here, the weight of zeta 5 is 5. And this quantity is 2 times 4 minus 3, which is 5.
01:06:40
Here, the weight is 6. And this quantity is 7. And this integral here has weight. This multiple zeta value has weight 8. But 2h minus 3 is 9.
01:07:00
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 the loop number minus 3. But in these examples here, the bound is not attained. And we say that these examples have weight drop.
01:07:21
So we see that even understanding the weights is a tricky business. But I should say that we actually understand the weights fairly well, at least in this setting.
01:07:42
So the weight is the first hint of something motivic going on. So the prototype and the inspiration for this entire subject is the following experiment, where we will try
01:08:00
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
01:08:31
of, in inverted commas, motivic periods. It's an abuse of the word motivic, but
01:08:43
I'll keep it anyway. So we have a ring that's called PM sub H, which I'll explain next time. And it comes with the following structures. It comes with a period homomorphism from these
01:09:11
elements of this ring to actual complex numbers. Then it comes with an action of an affine group scheme over
01:09:31
Q that I'll just call G. So this group
01:09:41
will act on this ring. And it has other structures, in particular, an increasing weight filtration. So in general, the weight is only filtration.
01:10:01
But in the examples that we're going to look at, like multiple zeta 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 if it were grading. But in general, one should remember that it is not.
01:10:30
So what is? H, I will define that next time. It will be a Tanakian category of realizations or something.
01:10:45
Yeah. I want to explain what a Galois theory of amplitudes means, and I now want to explain how it will explain a lot of the structure that we see. So I think these examples are very striking and were the
01:11:02
motivation for constructing this. So examples, there are things called motivic multiple zeta values, which I defined a few years ago. Again, they actually live in a sub-ring of another ring of
01:11:25
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 they're objects corresponding to multiple zeta values, and the period is the multiple zeta value.
01:11:52
OK, so then what can we do with this group? Well, we can define the Galois conjugates of an
01:12:04
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
01:12:25
points of this group. And therefore, in fact, every element will generate a
01:12:44
finite-dimensional representation that I will call v sub psi of this group. And we will get a representation, a row from
01:13:01
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
01:13:21
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
01:13:41
about these representations in the case of multiple zeta values. So let me give examples for this in the case of some simple multiple zeta values. So the first example is even zeta values.
01:14:08
So the Galois conjugates of an even zeta value are just itself. So v, in this case, will be q zeta-matific 2n.
01:14:24
And so how does the group g act on this matific zeta value? Well, it just multiplies it by some number to the 2n.
01:14:43
So what that means is that there exists a homomorphism from g to gm, which is called lambda. And every even zeta value gets scaled by lambda to its weight. So this is kind of a bit uninteresting.
01:15:04
To get something more interesting, we should look at odd zeta values.
01:15:22
So now the Galois conjugates form the vector space spanned by 1 and the odd zeta values.
01:15:42
So we get now a two-dimensional representation. And g of an odd zeta value is lambda g, it's the same lambdas before to the weight, plus g.
01:16:13
So this is in q. So this Galois group, matific Galois group, if you like,
01:16:21
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 this basis, so we get g to ought v, and g in this basis
01:16:41
will map to the matrix 1s 2n plus 1 lambda to the 2n plus 1 of g. So we should think of an odd, an even zeta value as a
01:17:01
one-dimensional representation, and an odd zeta value is a two-dimensional representation. Sorry? s is a function from g to q. It's a function from g to fine line. If you like, it's a function from s 2n plus 1 is a
01:17:22
function from gq to q. And this thing is a homomorphism from the additive group, semi-direct, the multiplicative group.
01:17:41
So these are functions on g, and this forms a group of matrices into this group of matrices. My apologies. I forgot the 0 here. What's that? Sorry? Zeta m1.
01:18:01
It's 0. Yeah, zeta m1 is it. Well, you can define it. So I'm down to 3 pounds. Yeah, I prefer it. How do you consider this?
01:18:21
Yeah, it's very useful in many contexts to consider it to be a parameter. Actually, here I prefer it to be 0. Sorry? How do you call the big data? Oh, the big data.
01:18:42
So then we can take tensor products of representations, as we know. So the first two examples tells us what the Galois of an odd zeta value and an even zeta value are.
01:19:06
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 did I call it?
01:19:29
s, 2n plus 1g, zeta into 2k. That just follows from multiplying the two previous examples together.
01:19:41
So in particular, the Galois conjugates of, let's just do an example, zeta 3, zeta 2, are elements of the vector space zeta 2 and zeta.
01:20:07
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. For the first really interesting example is zeta 3, 5.
01:20:22
And you have to trust me on this, that there's a way to compute the Galois conjugates. It's a big subject. It would take a very long time to explain, and we won't need everything. But the upshot is that the Galois conjugates are the
01:20:45
number itself, zeta 3, and 1. Absolutely. Yeah, the reason it's non-trivial is she follows
01:21:01
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 you can, yeah. So now we get a representation on this vector space. So g to ought v. And in this basis, we can
01:21:32
write it as a matrix. 1, 0, 0, lambda cubed, lambda to the 8.
01:21:45
This is S3. This is minus 5 times S5. So there's some computation to be done there to obtain this minus 5. And there's some new map from g, set theoretic map from
01:22:03
gq to q, which I call S3, 5, such that this thing forms a homomorphism of groups. So the convention is not, so 1 goes to 1 on the GDS. Yeah, so yes, 1 goes to 1. So they're going by the matrix where it depends on 1.
01:22:23
I think it should be a transpose, but it doesn't matter where it is. Do you think I should write, well, this is Durham. Maybe. I think you want the matrix to multiply column vector, but maybe I'm not doing it. I'm notoriously bad at getting my left and right mixed up. But yeah, we can discuss it.
01:22:42
We can discuss it. Convention. OK. OK, and then in the third lecture, I will try to set
01:23:10
this up so that all this has a rigorous meaning. And it works. In the third lecture, I will define a motivic period.
01:23:24
Again, abuse of the word motivic. I-M-G, in particular, for example, for any g of the type we're looking at here. So primitive, overall log divergent, and fighter 4.
01:23:46
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-M-G, in this same ring, and such that its period is the Feynman
01:24:04
amplitude we want, which by these assumptions converged, as I said earlier. And then the key observation, which is the whole purpose
01:24:24
of this series of talks, is that the subspace spanned by the motivic Feynman amplitude is, in brackets, nearly, in
01:24:53
some precise sense, but the story is slightly more complicated than it at first seems, is actually closed
01:25:01
under this Galois action. So the goal is to make this precise.
01:25:20
So in some sense, Calty's original dream was that there should be a group acting on Feynman amplitudes, and it should be the same group as the one acting on multiple utilities. But since we have all these counterexamples and these complications, that can't be true. But the point is then, there is such a group acting
01:25:43
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 Galois conjugate, it should still be an amplitude. And that is nearly the case, and is an absolutely astonishing fact.
01:26:02
And I will now try to explain to you why it's such an astonishing fact on these examples. So this fact has extraordinary predictive power for the amplitudes. So the purpose of this talk was really to try to
01:26:22
motivate this entire construction with the examples I'm going to give now. So to test out the validity of these ideas, let's take these amplitudes, which are computed as multiple zeta
01:26:41
values, and just put a little m everywhere. That's reasonable, because the standard transcendence conjecture for multiple zeta values suggests that the period map on multiple zeta values is injective, and
01:27:01
everyone believes this. So that's not a big price to play. So let us assume the transcendence conjecture for MZVs, and then do the following experiment.
01:27:31
So what Oliver Schnetz did is he took the vast data of amplitudes that were now known in the 4-theory, and he computed the Galois conjugates of all of them.
01:27:44
And he made the following conjecture based on that numerical evidence, which is that the periods, sorry, the vector space spanned by the material amplitudes for G as
01:28:06
above, primitive, overall log divergent, in fact, the 4-theory is closed under the Galois action of G. So
01:28:25
this has not appeared yet. It will be written up in a paper joined with Eric Panser, 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
01:28:41
consequences of this conjecture. So from 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 the multiple zeta value.
01:29:02
So in fact, there are theorems that will guarantee this in even infinite families of examples. So 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 Karen Schnetz in which we explain
01:29:22
how to predict the weight, at least in low degrees. OK, so I'm out of space.
01:29:40
OK, so let's assume that we know that these amplitudes are multiple zeta values, and we know their weights. That's not much to assume. So here's a table.
01:30:01
Here's a table. I've run out of board space. OK, I'll try and squeeze it on here. Here's a table of multiple zeta values up to weight 8. So the weight 2.
01:30:33
So these are the numbers which we expect to find at least low degrees coming out as amplitudes in 5 to 4 theory.
01:30:43
And here I will write a basis for the space of motivic MZVs in that weight. So in weight 0, there's just the number 1. In weight 1, there's nothing.
01:31:02
In weight 2, there's zeta motivic 2, nothing else. In weight 3, there's zeta motivic 3, nothing else. In weight 4, there's zeta motivic 4, nothing else. In weight 5, there's zeta motivic 5 and zeta motivic 3 times zeta motivic 2.
01:31:26
In weight 6, there's zeta motivic 6 and zeta motivic 3 squared. In weight 7, zeta motivic 7, zeta motivic 5, zeta
01:31:43
motivic 2, zeta motivic 3, zeta motivic 4. And in weight 8, we get zeta motivic 5, 3. This is interesting. Before I wrote 3, 5, but you can take 5, 3.
01:32:02
It's the same. Zeta motivic 5, zeta motivic 8 at the top. Then zeta motivic 3 squared, zeta motivic 2, zeta motivic
01:32:26
5, zeta motivic 3. And the first motivic multiple zeta value that is not a product of things which came before, 5, 3. OK. So here we go. We have this data of numbers, and we have this
01:32:44
correction conjecture, and we have these graphs. So let's put the pieces together. Let's assume this conjecture.
01:33:05
So first of all, there is no graph whose amplitude is a multiple of zeta 2. There can't be, because we know something about the
01:33:21
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 2.
01:33:41
So then the correction conjecture says that. But I think now you don't assume that you know the weight of this graph. No, I'm saying I just assume I know the weights. So I know that this graph has weight 3. And there are no graphs in between the one-loop graph and the three-loop graph. So there's no space to have a zeta 2.
01:34:01
There cannot be a zeta 2 in this theory. And that implies that there are no numbers whose conjugates are zeta 2 that can ever occur. So this implies that the number any zeta odd times zeta 2 can never occur to any new border.
01:34:27
So we take our table of MZVs. And having struck out the zeta 2, the correction conjecture tells us that this number can't occur, this number can't occur, and this number can't occur, and
01:34:43
so on ad infinitum. So let's do an example. So example in weight 5. Let's compute the four-loop amplitude by pure thought. We know we have a graph here, and we know
01:35:02
that its weight is 5. So a priori, the amplitude, since we're assuming we know it's a multiple zeta value, should be in this vector space of weight 5. So it's a rational multiple of zeta 5 plus a rational
01:35:21
multiple of zeta 3, zeta 2. So what would its conjugates be by the examples I worked out laboriously beforehand? The conjugates would be the number 1, beta zeta
01:35:41
motivic of 2, 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 2, so that forces beta equal to 0.
01:36:02
So it tells us that the amplitude cannot be an arbitrary linear combination. It has to be a multiple of zeta 5. OK? We keep going. Can you repeat the argument about weight dropping? So if you go to very high loop order, and so soon the
01:36:22
weight drop is very drastic, so you go to weight 2? So the weight is tricky. So we can understand when graphs don't have a weight drop. And in some infinite families of cases. And we can guarantee that there's one weight drop. So there are combinatorial criteria where you can just
01:36:41
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 general recipe where you give me a graph and I can write down the weight. So you don't have a bound? There's an upper bound. So there's an upper bound. The upper bound is it's always 2h minus 3.
01:37:00
And sometimes that bound goes, no, no lower bound. So you could have zeta 2 at the? Good point, yeah. So I'm assuming, I said I assumed you know the weights. But there are, yeah, you can get a lower bound using,
01:37:20
I will come to that in the fourth lecture. So in Hodge theory, there's an upper and lower bound on weights. So there are two different things. One is, can we predict when that weight drops? That's the question I just answered. But in Hodge theory, you know that the cohomology of Hn of a smooth variety has weights between n and 2n.
01:37:41
So there's a lower bound as well. You have to apply that. So in some sense, there is a lower bound. I'll explain that in the fourth lecture. Sorry? F between n and 2n is just consequence of Hodge theory. Consequence of Hodge theory. But in some sense, that gives a lower bound, which means that beyond a certain point, beyond a certain point,
01:38:04
the only way you could get a z to 2 is, you have to do the whole theory. Maybe just one remark that there is a conjecture that there is a lower bound which grows with the loop number, so we have this upper bound with two times loop number, roughly. And one of the many conjectures from the data
01:38:21
we have is that also there is a lower bound growing linearly with the number of loops. So there are examples where you have triple weight drops between the higher loop orders, but there's always each loop also, the upper bound grows by two, and the lower bound experimentally seems to grow by one. It's not trivial.
01:38:40
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 recipe that predicts that there's some theorems that tell you from the common talks of the graph
01:39:01
that there's one weight drop, but the two weight drops is out of reach. Now, I don't know whose question I'm answering. OK, so now continue. So likewise, by staring at the examples, there is no
01:39:27
graph corresponds to z to 4, z to 6, because there's just not room. So now let's look at six loops.
01:39:43
So let's look at this example here. So most six-loop graphs have weight 9, and they're going to be not very interesting amplitudes because of this correction structure, this Galois structure. But let's look at a weight drop case. It's going to be weight 8, and it's going to give
01:40:03
something interesting, so this graph here again. So a priority amplitude in this basis is going to be alpha z to 3 z to 5 plus beta z to 3, 5 plus gamma z to 8
01:40:27
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.
01:40:43
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
01:41:01
can be amongst the numbers 1, z to 3, z to 5, and the amplitude itself. But that's OK, because these numbers do occur as previous amplitudes.
01:41:27
One way to answer this weight thing is that the correction should respect the number of edges, so if you haven't seen a z to 2 so far, you're not going to see it now.
01:41:49
Finally, so these do occur as amplitudes, and let's do one final example.
01:42:09
At weight 11, so this would correspond to a seven loop graph, which until a few years ago was totally beyond what was computable or even foreseeably computable.
01:42:25
So at seven loops, a multiple zeta graph would involve the following periods. So here's a basis of the motivic multiple zeta values at weight 11, and this is wrong, this should be cubed.
01:43:11
9 plus 2 is 11, zeta motivic 9.
01:43:28
OK, so sorry for that. So this is a basis of motivic MZVs in weight 11, and it is nine dimensional.
01:43:43
And the correction conjecture tells us we don't see zeta 2s and zeta 4s and zeta 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
01:44:00
should say these are extraordinarily difficult to compute these amplitudes, but I pick a random seven loop graph, which is called p78, and you compute its amplitude and you find after a huge amount of work that it can be written in this basis of multiple zeta values, and you
01:44:25
get some coefficients out, and indeed you notice that the
01:44:56
coefficients of all these four quantities vanish in accordance with the correction conjecture.
01:45:02
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 amplitudes in 5 to 4 theory.
01:45:21
And if you work out what the correction conjecture tells you, or the Galois action conjecture implies that some constraint on these numbers, in fact it implies precisely
01:45:43
that 3,024 over 5 ratio should be exactly equal to the ratio of zeta 3 5 to zeta 8 over there, so 27 over 5 over 261 over 20, and you can check that that's indeed
01:46:04
correct. So I hope that this illustrates that this correction conjecture gives some extraordinarily subtle constraints on the possible amplitudes, which up to now
01:46:21
has not been considered at all. Yeah, so the correction, you take this quantity and you compute the correction, so zeta 11 is primitive. It's not going to give anything. So we're looking for things that will be conjugates of
01:46:44
the form zeta 5 3 or zeta 8. And the second one in this basis was chosen in such a way that its Galois conjugate was zeta 5 times zeta 3. And then so the only ones that can occur are the third one and the fourth one.
01:47:01
And so the Galois action sends the third one and the fourth one to zeta 5 3 and zeta 8 respectively. And the zeta 5 3 and zeta 8 have to occur in a combination that has previously occurred in the quantum field theory, because this weird combination of
01:47:21
numbers here, the only way it comes in to fight the fourth theory is that zeta 5 3 and zeta 8 always occur with this constant of proportionality. And so you do your six-loop calculation once, and that gives you some information at a completely different loop order by this conjecture.
01:47:44
So I find this absolutely astonishing that knowing something in low degrees gives you constraints to all orders in perturbation theory. So it was these calculations that motivated this story.
01:48:09
So there are about 250 examples that have been checked by Oliver Schnetz and Panzer of this type.
01:48:25
The conjecture has been up to 11 loops. The constraints, as I hope I explained, that this correction puts on the possible amplitudes actually
01:48:42
get 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.
01:49:01
Here we see something where it's getting more and more powerful as the order increases. And indeed, the dimension of the space of amplitudes is exactly 4, which sits inside the
01:49:22
dimension of space of MZVs. 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 Galois action.
01:49:41
Had a five-dimensional, but the point was that here we had a three-dimensional space of possible amplitudes, zeta 5, 3, zeta 5 times zeta 8. There's another graph that gives zeta 5, zeta 3.
01:50:00
And at six loops, you only get a two-dimensional space out of the four possible three dimensions. I forgot to say that. My apologies. I should have made that more clear. So the fact that that explains, yeah, so I should have said that more clearly here, that you have, yeah,
01:50:23
sorry, I omitted 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 then feeds. Is there any Zagit type conjecture about this
01:50:40
dimension in higher order weights? None whatsoever, because in general, we don't get multiple zeta values. 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
01:51:01
very small subspace. So there's a very striking example due to Schnetz and Panzer where you have an Euler sum. So you have minus 1 to some power here. And the space of Euler sums at that dimension is hundreds, I believe.
01:51:20
And 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 400 down to five or something. So if the numbers are not multiple zeta values, this becomes a stronger and stronger prediction, because you're in a bigger box of possible periods.
01:51:45
And so the purpose of the remaining lectures will be to prove a version, a weaker variant, of this conjecture. So it will be much weaker, but it will be valid for
01:52:06
arbitrary masses and momenta as well. Yes. So the problem will be that the Galois conjugate of a
01:52:21
graph in 5 to 4 theory, the way, if you do the mathematics, you expect graphs with higher in 5 to 5 theory or 5 to 6 theory. So 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,
01:52:44
because the amplitudes in higher degree vertices are no more complicated than the ones with four vertices at this loop order, up this far in the picture. So it could be that this conjecture goes wrong in higher loops.
01:53:04
But yeah, that's exactly precisely the statement of weakness. So it's not clear whether the conjecture is true. That's one reason why my name isn't on the conjecture, because I'm nervous about it. Oliver is a braver man than I am, and the eminence is phenomenal.
01:53:22
But we don't know yet whether such a strong sort of statement can be. And so very briefly, what is the plan for the sequel? In the second lecture, I will be just pure mathematics. I will talk about motivic periods. I will try to do everything in that lecture.
01:53:43
It will be completely independent from this one. The third lecture, I will talk about graph motives. There will 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 Galois group and small graphs.
01:54:12
That's it. Yeah, thank you.
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