Graph Complexes, Invariant Differential Forms and Feynman integrals
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Brown, Francis
Kontsevich introduced the graph complex GC2 in 1993 and raised the problem of determining its cohomology. This problem is of renewed importance following the recent work of Chan-Galatius-Payne, who related it to the cohomology of the moduli spaces Mg of curves of genus g. It is known by Willwacher that the cohomology of GC2 in degree zero is isomorphic to the Grothendieck-Teichmuller Lie algebra grt, but in higher degrees, there are infinitely many classes which are mysterious and have no such interpretation. In this talk, I will define algebraic differential forms on a moduli space of graphs (outer space). Such a form is a map which assigns to every graph an algebraic differential form of fixed degree, satisfying some compatibilities. Using the tropical Torelli map, I will construct an infinite family of such differential forms, which can in turn be integrated over cells. Surprisingly, these integrals are always finite, and therefore one can assign numbers to homology classes in the graph complex. They turn out to be Feynman periods in phi^4 theory, and can be used to detect graph homology classes. The upshot of all this is a new connection between graph cohomology, Feynman integrals and motivic Galois groups. I will conclude with a conjectural explanation for the higher degree classes in graph cohomology.
 Thank you very much, Eric. Of course, it's a great pleasure to speak at this conference. I'm very sad that we couldn't meet in person. It's been a while since I've seen Dirk and I think it would have been tremendous fun. I think it would have been fun because we could have done some of this. I wonder if you can see. Can you see my screen? Nope, you can't. That's very strange. Okay, let me find a different way. We could have done some of this. At the risk of embarrassing Dirk, this is a photo he sent me of a New Year's greetings in 2011 of a typical crime of family gathering. There's quite an extraordinary collection of bottles there. It's well worth close inspection. Okay, so now back to mathematics. I should say that I was in fact slightly relieved when the conference was delayed. I'm sorry to say because this project wasn't finished and I thought that when the time came I'd have made some progress. Of course, nearly a year has passed and I haven't worked on this for a single minute. This isn't so much. It touches on pretty much every single aspect of it. It touches on work of Dirk's. Dirk has done it over the previous few decades. With that said, let's begin. I'm going to talk about graph homology. We're going to take G, a connected graph, and I'm going to denote the loop number, also known as the genus of the graph, by HG, the standard. The number of edges is EG. The degree here is something slightly funny. It's the edges minus twice the loops. That will be the degree. This will be familiar to you as minus the superficial degree of divergence. It's like a superficial degree of convergence. Next an orientation on a graph will be essentially plus or minus a wedge product of symbols corresponding to the edges in the graph. What that means, it's just an ordering on the edges modulo the action of even permutations. If you flip two edges, you get the opposite orientation. That's an oriented graph. It's not oriented in the usual sense when you put arrows on the edges. It's an orientation on the ordering of the edges. The graph complex is defined by taking the Q vector space spanned by oriented graphs G, E, sorry, G, Eta. We assume that G has no tadpoles, so no self edges like this, and it has no vertices of degree less than or equal to two. Then you impose some very simple relations that a graph with a negative orientation is the negative of the graph with that orientation. That's fairly natural. You also impose a relation that the G of Eta is equivalent to the same graph where you permute the edges according to an automorphism of the graph. If a graph has an automorphism, it's equivalent to the new graph with the new orientation. In particular, if a graph has an automorphism which induces an odd permutation of the edges, then its class is zero. In equivalence classes, it will be denoted in square brackets. Then on this vector space of graphs, oriented graphs, you can define the following differential. This was done by Maxime many years ago. The differential of an oriented graph is a sum of all the edges in the graph. What you do is you contract each edge in the graph and have the induced orientation with the appropriate sign. This contraction, I use this double slash notation. I always use this to mean the contraction where if you contract a tadpole, if you contract a loop, that's zero. That's the empty graph. In case you're not allowed to contract tadpoles. This differential is well-defined and it squares to zero, so it defines a complex. Furthermore, you can check that it has degree minus one with respect to the degree. I remind you that the degree is edges minus twice the number of loops. Great. Here are some examples which I'm sure you'll, with this audience, will find extremely straightforward. Here's the differential at the top. The first remark is that any graph that has a doubled edge is zero in this graph complex GC2. That's because if you take a double edge here with edges one, two, with the orientation e1 wedge, e2 wedge, something else, e2, all the other edges, then you can flip the edges one and two. That gives you an isomorphic graph, but it will reverse the orientation. Therefore, it's minus itself and therefore it must vanish. Any graph with doubled edge vanishes. By the way, from now on, I'm going to drop the orientations. It's boring to keep writing the orientations. They'll just be implicit from now. From the previous remark, we get that any graph with the property that every edge lies in a triangle, it automatically has the property that its differential is zero. That's because if you take a triangle and contract an edge in a triangle, you get a doubled edge. As we've just seen, a doubled edge is zero. Graphs built out of triangles are going to have zero differential in the graph complex. Our favorite example is therefore the wheels. The wheels with n spokes and they satisfy, of course, d of the class of the wheel with any orientation, doesn't matter, is zero. Here's just a hammer at Harmon's shore. You don't need this. Here's another example of a differential in the graph complex. We're going to contract all the edges in this graph here on the left. There are two triangles. Contracting them gives zero, as we've already seen. The only edges that will do something when you contract them are the red ones. When you contract the top one, you get this graph here. When you contract the middle one, you get a wheel with four spokes. When you contract the bottom one, you get this graph here. The first and third graphs cancel out and you're left with a wheel with four spokes. The wheel with four spokes is exact. In fact, more is true. The wheels with even numbers of spokes are always zero because they have a symmetry that's odd and that forces them to be zero in the graph complex. Only the odd wheels survive. Graph homology is HNGC2 is the kernel of this differential, modulo the image. The homological degree is this degree edges minus twice number of loops. It's also graded by the genus, the loop number. We get a bunch of homology groups and they are in turn graded. What do we know? It's known that the homology vanishes in negative degrees for positive genus. It has a lot of extra structure. The first bit of extra structure, and here we see the first interaction with the work of Dirk and Anna, and that's that the graph homology has a leacoalgebra structure and it's induced by anti-symmetrizing the con-crime or coproduct. I'm sure everybody here knows, I've seen this before, you take the sum of a certain class of subgraphs, typically one particle irreducible, and on the left of the tensor you have the subgraph and on the right you have the quotient graph. I apologize, there's a typo here which I can't change. This should be a single slash and not a double slash. I'm sorry that formula is wrong. The single slash means you just contract the subgraph gamma even if it contains a loop, it's not zero. I'm sorry I've used the wrong notation, that should be a single slash. Then Wielwacher showed in 2014 a fantastic result that the zero degree homology of the graph complex is due to the Grotten-Diek-Teichmann-Lie algebra, which is something that's explicitly defined but quite tricky to understand. We don't know much about the Grotten-Diek-Teichmann-Lie algebra at all. What we do know is something predicted by Deline. This used to be called the Deline-Iharra conjecture but I was recently told by Iharra that it should be called the Deline conjecture. It states that there's a free Lie algebra with infinitely many generators, sigma 3, sigma 5, sigma 7 in degrees 3, 5, 7, every odd degree, which injects into Grt. That means that Grt has got this huge free Lie algebra inside it and you can show, I think Wielwacher showed that the sigmas end up being due to the odd wheels. Sigma 3 corresponds to the wheel with 3 spokes, sigma 5 the wheel with 5 spokes, etc. The first puzzle here is that this group is to do with the Motyvic-Gallore groups. This free Lie algebra is the Lie algebra of the Motyvic-Gallore group. We can rightfully ask what on earth have Motyvic-Gallore groups got to do with graph homology? That's the first mystery that I'd like to come back to. Here's a picture of graph homology in low degrees and I'd like to spend a little bit of time discussing this. Up to the left axis is the homological degree, H0, H1, H2. Along the right axis are the number of loops, one-lip, two-lip, three-lips, four-lips and so on. You can draw this picture different ways. You can grade by edges and that's actually probably a better thing to do as we'll see. There are different ways you can represent this. The first remark is that there's this red diagonal line and this red diagonal line is the line where, in fact, just immediately below this line are trivalent graphs. Every vertex has degree 3. Above that, that means on the red line or above, all these graphs have a two-valent vertex. There's zero in the graph complex. Your intuition is as you go up in this diagram, you have lots and lots of edges, very few loops. In quantum field there, that'd be a very convergent graph. As you go down, you get infrared divergences as you go vertically down in this picture. Very convergent. Anything above this red line is zero. Now, going across the H0, that's the only degree in which things are really well understood. So, this is the Grotten-Diegthe-Teichmeler, the algebra. Here, in degree 3, we have the wheel with three spokes and you can show that gives a non-trivial class. Here, we have the wheel with five spokes, seven spokes, nine spokes, eleven spokes. All these yellow classes are the ones that are understood of sorts. The first really interesting class here in degree 8 is G3,5. It's something whose co-product gives a wheel 3 and a wheel 5. It's a complicated linear combination of graphs. You can compute it on a computer. But the reason we know it exists is, unfortunately, because of my theorem. Likewise, with this class at 10 loops and 11 loops, there's an 1G335. Unfortunately, there isn't a purely graph-threated way to get these elements on. Then, it's conjectured that H1 is always zero. This line should always be zero. The next interesting stuff happens in H3. There's some green classes here and a class in H7. They don't always occur in odd homological degree. There's a class in H6 up here. We'll see that later on. These green classes are kind of mysterious. Somehow, the dual of Gottten-Dichter-Hemeler group, many of you will know that this is somehow you can think of this as formal, multiple zeta values, modulo products. Symbol satisfying associative relations, modulo products. Though, of course, we're just doing combinatorics here. There are no numbers anywhere in this picture just yet. The green classes, on the other hand, have no such interpretation. I want you to hold this in your mind if you can. Another reason why this is very interesting is a recent theorem by Chan, Galatius and Payne in 2018. They showed how to relate the homology of the graph complex to the co-homology of the moduli space of curves, mg, in genus G. In fact, the graph complex computes the very top weight graded piece of its co-homology. By Delin, the co-homology has a mixed-hot structure. The graph complex sees the very top piece, which is point-caridual to a trivial motive. The key thing here is that in this equality from an algebraic geometric point of view, the motives on the left are trivial. They're just vector spaces. They just take motives plus the data of a weight. There's no GRT or anything like that that comes out of this picture. That's a puzzle. This will with three spokes corresponds to an interesting class in M3. There are two questions that jump out. One is how do we think of these higher degree classes in graph homology, the green ones? How do we relate the graph complex to mixed motives? If we could relate it to mixed motives, then we would expect to see the Gallo groups acting. We might be lucky and find this Gallo group acting, which would explain the appearance of the Gottten-Dichtegmacher group. This theorem here, unfortunately, doesn't do the job because this weight graded piece of mg is a pure motive. It doesn't do that. Today what I'm going to do is I'm going to define a notion of differential forms on a modular space of metric graphs. That's some variant of the outer space that Akkaren talked about this morning. I want to do DRAM theory on outer space, DRAM homology. Using that, we'll be able to assign numbers or motives to classes in the graph complex. After explaining that, I will explain some conjectures about the meaning of these higher degree homology classes. Here's a battle plan for how the talk will proceed. We're going to take this graph complex and promote it to a modular space of metric graphs. A metric graph is a graph where all of its edges have a length. The contraction of edges has a real tangible meaning. You're letting the length go to zero. There's a space of all possible lengths on a graph. That gives a modular space of metric graphs. You can embed that as the real points of an algebraic variety, just a projective space in fact. Doing some shenanigans, some compactifications, which we first learned in work of Dirk, joined with Spencer and Ilen some years ago. This is a variant of that. You can glue all these algebraic varieties together to form a huge infinite dimensional co-simplical scheme. On that, you can take the Duram complex. What does that mean? That means that a differential form will be a collection of infinitely many differential forms for every graph. For every graph, you're going to get a differential form. They are going to have to fit together in a nice way that reflects the way you glue these metric graphs together. A differential form is an infinite collection of forms that satisfies some compatibilities. The problem is, how do we construct such a thing? Such a differential form, it's not obvious. The idea here is to imitate the Torelli map in algebraic geometry. There's a map from the space of metric graphs called the tropical Torelli map to a space of symmetric matrices. This came up this morning in Cairns talk. On the space of symmetric matrices, we know how to construct invariant differential forms. This goes back to work in the 1930s, I think. I think Brower, in the book of Elman Vial, he mentions papers of Brower in 1935. It may go back before that. We write down invariant differential forms on the space of symmetric matrices, and then we pull them back to the space of metric graphs, and then we check that they satisfy all the properties we need. That's the battle plan. First, metric graphs. A metric graph is a graph, it'll be a connected graph, plus the data of a length, positive length, to every edge. That length is L sub e. You normalize it so that the total length, the sum over all the edges, is 1. That means if you fix your graph and you allow all the edges to vary, the space of possible metrics is just a simplex. It's just the set L1 up to Ln, positive numbers such that the sum is equal to 1. It's just a hyperplane. The idea is that if you send a length to 0, that's the same thing as, that should topologically correspond to contracting the edge, which is a different graph, of course. That'll be a different simplex associated to a different graph. Of course, when you contract edge, well, I'll come to that in a minute. Here's a picture. Let's take this graph here with a sunrise diagram with two vertices and three edges. It has two loops, genus two. The edges have lengths L1, L2, L3. They are subject to the condition that their sum equals 1. That defines a Euclidean triangle, two simplex, and Euclidean three space. It's the open triangle. It's the interior of this triangle. Now, as one of the lengths goes to 0, that means we travel, I don't know if you can see my mouse, but let's contract this length L2, let that go to 0. We travel down the simplex and end up on the boundary L2 equals 0. The boundary is not in the simplex. It was an open simplex, but this is the boundary in its compactification. Now we get a new simplex in Euclidean two space where L1 plus L3 equals 1. That we identify with the simplex, the space of metric graphs of this type, now with two loops in which the middle edge has been contracted. Now, it's the space of all possible values of L1 and L3. We think of that as being a degenerate version of this graph for which the second edge has zero length. You take all these open simplices and you glue some of them into the boundaries of others. What you get then is this open simplex and the three faces, but you don't get the three corners. The three corners would correspond to contracting a tadpole, contracting a loop, and we're not allowed to do that. In outer space, for example, all these faces fit together in a stratified way, but you don't have the corners. That's very important. I've written here that you can assemble all these simplices to form what I think is called reduced outer space. I'm not 100% sure on the terminology here. There's lots of different variants. A caveat here is, of course, in outer space, you need marked graphs. The markings actually play very little role here, so I'm going to just ignore that. The next stage then, so that's the space of metric graphs. Now, to make it algebraic, it's very easy. We can just identify this simplex as the real points in a projective space. The set of LI positive reels that add up to one, we can just view that as the positive real coordinates in simplex inside projective space. Here comes an approximate definition. It's actually no K definition, but it's not going to be what we need, actually. Then an algebraic differential form then on the space of metric graphs, so a form of degree K will be an infinite connection of algebraic differential forms, omega g, for every graph. It's going to be a form on this open simplex. When I say it's algebraic, I actually mean it's a form on this bigger projective space that the simplex sits in. I'm going to allow it to have poles because we're going to need that. What we want is for it to extend, sorry, we want it to be smooth on the space of metric graphs. So it's smooth on this real Euclidean simplex. Then what we want is that we want it to extend to the boundary and in such a way that on the boundary, it agrees with the differential form corresponding to the contracted graphs that sit on the boundary. Finally, we want this to be functorial with respect to automorphisms of the graph. If the graph has an automorphism, then the differential form, so if you have isomorphic graphs, then that differential forms should be isomorphic. Here's a picture. The same simplex as before. If we want to, what will a differential form look like? Well, it'll be a differential k form here on the interior of the simplex. That form is indexed by the sunrise graph. It's a form in the three variables L1, L2, L3 corresponding to the three edge lengths. On each face, we have differential forms for each graph. The property should be that this form when restricted to the boundary should line up, should match with the forms corresponding to these caution graphs. Now, if you look at these three faces, in fact, they all correspond to the same graph. It's just that the edge labelings are different. Therefore, these three forms, in fact, should all be the same. They're just obtained from each other by changes of variables, and that's the third property. Now, the key point here is that it's not obvious to construct such a thing, because if you try to do it inductively, you start with the boundary of your simplex, and you might have defined something on the boundary, then you need to extend it into the middle of this simplex. You might be able to do that, but then you need to extend it into the larger cells and so on and so forth, infinitely many times. You've got to do that in a functorial way. It's not obvious. The way we'll do this is using the Torelli map. Let me skip that. Basically, we'll replace the graph with its Laplacian matrix and define an invariant form on matrices. Let me quickly remind you of the graph Laplacian. I take a connected metric graph, and we have the usual complex that calculates the simplicial homology of a graph. Let's say you have z to the number of edges, to the space of edges, z to the vertices, and there's a boundary map that to an edge gives you the endpoint minus the source. The kernel of this map is the homology of the graph. Because the graph is connected, the co-kernel is h0, which is z. What we do is define an inner product on the set of edges. This will be extremely familiar to everybody here. You just say that the inner product of two edges is zero if they're distinct edges, but it's the length of that edge if it's the same edge. The norms of the edges are the lengths. That's an inner product on this space. It restricts to an inner product on this space, on the homology. That's the graph Laplacian. Another way to say it is that you can just write it in terms of matrices in a very straightforward way. It's probably easiest if I just show an example. Here's the wheel with three spokes. It's got three loops. Here's a basis for loops. This loop, 156, this loop, 246, and this one, 345. You write down this incidence matrix where the rows are indexing all the edges. The loop 156 involves edge 1, edge 5, and edge 6 with appropriate signs, cross pointing to choice of orientation. The inner product I defined, I mentioned a minute ago, that gives the length l, i to edge i is just a diagonal matrix in this basis. The graph Laplacian, you just take this matrix, the incidence matrix transpose times a diagonal times epsilon. When you do that, you get this matrix whose entries are the lengths of the edges. If you think of the edge lengths as variables, this is a matrix whose entries are just polynomials, just linear forms in variables. This is called the graph Laplacian matrix. It's very, very classical. It depends on a choice of basis. What we want to do is construct invariance of this that don't depend on the choice of basis. As many of you know, you could take the determinant and that would give you the graph polynomial, which comes up all over the place in quantum field theory. That's not good enough. We want forms. Now we turn to this classical theory of invariant forms. How do we do invariant forms? This is, again, very old stuff. I'm just going to take an arbitrary graded commutative differential graded algebra. You just think of this as, in the examples, it'll just be polynomials in some determinants and they're cated differentials. If I take any matrix whose entries are, say, polynomials, and if the matrix is invertible, then you can do the following thing. You can do x inverse dx. Excuse me. Now x inverse dx is a more carton form. You raise it to the power n, you take its trace, and that produces a form of degree n for every n. Now very elementary properties of the trace shows that this vanishes when n is even. It's always closed. It's a closed differential form. It behaves nicely with respect to transposing matrices. So if you transpose a matrix, you get a sign. That shows in particular that when your matrix is symmetric and our graph Laplacian matrices are always symmetric, then half of these forms always vanish. Another key property is that it's bianvariant. If you multiply by a constant matrix on the left or on the right, then the corresponding invariant form is unchanged. That's what's called invariant. Another thing that's quite important is that if x is a k by k matrix, then all these forms vanish once the degree is bigger than twice the size of the matrix. Here's a plea. You can notice that in fact more is true. It's not just the trace, but the matrix itself identically vanishes in that range. If anyone knows a reference for that, please could you let me know. The first such form is just the logarithmic derivative of the determinant of the matrix. Then we have all these odd forms, 3, 5, 7, 9, 11, 13. If x is symmetric, which is our case, 3, 7, 11 vanish and we're just left with 5, 9, and 13. These in fact give you the classes in algebraic k theory, which came up in the last talk, interestingly enough. Here's some examples. Let's take a 2 by 2 generic matrix. Then the beta 3x, you take x inverse dx, multiply by itself three times, take its trace, and you get this nice differential 3 form. If you do the same thing with a 3 by 3 matrix, we're going to get a big mess. We have to assume x is symmetric, otherwise the formula won't fit on, well it'll fit on a sheet of paper, but it's kind of ugly. If I take x as symmetric matrix, of course beta 3 now vanishes, but we get an interesting beta 5. Here it is. It's something over the square of the determinant. You notice when you do that you get massive cancellations. Basically you're getting what's called condensation of determinants. Those of you who have been around this subject for a while will recognize that phrase due to Dodson. Here's a theorem that I'll prove in the write-up of these notes. I don't know, I'm sure it's sort of thing that must be known, I can't find it anywhere. That's that the denominators are much smaller than you expect. If you take this beta to the odd power, because you've taken an inverse of a matrix and you've raised it two n plus one times, you expect the determinant to appear two to the n plus one times. But actually it only appears with half that power. In the case when x is symmetric, it's even more spectacular, you only get one quarter of the power. Now the reason I say this is because we see something very similar in quantum field theory. So if x is the graph Laplacian, which it will be in a minute, what you're getting is actually if you write out a formula for this thing beta, you're getting what are called Dodson polynomials in the numerator divided by the graph polynomial to some high power. You're getting massive cancellations between the numerators and the denominators. This is exactly what happens in quantum electrodynamics in parametric form, which was formulated by Dirk and worked out in the thesis of Marcel Gohls. So I think that's an interesting connection to quantum field theory. Okay, canonical graph forms. So now we apply invariant forms to graphs. We take a connected graph and take lambda g, the graph Laplacian. As I mentioned before, the determinant is just the Kirchhoff graph polynomial. But now we want to define the canonical graph form, not the graph polynomial, is one of these invariant trace forms. So it's the trace of lambda inverse g, d lambda g, where lambda is just the graph Laplacian matrix. And then we can take exterior products of these forms and we get what I call the canonical algebra. So these forms are a function from graphs to forms. To every graph, every such form is a map, which to every graph assigns a differential form. So it's an infinite connection of forms. So here's an example for the wheel with three spokes and you will recognize this. If you take a graph with edges, you want up to E n, we typically call the edge lengths alpha now in the physics contents, not l, it'll be alpha. So the graph Laplacian has entries which are linear functions of the alphas. It's determinant is a polynomial, it's called graph polynomial. And you can work out this this first form of degree five and it gives exactly what many people will recognize as the Feynman integral Feynman differential form for the wheel with three spokes. Okay, first theorem then is that for any graph, this canonical form is well defined, doesn't depend on choices. It is closed. It is projective. And it has poles only along the graph hypersurface, which is the zero locus of the graph polynomial, which is the vanishing locus of the determinant of the graph Laplacian. It is functorial in G. So if you have a nice morphism of graphs, it induces nifl morphism of these forms. And it's compatible with edge contraction. So if you contract an edge, you get the corresponding canonical form of the quotient graph. So that is exactly what we want to define a form on outer space or on space of metric graphs. So in fact, the theorem is true for any any wedge product, take any exterior product of these omega five, omega nine, omega 13, and so on. Those of you who know graph homology will be pleased to see that it has nice vanishing properties. So when the when the graph, if you restrict this form to a graph of the right, the right dimension, whether the graph is simplex is the right diamond of the same dimension. Then this form vanishes when it on the graph as a two valent vertex, when it has multiple edges, when it has a tadpole, when it is one vertex reducible. That's exactly what you see in the graph complex. So that's very nice. It has lots and lots of other nice properties that I shall skip. Okay, so now on now we have an infinite family of differential forms on spaces of graphs. So the natural thing to do is to integrate them. So to do this, we view the space of metric graphs. Remember this this simplex, which whose points, whether the sets of possible lengths of edges on a connected graph, sigma g, as I said before, we embed that in the real points of a projective space. So the sigma g here, which is the same triangle we had earlier, it's this the the set of points in projective space, alpha one, alpha two, alpha three, in homogeneous coordinates where all the alpha is positive. I apologize. Again, I put bigger than or equal to zero. That's a mistake that should just be strictly bigger than zero. I apologize for that. So that's that's a typo that should be strictly bigger than zero because the simplex was the open Euclidean simplex, which is very important. Okay. And as many of you know, of course, that these forms have have have poles along along the boundary. So there's a whole tricky business here that was initially in the case of financials worked out by Dirk and Spencer and Ellen, where you've got to do some blowups. And there's a whole business that I'm not going to talk about for reasons of time. But many of you know that very well. Okay, so the first term is you take any connected graph and any canonical differential form, such this trace of invariant trace of the Laplace matrix. And I suppose that I assume that the form is of the right degree to integrate it. So it's a it's a k form on a case simplex. So that means the number of edges of the graph is one more than the degree of this form. Then you can try to integrate this form over the simplex. And at this point, all the physicists brace them adopt the brace position and take cover because as we know, Feynman integrals never converge. They are always infinite. Pretty much in all the interesting cases. But this is the opposite. It's always finite. So even if you take a graph with the most horrible sub divergences you can imagine, the integral will always converge. So that's kind of amazing because we don't see that very often in quantum field theory, ever, in fact. And in fact, this integral is a period of fight the fourth theory, because this integrand, the differential form is some numerator, some very complicated numerator over the graph polynomial, the first semantic polynomial. So it defines some period in fight the fourth theory. Okay, so here goes. I'll just a quick message to the organizers. I think I started a bit late. I started terminus late. So I'll take the liberty of talking till 20 past. Sure. Yes, please. Is that okay? Yeah. Okay. Thanks. I'm nearly done anyway. Okay, so here's some examples. The wheel with three spokes we've already seen. If you compute this canonical form, you get in fact 10 times the residue, the coefficient of one over epsilon in dim reg in fight the fourth theory. And so everybody knows that that is six, Zeta of three. I had that explained to me first of all by Dirk and David many fond years ago. This is sort of the cornerstone, the thing that got me interested in all of this in the first place. But now it gets interesting. So the wheel with five spokes is not what you think it's a complicated thing. It's the Feynman integral. Some omega g says just some, you know, some alpha i d alpha one dot dot dot and you omit a D alpha. So just the standard projective form. Plus some multiple times alpha one to alpha five is a product of all the edge variables corresponding to the internal spokes. And this integral is not Z to seven the Feynman period of this graph, which is integral of just this first time. So the final integral is just this piece and that would give Z to seven. But this integral gives a multiple of Z to five, it's weight drop. And this coefficient exactly conspires to cancel out that coefficient of Z to seven and gives you a Z to five. So again, this is incredibly reminiscent of quantum electrodynamics, where we know that the highest weight parts in the quantum field theory cancel out. So it strongly suggests that quantum electrodynamics or other gauge theories might have some matrix theoretic formulation in this spirit. Wheel with seven spokes now gets seriously hard to compute these forms. I watched it out. Sorry, I didn't didn't check all the signs. It's something times Z to seven. Another class of graphs that we know how to compute are the complete graphs. And you know then that this integral is some multiple of a product of odd Z to values, two to three Z to five Z to two minus one. The reason you know that is that because this is literally the Borel regulator in algebraic K theory. And this calculation, the calculation that this integral gives a product of Z to values goes back to Z goal in a very beautiful paper in which he invented the what's called the unfolding technique for which is used across the theory of modular forms. So it's the unfolding for maybe an orthogonal group or special linear group. And it's the beginnings, the birth of the whole subject of Tamagawa numbers is in this calculation. So it's great to be that we can give a motivic interpretation. You can write down a motive associated to this Borel regulator, and it fits in this whole graph complex story. Okay. Stokes formula. So you can tell what's coming next. If I take a canonical form, so actually the canonical forms form a hopfile to Brown. Don't worry about this in the first approximation. You just need to know that if you took one of these basic forms like omega five or omega nine or omega 13, they're primitive. So Delta of omegas does one tensor omega plus omega tensor one. Okay. And a lot of times will simplify on this formula. But in general, it's a hopfile to Brown. So and it's the obvious hopfile to be so omega three, so I'm gonna five wedge omega nine would map to omega five tensor omega nine minus omega nine tensor omega five. Okay. And by applying Stokes them to this compactification of the simplex in this blow up of projective spaces. And using properties of these forms, you show that you get that the, that the sum of integrals corresponding to every face of this that the compactified simplex is zero. So that means that the sum of graph integrals for every possible contraction of your graph plus a bunch of products corresponding to sub and quotient graphs that some vanishes. So you'll recognize on the right this co con climber co product. And the proof is once you've said everything up is very straightforward. So we can rewrite this Stokes formula by taking this co product here and writing it into the trivial part, omega tensor one plus one tensor omega plus the reduced co product. And when we do that and rearrange this, what we find is that the Stokes formula has three terms. It has the first term, which is literally the differential in the graph complex. It has a second term, which is what is sometimes called the second differential in the graph complex, which is the differential where you don't contract edges, but you delete edges. And that's also differential in the graph complex. And the third term is exactly the reduced co product. It's the co algebra structure on graph homology induced by the con climber co product that I mentioned way at the beginning of the talk. So this Stokes formula has a, gives a very nice geometric interpretation of all these structures that are built into the graph complex. Some applications, you can use these integrals to detect non vanishing graph homology classes. That's harder in practice than it, than it, than it looks. But in principle, it's possible and you can do it in some cases. We can associate a motive to any graph homology class. So that explains perhaps why we get Grottendiktachemala, Mettific Galois groups appearing in graph complexes. And this third point here, I'm a bit embarrassed by, but it is actually true that the cosmic Galois group acts on the differential forms on outer space. And it sounds like a big joke that I've concocted this. I promise you I didn't come up with the phrase cosmic Galois group. It was due to Cartier as we learned in the last talk. Outer space, it was coined by, by Kahn, Vokhtman. It just happens that these things are in fact very closely related in a meaningful way by, by this machinery. So here's an example of the Stokes theorem. So let's go back very quickly to, to this picture of graph homology. Remember there was this, this class here, which is the first non trivial sort of Liebracket across G3,5 and weight 8. And using it, we're going to calculate an integral on this class, this green class way up here. And the way we do that is we, we take this, this graph here was a linear combination of graphs, and we integrate zero over that graph. We apply the Stokes theorem to, to that. And what it does is it will produce, because its co-product is a wheel 3 and a wheel 5, which gave z to 3 and z to 5. We then do this, this differential D and delta, these different, these two differentials in the graph complex. And we zigzag our way up to this class. And using Stokes formula, we deduce that the integral of omega 5 wedge omega 9 on, on this class here is in fact z to 3 times z to 5. And that proves that this class, this green class is non vanishing in homology. So, sorry, that was a bit quick. But here it is. So, this class in H3 here, let's call it xi, you can show that this integral of a, of omega 5 wedge omega 9 over this class is a non trivial product of z to values. Okay, so now we, now let's look again at this picture of graph cohomology. So graph homology, I've actually switched to graph cohomology because I want to dualize because we're talking about forms. So it's, it's, it's easier to relate to graph cohomology, but it's the same. It's really the same thing. So now I've redrawn the conjectural, semi conjectural, or anyway, in any case, the computer calculations of non classes in graph cohomology, which is just the dual of graph homology. And I've, I've given them names and the names are these differential forms and lead brackets of these differential forms. So what we have omega 5 corresponds to the wheel with three spokes, omega 9 the wheel with five spokes, I'm the 13 new with seven spokes, etc. Then we have wedge products. And the first wedge product is this class in H3. We have a triple wedge product up here in H6, and so on and so forth. And then we have lead brackets of the original forms. And we also have lead brackets of the wedge products of these forms. And these classes line up exactly with all the known graph homology classes. So here's a conjecture. I conjectured that the free Lie algebra on this canonical algebra of differential forms injects into the graph cohomology. It does it in a funny way. So the grading here is the number of edges is the degree and it can correspond to the number of edges. We have to forget, I don't know they line up in funny, common, common logical degrees. I don't know how to predict those obvious question one is, is this map an isomorphism? I, I because I can later deny it. I will conjecture here during this talk that it is an isomorphism, but I may, I may deny that later on. I reserve the right. But anyway, it's a conjecture, but it's a weaker conjecture, a more tentative conjecture than the previous one. And I was going to say something about the, the dual since I'm out of time, I'll stop and, and leave you with this, this picture so you can, you can, you can contemplate it if there are any questions. Thank you very much for listening. Thank you for answers. This was amazing. Thank you. I have a million questions, but if anybody else wants to go first, I'll go first. So the G three five linear combination, which you said was hideous. Do you have a handy, I'd love to know just. Oh, it's known. I mean, I think it's, it's written down in some papers. That that one's okay, you can compute that one. But I think, I think the limits of the computer calculations are this column. I'm not 100% sure. And I'm out of date. As I said, I got this from, from slides of talks a year ago. I don't know if things have moved on, but I think the 10 column and everything to the left of it is checked by computer and this column 11 is possibly not. So, so you'll get your three five graph, but you'll be in a mess when you, if you want to go much further than that, you, you're going to have to give up. I think if you just want to compute the graph, you can take G three G five and just compute the graphically bracket. I mean, the non-true question is that this is a non zero class. Yeah, but also it's, so there's difference when you work in graph homology or graph homology. So if you work with graph homology, the classes are not wheels. They're wheels plus a long tail that's not known. And you've got to take the lead brackets of those. But you don't have a question of David. Yes. I didn't mind you throwing away half of the wheels. I thought that was quite fun. You got back the zeta values decreased. But as it's the week of the expert take, do you have any thoughts about the zigzag diagrams? Those are after all the only infinite class of diagrams on quantum field theory, which have a closed formula, which they can I conjectured and you and Oliver proved that that's a great question. Actually, when I'm new to this subject, when I first learn about this at a conference a year ago, I asked exactly the same question to the experts and they said they didn't know. That's a great question. Thank you. I wanted to ask a question about the weights. I mean, you mentioned these weight drops and you have these. I found it quite striking that you had these pure weight evaluations of these wheels with these complicated into grants and you just got a zeta seven or just got a zeta five. So is there like a understanding from the structure of the blow up that you get some bounds on the weights that would predict this? It's not obvious from an algebraic geometry perspective. Not that I know of, but I kind of have the feeling that all these these integral come from this Burrell regulator. So you know that morally, beta five wedge beta nine corresponds to zeta three zeta five. And so somehow what I would like to be able to do, but I can't do it, is relate this complete graph using Stokes formula and identities to wheel three tensor wheel five. And that would give that would give it immediately. But I don't know how to do that. So I did that argument in the opposite direction to compute the one of the examples. Thank you very much. Let's thank friends again. I'll see you in other questions at the moment. Thank you.
 Thank you very much, Eric. Of course, it's a great pleasure to speak at this conference.  I'm very sad that we couldn't meet in person. It's been a while since I've seen Dirk and  I think it would have been tremendous fun. I think it would have been fun because we  could have done some of this. I wonder if you can see. Can you see my screen? Nope, you  can't. That's very strange. Okay, let me find a different way. We could have done some of  this. At the risk of embarrassing Dirk, this is a photo he sent me of a New Year's greetings  in 2011 of a typical crime of family gathering. There's quite an extraordinary collection  of bottles there. It's well worth close inspection. Okay, so now back to mathematics. I should  say that I was in fact slightly relieved when the conference was delayed. I'm sorry to  say because this project wasn't finished and I thought that when the time came I'd have  made some progress. Of course, nearly a year has passed and I haven't worked on this for  a single minute. This isn't so much. It touches on pretty much every single aspect of it. It  touches on work of Dirk's. Dirk has done it over the previous few decades. With that said,  let's begin. I'm going to talk about graph homology. We're going to take G, a connected  graph, and I'm going to denote the loop number, also known as the genus of the graph, by HG,  the standard. The number of edges is EG. The degree here is something slightly funny. It's  the edges minus twice the loops. That will be the degree. This will be familiar to you  as minus the superficial degree of divergence. It's like a superficial degree of convergence.  Next an orientation on a graph will be essentially plus or minus a wedge product of symbols corresponding  to the edges in the graph. What that means, it's just an ordering on the edges modulo  the action of even permutations. If you flip two edges, you get the opposite orientation.  That's an oriented graph. It's not oriented in the usual sense when you put arrows on  the edges. It's an orientation on the ordering of the edges. The graph complex is defined  by taking the Q vector space spanned by oriented graphs G, E, sorry, G, Eta. We assume that  G has no tadpoles, so no self edges like this, and it has no vertices of degree less than  or equal to two. Then you impose some very simple relations that a graph with a negative  orientation is the negative of the graph with that orientation. That's fairly natural.  You also impose a relation that the G of Eta is equivalent to the same graph where you  permute the edges according to an automorphism of the graph. If a graph has an automorphism,  it's equivalent to the new graph with the new orientation. In particular, if a graph  has an automorphism which induces an odd permutation of the edges, then its class is zero. In equivalence  classes, it will be denoted in square brackets. Then on this vector space of graphs, oriented  graphs, you can define the following differential. This was done by Maxime many years ago. The  differential of an oriented graph is a sum of all the edges in the graph. What you do  is you contract each edge in the graph and have the induced orientation with the appropriate  sign. This contraction, I use this double slash notation. I always use this to mean  the contraction where if you contract a tadpole, if you contract a loop, that's zero. That's  the empty graph. In case you're not allowed to contract tadpoles. This differential is  well-defined and it squares to zero, so it defines a complex. Furthermore, you can check  that it has degree minus one with respect to the degree. I remind you that the degree  is edges minus twice the number of loops. Great. Here are some examples which I'm sure  you'll, with this audience, will find extremely straightforward. Here's the differential at  the top. The first remark is that any graph that has a doubled edge is zero in this graph  complex GC2. That's because if you take a double edge here with edges one, two, with  the orientation e1 wedge, e2 wedge, something else, e2, all the other edges, then you can  flip the edges one and two. That gives you an isomorphic graph, but it will reverse the  orientation. Therefore, it's minus itself and therefore it must vanish. Any graph with  doubled edge vanishes. By the way, from now on, I'm going to drop the orientations. It's  boring to keep writing the orientations. They'll just be implicit from now. From the previous  remark, we get that any graph with the property that every edge lies in a triangle, it automatically  has the property that its differential is zero. That's because if you take a triangle  and contract an edge in a triangle, you get a doubled edge. As we've just seen, a doubled  edge is zero. Graphs built out of triangles are going to have zero differential in the  graph complex. Our favorite example is therefore the wheels. The wheels with n spokes and they  satisfy, of course, d of the class of the wheel with any orientation, doesn't matter,  is zero. Here's just a hammer at Harmon's shore. You don't need this. Here's another  example of a differential in the graph complex. We're going to contract all the edges in this  graph here on the left. There are two triangles. Contracting them gives zero, as we've already  seen. The only edges that will do something when you contract them are the red ones. When  you contract the top one, you get this graph here. When you contract the middle one, you  get a wheel with four spokes. When you contract the bottom one, you get this graph here.  The first and third graphs cancel out and you're left with a wheel with four spokes.  The wheel with four spokes is exact. In fact, more is true. The wheels with even numbers  of spokes are always zero because they have a symmetry that's odd and that forces them  to be zero in the graph complex. Only the odd wheels survive. Graph homology is HNGC2  is the kernel of this differential, modulo the image. The homological degree is this  degree edges minus twice number of loops. It's also graded by the genus, the loop number.  We get a bunch of homology groups and they are in turn graded. What do we know? It's  known that the homology vanishes in negative degrees for positive genus. It has a lot of  extra structure. The first bit of extra structure, and here we see the first interaction with  the work of Dirk and Anna, and that's that the graph homology has a leacoalgebra structure  and it's induced by anti-symmetrizing the con-crime or coproduct. I'm sure everybody  here knows, I've seen this before, you take the sum of a certain class of subgraphs, typically  one particle irreducible, and on the left of the tensor you have the subgraph and on  the right you have the quotient graph. I apologize, there's a typo here which I can't change.  This should be a single slash and not a double slash. I'm sorry that formula is wrong. The  single slash means you just contract the subgraph gamma even if it contains a loop, it's not  zero. I'm sorry I've used the wrong notation, that should be a single slash. Then Wielwacher  showed in 2014 a fantastic result that the zero degree homology of the graph complex  is due to the Grotten-Diek-Teichmann-Lie algebra, which is something that's explicitly defined  but quite tricky to understand. We don't know much about the Grotten-Diek-Teichmann-Lie  algebra at all. What we do know is something predicted by Deline. This used to be called  the Deline-Iharra conjecture but I was recently told by Iharra that it should be called the  Deline conjecture. It states that there's a free Lie algebra with infinitely many generators,  sigma 3, sigma 5, sigma 7 in degrees 3, 5, 7, every odd degree, which injects into Grt.  That means that Grt has got this huge free Lie algebra inside it and you can show, I  think Wielwacher showed that the sigmas end up being due to the odd wheels. Sigma 3 corresponds  to the wheel with 3 spokes, sigma 5 the wheel with 5 spokes, etc. The first puzzle here  is that this group is to do with the Motyvic-Gallore groups. This free Lie algebra is the Lie  algebra of the Motyvic-Gallore group. We can rightfully ask what on earth have Motyvic-Gallore  groups got to do with graph homology? That's the first mystery that I'd like to come back  to. Here's a picture of graph homology in low degrees and I'd like to spend a little  bit of time discussing this. Up to the left axis is the homological degree, H0, H1, H2.  Along the right axis are the number of loops, one-lip, two-lip, three-lips, four-lips and  so on. You can draw this picture different ways. You can grade by edges and that's actually  probably a better thing to do as we'll see. There are different ways you can represent  this. The first remark is that there's this red diagonal line and this red diagonal line  is the line where, in fact, just immediately below this line are trivalent graphs. Every  vertex has degree 3. Above that, that means on the red line or above, all these graphs  have a two-valent vertex. There's zero in the graph complex. Your intuition is as you  go up in this diagram, you have lots and lots of edges, very few loops. In quantum field  there, that'd be a very convergent graph. As you go down, you get infrared divergences  as you go vertically down in this picture. Very convergent. Anything above this red line  is zero. Now, going across the H0, that's the only degree in which things are really  well understood. So, this is the Grotten-Diegthe-Teichmeler, the algebra. Here, in degree 3, we have  the wheel with three spokes and you can show that gives a non-trivial class. Here, we have  the wheel with five spokes, seven spokes, nine spokes, eleven spokes. All these yellow  classes are the ones that are understood of sorts. The first really interesting class  here in degree 8 is G3,5. It's something whose co-product gives a wheel 3 and a wheel  5. It's a complicated linear combination of graphs. You can compute it on a computer.  But the reason we know it exists is, unfortunately, because of my theorem. Likewise, with this  class at 10 loops and 11 loops, there's an 1G335. Unfortunately, there isn't a purely  graph-threated way to get these elements on. Then, it's conjectured that H1 is always zero.  This line should always be zero. The next interesting stuff happens in H3. There's some  green classes here and a class in H7. They don't always occur in odd homological degree.  There's a class in H6 up here. We'll see that later on. These green classes are kind of  mysterious. Somehow, the dual of Gottten-Dichter-Hemeler group, many of you will know that this is somehow  you can think of this as formal, multiple zeta values, modulo products. Symbol satisfying  associative relations, modulo products. Though, of course, we're just doing combinatorics  here. There are no numbers anywhere in this picture just yet. The green classes, on the  other hand, have no such interpretation. I want you to hold this in your mind if you  can. Another reason why this is very interesting is a recent theorem by Chan, Galatius and  Payne in 2018. They showed how to relate the homology of the graph complex to the co-homology  of the moduli space of curves, mg, in genus G. In fact, the graph complex computes the  very top weight graded piece of its co-homology. By Delin, the co-homology has a mixed-hot  structure. The graph complex sees the very top piece, which is point-caridual to a trivial  motive. The key thing here is that in this equality from an algebraic geometric point  of view, the motives on the left are trivial. They're just vector spaces. They just take  motives plus the data of a weight. There's no GRT or anything like that that comes out  of this picture. That's a puzzle. This will with three spokes corresponds to an interesting  class in M3. There are two questions that jump out. One is how do we think of these  higher degree classes in graph homology, the green ones? How do we relate the graph complex  to mixed motives? If we could relate it to mixed motives, then we would expect to see  the Gallo groups acting. We might be lucky and find this Gallo group acting, which would  explain the appearance of the Gottten-Dichtegmacher group. This theorem here, unfortunately, doesn't  do the job because this weight graded piece of mg is a pure motive. It doesn't do that.  Today what I'm going to do is I'm going to define a notion of differential forms on a  modular space of metric graphs. That's some variant of the outer space that Akkaren talked  about this morning. I want to do DRAM theory on outer space, DRAM homology. Using that,  we'll be able to assign numbers or motives to classes in the graph complex. After explaining  that, I will explain some conjectures about the meaning of these higher degree homology  classes. Here's a battle plan for how the talk will proceed. We're going to take this  graph complex and promote it to a modular space of metric graphs. A metric graph is  a graph where all of its edges have a length. The contraction of edges has a real tangible  meaning. You're letting the length go to zero. There's a space of all possible lengths on  a graph. That gives a modular space of metric graphs. You can embed that as the real points  of an algebraic variety, just a projective space in fact. Doing some shenanigans, some  compactifications, which we first learned in work of Dirk, joined with Spencer and  Ilen some years ago. This is a variant of that. You can glue all these algebraic varieties  together to form a huge infinite dimensional co-simplical scheme. On that, you can take  the Duram complex. What does that mean? That means that a differential form will be a collection  of infinitely many differential forms for every graph. For every graph, you're going  to get a differential form. They are going to have to fit together in a nice way that  reflects the way you glue these metric graphs together. A differential form is an infinite  collection of forms that satisfies some compatibilities. The problem is, how do we construct such  a thing? Such a differential form, it's not obvious. The idea here is to imitate the  Torelli map in algebraic geometry. There's a map from the space of metric graphs called  the tropical Torelli map to a space of symmetric matrices. This came up this morning in Cairns  talk. On the space of symmetric matrices, we know how to construct invariant differential  forms. This goes back to work in the 1930s, I think. I think Brower, in the book of Elman  Vial, he mentions papers of Brower in 1935. It may go back before that. We write down  invariant differential forms on the space of symmetric matrices, and then we pull them  back to the space of metric graphs, and then we check that they satisfy all the properties  we need. That's the battle plan. First, metric graphs. A metric graph is a graph, it'll be  a connected graph, plus the data of a length, positive length, to every edge. That length  is L sub e. You normalize it so that the total length, the sum over all the edges, is 1.  That means if you fix your graph and you allow all the edges to vary, the space of possible  metrics is just a simplex. It's just the set L1 up to Ln, positive numbers such that the  sum is equal to 1. It's just a hyperplane. The idea is that if you send a length to 0,  that's the same thing as, that should topologically correspond to contracting the edge, which is  a different graph, of course. That'll be a different simplex associated to a different  graph. Of course, when you contract edge, well, I'll come to that in a minute. Here's  a picture. Let's take this graph here with a sunrise diagram with two vertices and three  edges. It has two loops, genus two. The edges have lengths L1, L2, L3. They are subject  to the condition that their sum equals 1. That defines a Euclidean triangle, two simplex,  and Euclidean three space. It's the open triangle. It's the interior of this triangle.  Now, as one of the lengths goes to 0, that means we travel, I don't know if you can  see my mouse, but let's contract this length L2, let that go to 0. We travel down the simplex  and end up on the boundary L2 equals 0. The boundary is not in the simplex. It was an  open simplex, but this is the boundary in its compactification. Now we get a new simplex  in Euclidean two space where L1 plus L3 equals 1. That we identify with the simplex, the  space of metric graphs of this type, now with two loops in which the middle edge has been  contracted. Now, it's the space of all possible values of L1 and L3. We think of that as being  a degenerate version of this graph for which the second edge has zero length. You take  all these open simplices and you glue some of them into the boundaries of others. What  you get then is this open simplex and the three faces, but you don't get the three corners.  The three corners would correspond to contracting a tadpole, contracting a loop, and we're not  allowed to do that. In outer space, for example, all these faces fit together in a stratified  way, but you don't have the corners. That's very important. I've written here that you  can assemble all these simplices to form what I think is called reduced outer space. I'm  not 100% sure on the terminology here. There's lots of different variants. A caveat here  is, of course, in outer space, you need marked graphs. The markings actually play very little  role here, so I'm going to just ignore that. The next stage then, so that's the space of  metric graphs. Now, to make it algebraic, it's very easy. We can just identify this simplex  as the real points in a projective space. The set of LI positive reels that add up to  one, we can just view that as the positive real coordinates in simplex inside projective  space. Here comes an approximate definition. It's actually no K definition, but it's not  going to be what we need, actually. Then an algebraic differential form then on the space  of metric graphs, so a form of degree K will be an infinite connection of algebraic differential  forms, omega g, for every graph. It's going to be a form on this open simplex. When I  say it's algebraic, I actually mean it's a form on this bigger projective space that  the simplex sits in. I'm going to allow it to have poles because we're going to need  that. What we want is for it to extend, sorry, we want it to be smooth on the space of metric  graphs. So it's smooth on this real Euclidean simplex. Then what we want is that we want  it to extend to the boundary and in such a way that on the boundary, it agrees with the  differential form corresponding to the contracted graphs that sit on the boundary. Finally,  we want this to be functorial with respect to automorphisms of the graph. If the graph  has an automorphism, then the differential form, so if you have isomorphic graphs, then  that differential forms should be isomorphic. Here's a picture. The same simplex as before.  If we want to, what will a differential form look like? Well, it'll be a differential k  form here on the interior of the simplex. That form is indexed by the sunrise graph.  It's a form in the three variables L1, L2, L3 corresponding to the three edge lengths.  On each face, we have differential forms for each graph. The property should be that this  form when restricted to the boundary should line up, should match with the forms corresponding  to these caution graphs. Now, if you look at these three faces, in fact, they all correspond  to the same graph. It's just that the edge labelings are different. Therefore, these  three forms, in fact, should all be the same. They're just obtained from each other by changes  of variables, and that's the third property. Now, the key point here is that it's not  obvious to construct such a thing, because if you try to do it inductively, you start  with the boundary of your simplex, and you might have defined something on the boundary,  then you need to extend it into the middle of this simplex. You might be able to do that,  but then you need to extend it into the larger cells and so on and so forth, infinitely many  times. You've got to do that in a functorial way. It's not obvious. The way we'll do this  is using the Torelli map. Let me skip that. Basically, we'll replace the graph with its  Laplacian matrix and define an invariant form on matrices. Let me quickly remind you of  the graph Laplacian. I take a connected metric graph, and we have the usual complex that  calculates the simplicial homology of a graph. Let's say you have z to the number of edges,  to the space of edges, z to the vertices, and there's a boundary map that to an edge  gives you the endpoint minus the source. The kernel of this map is the homology of the  graph. Because the graph is connected, the co-kernel is h0, which is z. What we do is  define an inner product on the set of edges. This will be extremely familiar to everybody  here. You just say that the inner product of two edges is zero if they're distinct  edges, but it's the length of that edge if it's the same edge. The norms of the edges  are the lengths. That's an inner product on this space. It restricts to an inner product  on this space, on the homology. That's the graph Laplacian. Another way to say it is  that you can just write it in terms of matrices in a very straightforward way. It's probably  easiest if I just show an example. Here's the wheel with three spokes. It's got three  loops. Here's a basis for loops. This loop, 156, this loop, 246, and this one, 345. You  write down this incidence matrix where the rows are indexing all the edges. The loop  156 involves edge 1, edge 5, and edge 6 with appropriate signs, cross pointing to choice  of orientation. The inner product I defined, I mentioned a minute ago, that gives the length  l, i to edge i is just a diagonal matrix in this basis. The graph Laplacian, you just  take this matrix, the incidence matrix transpose times a diagonal times epsilon. When you do  that, you get this matrix whose entries are the lengths of the edges. If you think of  the edge lengths as variables, this is a matrix whose entries are just polynomials, just linear  forms in variables. This is called the graph Laplacian matrix. It's very, very classical.  It depends on a choice of basis. What we want to do is construct invariance of this that  don't depend on the choice of basis. As many of you know, you could take the determinant  and that would give you the graph polynomial, which comes up all over the place in quantum  field theory. That's not good enough. We want forms. Now we turn to this classical theory  of invariant forms. How do we do invariant forms? This is, again, very old stuff. I'm  just going to take an arbitrary graded commutative differential graded algebra. You just think  of this as, in the examples, it'll just be polynomials in some determinants and they're  cated differentials. If I take any matrix whose entries are, say, polynomials, and if  the matrix is invertible, then you can do the following thing. You can do x inverse  dx. Excuse me. Now x inverse dx is a more carton form. You raise it to the power n, you  take its trace, and that produces a form of degree n for every n. Now very elementary  properties of the trace shows that this vanishes when n is even. It's always closed. It's a  closed differential form. It behaves nicely with respect to transposing matrices. So if  you transpose a matrix, you get a sign. That shows in particular that when your matrix  is symmetric and our graph Laplacian matrices are always symmetric, then half of these forms  always vanish. Another key property is that it's bianvariant. If you multiply by a constant  matrix on the left or on the right, then the corresponding invariant form is unchanged.  That's what's called invariant. Another thing that's quite important is that if x is a k  by k matrix, then all these forms vanish once the degree is bigger than twice the size of  the matrix. Here's a plea. You can notice that in fact more is true. It's not just the trace,  but the matrix itself identically vanishes in that range. If anyone knows a reference  for that, please could you let me know. The first such form is just the logarithmic derivative  of the determinant of the matrix. Then we have all these odd forms, 3, 5, 7, 9, 11,  13. If x is symmetric, which is our case, 3, 7, 11 vanish and we're just left with 5,  9, and 13. These in fact give you the classes in algebraic k theory, which came up in the  last talk, interestingly enough. Here's some examples. Let's take a 2 by 2 generic matrix.  Then the beta 3x, you take x inverse dx, multiply by itself three times, take its trace, and  you get this nice differential 3 form. If you do the same thing with a 3 by 3 matrix,  we're going to get a big mess. We have to assume x is symmetric, otherwise the formula  won't fit on, well it'll fit on a sheet of paper, but it's kind of ugly. If I take x  as symmetric matrix, of course beta 3 now vanishes, but we get an interesting beta 5.  Here it is. It's something over the square of the determinant. You notice when you do  that you get massive cancellations. Basically you're getting what's called condensation  of determinants. Those of you who have been around this subject for a while will recognize  that phrase due to Dodson. Here's a theorem that I'll prove in the  write-up of these notes. I don't know, I'm sure it's sort of thing that must be known,  I can't find it anywhere. That's that the denominators are much smaller than you expect.  If you take this beta to the odd power, because you've taken an inverse of a matrix and you've  raised it two n plus one times, you expect the determinant to appear two to the n plus  one times. But actually it only appears with half that power. In the case when x is symmetric,  it's even more spectacular, you only get one quarter of the power.  Now the reason I say this is because we see something very similar in quantum field theory.  So if x is the graph Laplacian, which it will be in a minute, what you're getting is actually  if you write out a formula for this thing beta, you're getting what are called Dodson  polynomials in the numerator divided by the graph polynomial to some high power. You're  getting massive cancellations between the numerators and the denominators. This is exactly  what happens in quantum electrodynamics in parametric form, which was formulated by  Dirk and worked out in the thesis of Marcel Gohls. So I think that's an interesting connection  to quantum field theory.  Okay, canonical graph forms. So now we apply invariant forms to graphs. We take a connected  graph and take lambda g, the graph Laplacian. As I mentioned before, the determinant is  just the Kirchhoff graph polynomial. But now we want to define the canonical graph form,  not the graph polynomial, is one of these invariant trace forms. So it's the trace of  lambda inverse g, d lambda g, where lambda is just the graph Laplacian matrix. And then  we can take exterior products of these forms and we get what I call the canonical algebra.  So these forms are a function from graphs to forms. To every graph, every such form is  a map, which to every graph assigns a differential form. So it's an infinite connection of forms.  So here's an example for the wheel with three spokes and you will recognize this. If you  take a graph with edges, you want up to E n, we typically call the edge lengths alpha  now in the physics contents, not l, it'll be alpha. So the graph Laplacian has entries  which are linear functions of the alphas. It's determinant is a polynomial, it's called  graph polynomial. And you can work out this this first form of degree five and it gives  exactly what many people will recognize as the Feynman integral Feynman differential  form for the wheel with three spokes. Okay, first theorem then is that for any graph,  this canonical form is well defined, doesn't depend on choices. It is closed. It is projective.  And it has poles only along the graph hypersurface, which is the zero locus of the graph polynomial,  which is the vanishing locus of the determinant of the graph Laplacian. It is functorial in  G. So if you have a nice morphism of graphs, it induces nifl morphism of these forms. And  it's compatible with edge contraction. So if you contract an edge, you get the corresponding  canonical form of the quotient graph. So that is exactly what we want to define a form on  outer space or on space of metric graphs. So in fact, the theorem is true for any any  wedge product, take any exterior product of these omega five, omega nine, omega 13, and  so on. Those of you who know graph homology will be pleased to see that it has nice vanishing  properties. So when the when the graph, if you restrict this form to a graph of the right,  the right dimension, whether the graph is simplex is the right diamond of the same dimension.  Then this form vanishes when it on the graph as a two valent vertex, when it has multiple  edges, when it has a tadpole, when it is one vertex reducible. That's exactly what you  see in the graph complex. So that's very nice. It has lots and lots of other nice properties  that I shall skip. Okay, so now on now we have an infinite family of differential forms  on spaces of graphs. So the natural thing to do is to integrate them. So to do this, we  view the space of metric graphs. Remember this this simplex, which whose points, whether  the sets of possible lengths of edges on a connected graph, sigma g, as I said before,  we embed that in the real points of a projective space. So the sigma g here, which is the same  triangle we had earlier, it's this the the set of points in projective space, alpha one,  alpha two, alpha three, in homogeneous coordinates where all the alpha is positive. I apologize.  Again, I put bigger than or equal to zero. That's a mistake that should just be strictly  bigger than zero. I apologize for that. So that's that's a typo that should be strictly  bigger than zero because the simplex was the open Euclidean simplex, which is very important.  Okay. And as many of you know, of course, that these forms have have have poles along  along the boundary. So there's a whole tricky business here that was initially in the case  of financials worked out by Dirk and Spencer and Ellen, where you've got to do some blowups.  And there's a whole business that I'm not going to talk about for reasons of time. But  many of you know that very well. Okay, so the first term is you take any connected graph  and any canonical differential form, such this trace of invariant trace of the Laplace  matrix. And I suppose that I assume that the form is of the right degree to integrate it.  So it's a it's a k form on a case simplex. So that means the number of edges of the graph  is one more than the degree of this form. Then you can try to integrate this form over  the simplex. And at this point, all the physicists brace them adopt the brace position and take  cover because as we know, Feynman integrals never converge. They are always infinite.  Pretty much in all the interesting cases. But this is the opposite. It's always finite.  So even if you take a graph with the most horrible sub divergences you can imagine, the  integral will always converge. So that's kind of amazing because we don't see that very  often in quantum field theory, ever, in fact. And in fact, this integral is a period of fight  the fourth theory, because this integrand, the differential form is some numerator, some  very complicated numerator over the graph polynomial, the first semantic polynomial.  So it defines some period in fight the fourth theory. Okay, so here goes. I'll just a quick  message to the organizers. I think I started a bit late. I started terminus late. So I'll  take the liberty of talking till 20 past. Sure. Yes, please. Is that okay? Yeah. Okay.  Thanks. I'm nearly done anyway. Okay, so here's some examples. The wheel with three spokes  we've already seen. If you compute this canonical form, you get in fact 10 times the residue,  the coefficient of one over epsilon in dim reg in fight the fourth theory. And so everybody  knows that that is six, Zeta of three. I had that explained to me first of all by Dirk  and David many fond years ago. This is sort of the cornerstone, the thing that got me  interested in all of this in the first place. But now it gets interesting. So the wheel  with five spokes is not what you think it's a complicated thing. It's the Feynman integral.  Some omega g says just some, you know, some alpha i d alpha one dot dot dot and you omit  a D alpha. So just the standard projective form. Plus some multiple times alpha one to  alpha five is a product of all the edge variables corresponding to the internal spokes. And this  integral is not Z to seven the Feynman period of this graph, which is integral of just this  first time. So the final integral is just this piece and that would give Z to seven.  But this integral gives a multiple of Z to five, it's weight drop. And this coefficient  exactly conspires to cancel out that coefficient of Z to seven and gives you a Z to five.  So again, this is incredibly reminiscent of quantum electrodynamics, where we know that  the highest weight parts in the quantum field theory cancel out. So it strongly suggests  that quantum electrodynamics or other gauge theories might have some matrix theoretic formulation  in this spirit. Wheel with seven spokes now gets seriously hard to compute these forms.  I watched it out. Sorry, I didn't didn't check all the signs. It's something times Z to seven.  Another class of graphs that we know how to compute are the complete graphs. And you know  then that this integral is some multiple of a product of odd Z to values, two to three  Z to five Z to two minus one. The reason you know that is that because this is literally  the Borel regulator in algebraic K theory. And this calculation, the calculation that  this integral gives a product of Z to values goes back to Z goal in a very beautiful paper  in which he invented the what's called the unfolding technique for which is used across  the theory of modular forms. So it's the unfolding for maybe an orthogonal group or  special linear group. And it's the beginnings, the birth of the whole subject of Tamagawa  numbers is in this calculation. So it's great to be that we can give a motivic interpretation.  You can write down a motive associated to this Borel regulator, and it fits in this whole  graph complex story. Okay. Stokes formula. So you can tell what's coming next. If I take  a canonical form, so actually the canonical forms form a hopfile to Brown. Don't worry  about this in the first approximation. You just need to know that if you took one of  these basic forms like omega five or omega nine or omega 13, they're primitive. So Delta  of omegas does one tensor omega plus omega tensor one. Okay. And a lot of times will simplify  on this formula. But in general, it's a hopfile to Brown. So and it's the obvious hopfile  to be so omega three, so I'm gonna five wedge omega nine would map to omega five tensor  omega nine minus omega nine tensor omega five. Okay. And by applying Stokes them to this  compactification of the simplex in this blow up of projective spaces. And using properties  of these forms, you show that you get that the, that the sum of integrals corresponding  to every face of this that the compactified simplex is zero. So that means that the sum  of graph integrals for every possible contraction of your graph plus a bunch of products corresponding  to sub and quotient graphs that some vanishes. So you'll recognize on the right this co con  climber co product. And the proof is once you've said everything up is very straightforward.  So we can rewrite this Stokes formula by taking this co product here and writing it into the  trivial part, omega tensor one plus one tensor omega plus the reduced co product. And when  we do that and rearrange this, what we find is that the Stokes formula has three terms.  It has the first term, which is literally the differential in the graph complex. It has  a second term, which is what is sometimes called the second differential in the graph  complex, which is the differential where you don't contract edges, but you delete edges.  And that's also differential in the graph complex. And the third term is exactly the reduced  co product. It's the co algebra structure on graph homology induced by the con climber  co product that I mentioned way at the beginning of the talk. So this Stokes formula has a,  gives a very nice geometric interpretation of all these structures that are built into  the graph complex. Some applications, you can use these integrals to detect non vanishing  graph homology classes. That's harder in practice than it, than it, than it looks. But in principle,  it's possible and you can do it in some cases. We can associate a motive to any graph homology  class. So that explains perhaps why we get Grottendiktachemala, Mettific Galois groups  appearing in graph complexes. And this third point here, I'm a bit embarrassed by, but  it is actually true that the cosmic Galois group acts on the differential forms on outer  space. And it sounds like a big joke that I've concocted this. I promise you I didn't  come up with the phrase cosmic Galois group. It was due to Cartier as we learned in the  last talk. Outer space, it was coined by, by Kahn, Vokhtman. It just happens that these  things are in fact very closely related in a meaningful way by, by this machinery. So  here's an example of the Stokes theorem. So let's go back very quickly to, to this picture  of graph homology. Remember there was this, this class here, which is the first non trivial  sort of Liebracket across G3,5 and weight 8. And using it, we're going to calculate an  integral on this class, this green class way up here. And the way we do that is we, we  take this, this graph here was a linear combination of graphs, and we integrate zero over that  graph. We apply the Stokes theorem to, to that. And what it does is it will produce,  because its co-product is a wheel 3 and a wheel 5, which gave z to 3 and z to 5. We  then do this, this differential D and delta, these different, these two differentials in  the graph complex. And we zigzag our way up to this class. And using Stokes formula,  we deduce that the integral of omega 5 wedge omega 9 on, on this class here is in fact  z to 3 times z to 5. And that proves that this class, this green class is non vanishing  in homology. So, sorry, that was a bit quick. But here it is. So, this class in H3 here,  let's call it xi, you can show that this integral of a, of omega 5 wedge omega 9 over this class  is a non trivial product of z to values. Okay, so now we, now let's look again at this picture  of graph cohomology. So graph homology, I've actually switched to graph cohomology because  I want to dualize because we're talking about forms. So it's, it's, it's easier to relate  to graph cohomology, but it's the same. It's really the same thing. So now I've redrawn  the conjectural, semi conjectural, or anyway, in any case, the computer calculations of  non classes in graph cohomology, which is just the dual of graph homology. And I've, I've  given them names and the names are these differential forms and lead brackets of these differential  forms. So what we have omega 5 corresponds to the wheel with three spokes, omega 9 the  wheel with five spokes, I'm the 13 new with seven spokes, etc. Then we have wedge products.  And the first wedge product is this class in H3. We have a triple wedge product up here  in H6, and so on and so forth. And then we have lead brackets of the original forms.  And we also have lead brackets of the wedge products of these forms. And these classes  line up exactly with all the known graph homology classes. So here's a conjecture. I conjectured  that the free Lie algebra on this canonical algebra of differential forms injects into  the graph cohomology. It does it in a funny way. So the grading here is the number of  edges is the degree and it can correspond to the number of edges. We have to forget,  I don't know they line up in funny, common, common logical degrees. I don't know how to  predict those obvious question one is, is this map an isomorphism? I, I because I can  later deny it. I will conjecture here during this talk that it is an isomorphism, but I  may, I may deny that later on. I reserve the right. But anyway, it's a conjecture, but  it's a weaker conjecture, a more tentative conjecture than the previous one. And I was  going to say something about the, the dual since I'm out of time, I'll stop and, and  leave you with this, this picture so you can, you can, you can contemplate it if there are  any questions. Thank you very much for listening.  Thank you for answers. This was amazing. Thank you. I have a million questions, but if anybody  else wants to go first,  I'll go first. So the G three five linear combination, which you said was hideous. Do  you have a handy, I'd love to know just.  Oh, it's known. I mean, I think it's, it's written down in some papers. That that one's  okay, you can compute that one. But I think, I think the limits of the computer calculations  are this column. I'm not 100% sure. And I'm out of date. As I said, I got this from, from  slides of talks a year ago. I don't know if things have moved on, but I think the 10 column  and everything to the left of it is checked by computer and this column 11 is possibly  not. So, so you'll get your three five graph, but you'll be in a mess when you, if you want  to go much further than that, you, you're going to have to give up.  I think if you just want to compute the graph, you can take G three G five and just compute  the graphically bracket. I mean, the non-true question is that this is a non zero class.  Yeah, but also it's, so there's difference when you work in graph homology or graph  homology. So if you work with graph homology, the classes are not wheels. They're wheels  plus a long tail that's not known. And you've got to take the lead brackets of those. But  you don't have a question of David. Yes. I didn't mind you throwing away half of the  wheels. I thought that was quite fun. You got back the zeta values decreased. But as  it's the week of the expert take, do you have any thoughts about the zigzag diagrams? Those  are after all the only infinite class of diagrams on quantum field theory, which have a closed  formula, which they can I conjectured and you and Oliver proved that that's a great  question. Actually, when I'm new to this subject, when I first learn about this at a conference  a year ago, I asked exactly the same question to the experts and they said they didn't know.  That's a great question. Thank you. I wanted to ask a question about the weights. I mean,  you mentioned these weight drops and you have these. I found it quite striking that you  had these pure weight evaluations of these wheels with these complicated into grants  and you just got a zeta seven or just got a zeta five. So is there like a understanding  from the structure of the blow up that you get some bounds on the weights that would  predict this? It's not obvious from an algebraic geometry perspective. Not that I know  of, but I kind of have the feeling that all these these integral come from this Burrell  regulator. So you know that morally, beta five wedge beta nine corresponds to zeta  three zeta five. And so somehow what I would like to be able to do, but I can't do it,  is relate this complete graph using Stokes formula and identities to wheel three tensor  wheel five. And that would give that would give it immediately. But I don't know how  to do that. So I did that argument in the opposite direction to compute the one of the  examples. Thank you very much. Let's thank friends again. I'll see you in other questions  at the moment. Thank you.
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