 I apologize this is work that was done really a year ago but the dynamics of the virus were such that those of us who didn't have a natural place to go found ourselves working with whomever we could and I was working with some mathematicians and so I didn't think about this what's involved here is goes back to a paper of dirt can you see my little arrow when I move it on the yes yes yes so I'm pointing to a paper here of primer which is an old paper physicists never look at old papers but mathematicians do this one goes back to 1991 and basically it talks about the amplitude for two loop two loop graphs and from this this this paper had really a lot of influence in my in my thinking but not necessarily in the way that I think dirt would have would have wanted it to basically the takeaways the first takeaway which I think is is worth keeping in mind when you talk to a mathematician is you want to convey what's interesting to physics and the takeaway which was conveyed by that paper was that two loops are already an interesting physical problem that is the amplitude for two loop graphs it's an interesting physical problem and and so I added that the fact that the algebraic geometers love cubic hyper cubic hyper-surfaces it's a little bit like the story of Goldilocks and the three bears I mean degree two hyper-surfaces are too cold and degree four hyper-surfaces are for many purposes too hot but degree three hyper-surfaces are just right and and when a mathematician an algebraic geometer feels they can do something with a cubic hyper-surfaces so even though the story is complicated it's it's it's approachable the second take away which is a natural thing for an algebraic geometer to do we're dealing with the second semester we're dealing with the the amplitude as a as a function of external parameters like masses and momenta and the physicist immediately wants to specialize to the situations of interest and being physicists they know the situations of interest but mathematicians have no idea which configurations of masses and momenta are a physical interest and so the natural thing for a mathematician to do particularly an algebraic geometer is to take generic parameters which has actually a technical meaning but it is kind of obvious and when you take generic parameters for example I will be talking about the the the double box so let's see if I can find the double box here yeah there it is up in the upper left corner there my double box is going to have external momenta in the middle which I don't think a physicist would tend to do because physicists will tend to want to talk about trivalent vertices but that's okay having put my momenta there and worked out the answer then at the end I can hope to specialize to where those external momenta are zero and to see whether the general picture which you'll see is is quite quite simple and nice uh specializes as well so anyway that's the point of view that I will adapt um there let me see there there's one other thing that I don't seem to have written on this slide did I write it on the slide yet um let's let's go to the next page maybe remember the next page here um yeah so and again an algebraic geometer confronted with this kind of problem tends to think of it in two parts the amplitude is a relative period that is the chain of integration is not topologically closed it bounds on the on the where the coordinate hyperplanes are zero um and that means that um you you're it's a it's a that adds a complexity to the situation which is not inherent in the in the pure algebraic geometry of the second semantic hypersurface uh so for an algebraic geometer it's kind of natural to to break down the problem of understanding two loop graphs into two parts first of all you try to understand the the algebraic geometry of just the hypersurface itself uh and understanding the algebraic geometry uh means uh first of all resolving the singularities of the second semantic hypersurface and then uh computing the motive which is the the middle dimensional comology of that non-singular variety um and I have to admit that early in my in my career that I sort of saw that as the whole story but in fact of course it's not because having done that you next faced the problem of understanding the the the amplitude itself which is not a period in the in the pure sense it's a relative period um and there are two ways to think about these relative periods uh and this is something that's sort of grown on me only over the years for one thing uh the work of Matt Kerr in recent years uh for the three or four banana graphs relates this relative period um to uh a what's called a motivic comology and motivic comology is is a mathematical uh theory uh that sort of generalizes it yields uh what are called normal functions which are um periods but they are relative periods I mean they're sort of is better they're functions which take values which are relative periods uh sorry I think I've lost the thread here so um these the there's a real gain I mean the the thing is that the integral the Feynman integral uh is very complicated I'm thinking now of the parameterized version it's very complicated because the chain of integration first of all is not topologically closed and second of all it um it it can meet the polar focus uh and and and the whole thing is really something of a mess uh so um the hope would be to find some to reach out beyond the integral the integrate itself and find some method to produce uh the the the amplitude uh from some other from some other ideas and that's what Kerr has done in the context of three or four banana graphs which are simpler I mean in many ways simpler and another consequence of of of this idea is that the motivic homology is related to balancing conjectures and these are this these conjectures give nice arithmetic interpretations of special values of of these these functions and so there's the possibility of a nice picture now when you try to do this uh I can't do it let me let me be blunt I can't go beyond where where Kerr went but there's hope and the hope is again goes back to something Dirk explained to me over the years if you think about the banana graphs for example you uh if you when you think about the banana graphs um you uh think of momentum flow and the second semantic of a banana graph is just the product of the of the various coordinates and um you you imagine topologically if you think of the graph you imagine cutting those edges and um so lost the thread here so you you want to cut those edges and the fact that that picture is so simple is what enables Kerr to to make this relation between the final amplitude and the the balance and type normal function construction but now if you take a more general graph you have momentum flow but momentum flow is kind of flowing in all kinds of different ways between you choose any two uh vertices and you look at how the momentum flows between those vertices or more generally you choose uh as you do when you construct the second semantic uh polynomial you you look at all ways of cutting the graph into two simply connected pieces and how momentum can flow between the the two simply connected pieces um and so the problem becomes one of generalizing the motivic homology construction which if i have time i'll say a few words about uh to situation where you have not just one cut but but sort of a whole chaotic range of cuts so anyway that's the second uh what i want to spend most of the the arm because i i have actually some results on is the first problem which is just understanding the the motive the pure motive that you get by resolving the second semantic and and then see see what we can say about about that pure motive okay so i'm going to focus on the double box which i've drawn over here um simply because that's where i have the best results i have results also for um for the kite which but the the kite is sort of simpler you just contract the two the two edges here and here um so let me just focus on the on the double box so uh again as i say i take the external momenta and masses all to be generic and if you do that um the the uh the hyperservice is defined by cubic in the seven edges one two three four five six seven the seven edges so it's a cubic in seven variables and if you work it out and this is the basic thing is it's um it has this this shape i have q capital q and capital q and capital q prime are our quadrics there are degree two in in three variables here q is an x five x six and x seven and i multiply it by the linear by the just the sum of x one x two x four and then q prime is another quadric this time in x one x two x three and i multiply it by by the others and then i add a sort of catchall term which is divisible by x four so x four is the guy in the middle here okay so he's divisible by x four and uh you can write it this way as as a sort of uh you take a matrix where i've taken generic uh it's a four by four uh four by four matrix with generic entries except that the lower left hand corner here is zero and then i just multiply by the row and column vectors as as indicated and and then i multiply the whole mess uh by again by x four so uh that means that f will have no uh term in x four cubed we'll have some terms in x four squared given by x four and this will have linear terms in x four so there will be terms in x four squared but not an x four okay so what uh what can we say well here we're going to use the fact that everything is generic we've taken all our parameters to be generic and in that case you you easily check that the singularities of your x are a disjoint union of two uh quadric uh two chronic uh curves actually so c and c prime are um are curves in in p two it's a little complicated p two for example is the the p two with parameters uh with coordinates x five x six six seven and then you you set the x one x two x three and x four all to zero so the singular it's a little confusing here because here you're taking a a sum but here you literally take the the total is equal to zero and you work out that the singular locus of this expression is just this disjoint unit okay so what happens then if i blow up these two conic curves i blow them up in the ambient projective space not in in the singular x but i blow them up in the ambient uh projective six space so this is all happening x is a five fold i should have said that in projective six space so i blow these up in projective six space and then the resulting blow up i call capital v uh so it's birational with p six uh and they're two uh disjoint uh the blow up uh are two disjoint uh smokestacks that stick up out of p six and then i take uh and this is the standard game for resolution singularities i take the strict transform of my y so y sits in p six here i take the strict transform uh sorry not y it's x it sits in p six i take its strict transform here and then i get then i take its strict transform and i get y uh inside v and then one easily checks again using this generic condition that this y is a resolution of singularities okay so um so here is then the picture of this is the this is the picture and so y is the resolution so we want to understand the motive associated to h five of this smooth variety y okay and um what we want to show it's not very well written here but i hope you can read it is that h five of y just the the cosmology of y is identified with the cosmology of a certain elliptic curve h one of an elliptic curve uh with a minus two twist this has it just says weight five so the h one of the elliptic curve has weight one so i have to goose that up to get to weight five so i twist by minus two which has the effect of adding four to the to the weight so this at least from the point of view of weights this kind of identity makes sense and so how can i show this well let me first of all say that this elliptic curve is kind of mysterious i've talked a lot with pierran hove about trying to figure out and i i think he can actually do it but it was certainly six months ago since we've talked and he was kind of displaced by the virus as well so uh we should we should check again to i think he knows how to compute the elliptic curve in terms of the coordinates of y but uh i'm not 100 sure but in any case it it's an elliptic curve and if we just say okay we want to see that it's an elliptic curve the basic thing that we have to do is we have simply to look at the hodge structure the hodge filtration on this h five of y and what we have to show well i mean a little bit more but the crucial thing is that the f three the third hodge filtration level is one dimension because what will then happen is is it that f two will then be symmetric so it will be two-dimensional and um so that will then be the whole story they'll um yeah so this is this is the crucial thing that has to be has to be shown okay so how how does one do this and one one doesn't want to do anything special because one wants a machine that ultimately you can hope to apply to other two loop graphs um and i will say if i have time at the end how to generalize this this this basic setup and and the basic construction of the of the y um um so the first thing you remark is that y sits in v right that's how we we got it uh and uh by long exact sequence of corpology you can identify uh oh and also the fact that v is very simple v has gotten just by blowing up uh uh conic curves uh in projective space so so the the corpology of v there's no um it's all hodge and so there's no no f four which then tells you that f four of the complement of y is f three of y which is the we want to show this as one dimensional so we need to calculate this and now there's a classical and very powerful theory uh due to deline and and sim sim cars sim cars the car uh there's another author i'm sorry forgot his name uh something called the pole filtration which deals exactly with the problem of calculating the hodge structure on a complement of a hypersurface in a in a variety um and what it tells you in this in this is that you can calculate this thing as the sort of hypercomology of a certain complex uh where i look at four forms on on v now uh with single order poles and then when you differentiate such a thing you get a five form on the but the pole order increases and again like this and so you get a complex here and the hypercomology in degree two of this complex is the relevant thing that's identified with this and this is identified with the thing that should be okay so now we come to an interesting problem in in how to present mathematics um the actual proof of this is quite complicated uh so i i kind of despair of trying to explain it by by zoom i do have a a letter that i wrote to uh matt kerr and pierre van hove uh with the full details which i am willing to share but not promiscuously so to speak so um if anybody really seriously wants to work on this problem i would be delighted and i'm happy to communicate this this letter to them but for the for the the masses so to speak i think it's best that we uh we just uh i'll explain some of the ideas but i won't be able to give the full details it's a it's a complicated linear algebra story okay but anyway uh let me stress again this pole order theorem that i makes this identification so now what we have to do is we have to identify this this uh thing which we can do sort of piece by piece and as i say the full story is complicated but at least we can look at the the last piece here so that that is what i propose to do and there's no reason not to do this more more more generally so instead of a double box let's consider a double n god okay now um how am i doing time wise can somebody tell me uh when i'm supposed to stop uh i think um at least until 20 pass or 15 20 pass i think that's certainly fine okay so 17 pass 17 and a half very good okay so um let's consider a more general double n god um then uh such an x is in projective two n minus two space and the the conics that we had before become general quadrics in n minus one variables the story is is is very similar to what i explained already you do the blow up again you blow up the two uh the two quadrics where you set the other variables to zero uh this gives two disjoint uh quadrics and we get the same kind of story and um we define the omega two n minus two that's the top dimensional comology but with a tilde and the tilde of a sort of it obeys block's law which says that always the most innocuous looking bit of any symbol is the most important uh so the tilde means that these are sections uh so with poles two n minus two forms with with some poles along x but they have the property that when you pull them back to v uh they don't get any poles along e so if i have here a form on projective space with poles on x and i pull it back to v a priori it can get some poles on on on on these e and e prime but some of the some of the some of the sections don't and these will be the tilde so that's a tilde and uh these are the ones that compute uh the uh the relevant piece of the end piece here and uh the proposition is in our case that is in the case of the double n god this fellow here is one dimension and that's good because that's exactly what we what we want because notice here we're interested in h2 so h2 this is a complex so h2 the relevant pieces are h2 of this h1 of this and h0 of this so the fact that that h0 is one dimensional that's cool that gives us exactly what we want what the rest of it sort of becomes a question of showing that these guys don't come in and mess up our nice our nice our nice one fellow that we we've got here so so that's that's as much as i'm going to say about the general argument um now there is one further remark um we we we'd like to look as i said the the game becomes to now understand all two graphs and by all two little graphs i i i sort of think of them as triples uh where i take the the two banana graph and i and i subdivide the three edges so the the pqr graph i have p edges up here uh q edges in the middle and r edges underneath okay um and empirically i can say with some confidence that the and the hard case uh the case where things are going to be different um is the case where pq and r are all at least two so that another way of saying that is that's the case of a non-planar case i mean in other words you see if if if i have this graph i cannot connect this vertex to infinity without without coming through the through the through the edge here so that's a sort of non-planar uh i mean this of course is planar but but but when i try to connect it to infinity i can't do that in planar right so um those empirically are this is not going to be true uh this proposition is is is false for the non-planar guys but notice here this is one and i i took the same number of edges for the for the two things here but i i'm i'm sure i guess for some reason i didn't do the calculation but i'm sure that it doesn't depend on the on the numbers of edges it simply means that one of the one of the three has to be left uh in one piece um okay so so the then more more generally um uh i want to describe uh in closing how to uh set up the the machinery um and let's take the hardest case first assume p q and r are all at least two then you choose quadrics uh which are labeled q sub p parenthesis and q sub q parenthesis and q sub r parenthesis in the indicated variables and with general coefficients and uh i write f p to be q sub p times the linear combination uh as indicated from p plus one to p plus q plus r and the same f q uh so all the omitted variables appear linearly in this in this extra factor here and f r and then i have a fourth uh term here which i call f p q r which is a generic linear combination of of these these fellows um and then f is um the the the sum of these of these uh these four terms so notice here i've assumed that all of these are at least two so this description does not work for for the double end we need another another description for that but this end is the is the uh the second semantic polynomial and so what has to be done is to go through the linear algebra using the the pole filtration as as i indicated and compute the the comology uh in the middle degree of this resolved uh hyperservious okay what i'm doing here yeah so uh there are two special cases which are the cases that you know i've already been talking about so this is when you have one uh edge in the middle r equals one then the the the thing uh has the shape that i already indicated for the double box again with generic quadrics now because p and q are can be larger um and then the extra term is uh divisible by x p plus q plus one x p plus q plus one is the is the the single edge that's the edge variable corresponding to the single edge and it divides all these all these fellows and then finally there where i actually have two edges that are left uncut and then the f has has the shape here um okay so um maybe i should say i have a few more minutes here so maybe i can say a few more words about the yeah so let me say a few words about the passing to the amplitude and and this is all completely uh fantasy well except that matt curr has made it work in the case of the banana graphs or at least the the two and three banana graphs um and so the game here involves uh as i said something called a motivic comology and motivic comology is um i mean i'm not going to go through the the details but here's an example um i have my hyper surface x and i remove the the loci where where one or more of the coordinate hyper planes meet x and so let me call that the resulting thing x star so on x star i have what's called a tame symbol that is one problem with this thing is it jumps i have the tame symbol that is these ti over tj's and i put always t0 in the denominator are well-defined units on x star because i've removed the the zeros so these are functions with zeroes of poles on x star and so it's such a it's such a tuple represents a class in the motivic comology of x star and the the indices are n notice the n plus one capital t's but but i i break symmetry by choosing one of them to be the denominator and so i have n actual functions here and so i have n tuple of of units and that defines for me a class in motivic comology uh h and with z of n twist um so that's a whole story in in itself um but we want something on all of x we don't want just something on x star and matt has developed a theory along with uh well i'm not sure i should give him the whole credit uh there's another name that escapes me who you worked with but in any case um they worked they developed a notion of tempered uh hypersurface and tempered means that this class which our priority is to find an x star actually lists to the whole motivic comology of x and that's an interesting business uh let me just say a few words because this is all in some mysterious way linked to the well-known properties of the second semantic polynomial with respect to contraction or or or cutting edges um and the fact that these classes extend is not a trivial fact and it's linked to the behavior of the polynomial the second semantic polynomial under uh edge contraction and so on um but once you've gotten it to to be a class in motivic comology then there there is a um numerical invariant associated to this class so to speak a cycle class which in this case sits in the comology of x with c mod z twisted by n uh coefficients so this is sort of a circle with a with a twist uh or it's not a circle it's c mod z it is c mod z it is what it is uh but it maps uh if we sort of throw out the compact piece of this thing to the comology uh with real coefficients with an n minus one twist and this is the this is the so-called cycle class and this is the class which cur relates to the Feynman amplitude the relation is not direct it's complicated and it's not as simple as one would hope but this is the in the banana graph this is the thing that cur uses so the second step in my program here which is to pass from the pure motive of the hypersurface to the mixed period which is the Feynman amplitude revolves around understanding in some much deeper way just as the second semantic is built up out of these cutting the various ways you can cut the graph into two pieces in this particular case in the banana case there's only one way you can cut the graph into two pieces and that's what makes this work but suppose that we have the full complicated situation with it in many ways then the challenge is to try to recreate the Feynman amplitude from some sort of analogous motivic comology okay so i i that's really all i want to say i say congratulations to Dirk for for living such a constructive life and being lucky enough to be a pro in where there are no virus cases unlike the rest of us who Chicago at this moment is absolutely horrible i tell my life not to go outside but anyway thanks for the opportunity to talk thank you very much thanks Spencer we had a question from Matt uh under the participants Karen can you unmute him because i can't seem to be able to do that yes and i will fix your situation too yes thank you uh so i was curious um for the double box and on you seem to be mostly describing the f polynomial unlike for the sunrise of course the u polynomial is also typically necessary even in integer dimensions to express these things is there a reason why you only consider the f polynomial um well so the f yeah so the well first of all in how many dimensions as you know the the the integrand uh the Feynman integrand is it depends on on what dimension space time uh but i take a dimension of space time where where u is so first of all let me say that u is very simple uh in in in this kind of case uh let me see if i can explain that statement just a sec here um yeah yeah so here if you take the double n gone so let's we want to compute the first semantic okay so what is the first semantic well we we're looking at uh pairs of edges first of all there are no coefficients it's just uh pairs of edges which cut this diagram which when i remove them uh disc i mean don't disconnect but but kill the the loops in this two loop picture okay so how can i kill the loops in this two loop picture well one way is i cut one edge from one of the loops and one edge from the other uh another way is i cut the guy in the middle here and then i just cut some other edge now if you think about it that's that's the end of the those are the only ways so the the first semantic then is just going to be a sum of uh those two kinds of quadratic terms one which is divisible by this edge variable and then divisible well in fact we can say what it is it's this edge variable times the sum of all the other edge variables that will capture all the monomials of degree two that enter in that part and then we have the other guy which is the product of the um the the sum of the edge variables in this loop but not this one and the sum of the edge variables in this loop so in other words i've written this quadratic as really a sum of two terms uh one i mean you see what i'm saying not so the algebraic geometry of the first semantic in this kind of situation is very very simple that's that's the takeaway i see thank you you had a question from david yes spencer i just wanted to remark that your um your box diagram with those extra lines in the middle is called a rail track diagram by people who work in gauge theories they are far from generic because all of their internal masses are equal to zero and they thought for a long while that will protect them from elliptic obstructions that uh they would evaluate in terms of multiple polylograms as a whole ethos built on that philosophy but in fact it's precisely that diagram where they encounter their first obstruction and it's recently been related uh one of the authors is matt von hippelge asked you to a single integral of a perfectly explicable trial algorithm over the square root of a quartic that's that so david you said that all the masses are zero is that the internal masses are zero internal masses yeah uh the four external masters at the corners are non-zero um they uh the lines in the middle of the railway track you can put equal to zero and in that situation the kinematics is enormous as simplifies because there are only a certain number of cross ratios that matt will i mean i'm old fashioned where i say mass i mean the mass terms as opposed to the momentum terms so there still are mass terms so to speak along the tracks no not the not the rarotides but no no that those are just gluons they're completely massless uh okay so we can't get gluons and mass just for last no no it's far from generic but what i'm saying is they have a perfectly explicit elliptic curve they can calculate its via stress invariance and it would be interesting to see in their non-generic this is for the box case or this is the double box yes fantastic uh so so uh so if this was a non-generic kind for today because we could meet in the hall and you could tell me more so your diagram is that i heard about at at least three major conferences as the first place that gauged theorists even n equals four super young mills in the planar limit meeting electric obstruction and you've just sat down by pure thought and told them that in advance yeah except that i can't tell you what the elliptic curve is i mean maybe maybe pier can i'm hopeful that pier can when i can well they can tell you in that simplified kinematic separation precisely what mass stress invariance of that elliptic curve or in terms of the uh i mean explain me toward a paper or yeah yeah i'll i'll i'll send you it's published okay okay if you get a chance that'd be beautiful yeah it seems to me like this is a perfect thing for a dinner conversation uh later on so what time is dinner can i well we we had planned it for uh for just in in eight minutes but because we're behind and i mean people will have to get the dinner from somewhere so you know it's paris time dinner so okay so okay and i would note that in the gather some of the tables have whiteboards at them so if you think you might want to talk math including leitech on the whiteboard you can pick a table that has a whiteboard i have a very nice story about maxine giving a whiteboard talk a week or so ago and it was a beautiful talk and the whole thing was lost at the end because the whiteboard turned white and nobody could recover the things so maxine said that's okay and he just wrote the talk out again wow i think nothing in gather gets lost i think everybody said that everybody said nothing nothing gets lost but nobody could recover anyway okay for now uh let's say thanks benzer again for his nice talk