OK, so thank you very much for the invitation
to talk at this conference.
Actually, I may use to talk to the people at DESI
as I'm essentially a physicist who
wants to understand Feynman integrals.
But what I want to explain in this talk
is some approach that we are following
with Spencer Block and Matt Gare on how
to try to understand the nature of these Feynman integrals.
In a sense, deriving the differential equation
they satisfy, evaluating them, understanding the function
to which they evaluate, using various tools,
including motifs.
And what we discovered actually in that paper
is that actually there's some nice application of mirror
symmetry that we can find in a class of Feynman integral
that I will describe in that talk.
So actually, in this talk I will first
start slowly from the motivation of why
I'm studying that problem, where I come from as a physicist.
And I will start building slowly toward more mathematical and more
abstract notions.
So essentially, what we want to do in physics and quantum theory,
so we want to compute scattering amplitude.
So the scattering amplitude are essentially
the object you want to compute if you
want to understand, for example, the physics of the eggs.
You can use scattering amplitude to compute values,
classical and quantum gravity effect.
So it's an essential tool that physicists
use in order to connect our model to reality.
And in this talk, I will essentially
be talking about quantum field theory,
but there is as well scattering amplitude in strength theory.
And so in general, what we use in physics these days
is that this is the amplitude you want to compute.
And we essentially understood that it's
pretty generic to try to expand any scattering
amplitude on what I call a basis of integrals
with some coefficient.
And depending on whether you do QED, QCD gravity,
if you at a given order in perturbation
know your basis of master integrals,
the difference between the various theories
come into the coefficient that are here.
So in a sense, essentially, the problem that today,
the way we approach one approach to quantum field theory
and the computation of scattering amplitude
is to understand at a given loop order, so one loop, two loop,
three loops, what are what we call the master integrals,
what kind of object we have here.
And then you just then understand the coefficient
on that basis of integrals depending
on the physical phenomena you want to study.
So my talk will be concerned about understanding
these integrals and actually trying
to do as much as possible without computing them
and using mathematical notions to tell you
what are these master integrals.
So Feynman integrals generally are objects like that.
So an L loop Feynman integrals in d-dimension,
so you integrate over a d-dimensional vector.
You have L of them.
And typically, you have, here, propagators.
These propagators are quadratic forms.
And they can be raised.
So the standard Feynman rules generally
gives you this propagator raised to the power 1.
But when you want to look at the master integrals,
you are then naturally to consider
higher powers of propagators.
So here, you can have new i that is an integer, positive integer.
So this form is not very practical for what I'm going to say.
So I'm switching to this so-called parametric form,
where you introduce as many parameters as that are conjugated
to your propagators.
And you get an object.
This is a total equivalence that tells you
that the same integral is given by a ratio of,
so there's a polynomial that u and v are
semantic polynomials that are homogeneous in your n variables.
n is the number of propagators.
u is of homogeneous of degree L. L is the loop order.
v is homogeneous of degree L plus 1.
And you see here, u is multiplied by something of degree 1.
So homogeneity works here.
And the sum of the indices is what I call new here.
The loop order is here.
The dimension of spacetime is here.
So actually, this formula looks like a very,
and this is here, you have a rational function
of your physical parameters.
So here are the masses that you have inside the loop.
And you can have external momenta and v,
I'm going to explain to you.
And the prescription is that you integrate
over the cycle that is all positive quadrant.
So in a sense, I mean, you know,
generally Feynman integrals diverge.
They have infrared or UV divergences.
But no, you know.
So one way to treat the divergences
is to analytic continue in d.
So you see d, the loop, are no parameters.
You can do some nice expansion.
What's this like?
The micron.
Well, that is going to be tricky.
No, can I do it?
It works?
OK.
OK, so what are these u and v polynomials?
So in a sense, I mean, there are many ways to describe them.
But one way I like is to think that the u polynomial is so,
suppose you have a two-loop graph.
It's a vacuum graph with no legs attached.
So you have naturally a nomology.
You have two cycles.
So as a string theory, the natural thing
I want to write is a period matrix.
And for all these graphs, there are two topologies
here at free loop.
So you can naturally write something
that is the integral over the cycles of vector
that you define on your graph.
So then you define a matrix, omega.
So if you do it this way, you will
get for these two-loop an object that is symmetric, like that.
For this one, you get something that has some 0 here.
And this one, the Mercedes, has no 0 anywhere.
And the semantic polynomial is exactly the determinant of that.
So for the people who prefer a string theory language,
it's really what you get when you degenerate the genus 2,
genus 3 human surface.
You have to tell exactly how you pinch the various cycles.
And you follow what Faye is telling you in his book.
The other piece of my parametric representation is v over u.
v over u actually is exactly the thing that
knows about the external state.
So here, I had the external momenta.
Oh, this is p.
This is k.
Sorry.
But so this is an object that is quadratic in the scalar
product of the p.
So this is pk.
You can have the self-product as well, if you want.
And here, you have a green function
that connects two points on here.
I just wrote it on the circle.
So here, you have just the natural green function.
Again, if you use the natural green function,
you have on a genus 1 Riemann surface between two punctures.
This is what you get by the degeneration limit.
And I know you can do it if you want.
If you have a state here, a state here, a two loop,
you get as well the green function this way.
So these objects are very, very natural objects
that you build out of the topology of the graph.
So essentially, what I want to tell you
is that by this graph polynomial that I call phi gamma, that
is the product of the semantic polynomial.
So this has to do with the vacuum graph.
The thing that knows about the internal masses,
so the mi would be the masses between here on the values
inside on the internal legs.
And v knows about the external state or the punctures
if you want.
If I know this, I can, of course,
completely recover u and v. You just
have to look for the term linear in the masses.
I can get what is u, and then I can guess what is v.
So from an homogeneous polynomial of certain degree,
from the number of variable, I know
the number of propagators.
From the degree, I know the loop order,
because the loop order is the degree minus 1.
And from the other characteristic of the graph,
I know the number of vertices is 1 plus n minus l.
So if you give me an homogeneous polynomial of certain degree,
I can tell you if it can correspond to a Feynman graph.
So suppose I take a most general quadratic in P2.
So here I have something that is a free variable, x1, x2, x3.
But it's homogeneous.
It's in P2, because normally, I mean, as I, yeah,
this has to do with the fact that normally one variable,
I can set to 1.
But I decide to write it this way.
It is of degree 2.
So I want to see what are the conditions on the coefficient
so that it corresponds to a Feynman graph.
So if you do the little exercise,
you realize that there is only one solution that
is a triangle graph.
So it's only for the precise identification
of the coefficient here that are related
to what I call the internal masses that
will be the mass of the propagators here.
And here, the mass square of the propagators here,
that you have that this guy is associated
is a graph polynomial.
So in a sense, not all homogeneous polynomial
are associated to a graph polynomial,
but graph polynomial, homogeneous polynomial.
So if I now look at a cubic, then I do the same.
And I find this guy that is no, again,
the linear part still have free propagator, free masses.
I have exactly the two-loop determinant that I showed you.
And here, there is only one kinematic variable,
like k square or p square, that is multiplied by that.
And this is the only solution I can find here.
And this is what I call the sunset or the two-loop sunset
if I want to make sure about the two-loop order.
And in general, if I want to look at the homogeneous
polynomial of degree n in pn, I find that the only solution
are the so-called n-loop sunset graph, which is just given
by this homogeneous polynomial.
OK?
So in general, when I have over a couple of graphs,
double-box, for example, that are of two-loop double-box,
that are of very special interest for the x physics,
the polynomial looks different.
But here, what I want to do is that I
want to study that family.
I want to study, actually, the Feynman integrals that
are associated with that family of graph.
So what I want to do now is that I
want to study that integral, seeing no this way.
I have a top form, so I write it a more general way.
For the graph that I just mentioned,
all this sunset graph in two dimensions,
it turns that the sum of the new is equal to l plus 1.
So there is no numerator, and the power is 1,
and there is no power.
So you will see in a moment.
But anyway, in general, any Feynman graph,
what I'm going to say is very general.
So any Feynman integrals, in this representation,
you have a top form, and it is sitting in there.
So it is sitting in the cohomology,
where that is on the complement on the I-plus surface defined
by the graph polynomial.
OK?
And then I'm integrating over this positive quadrant.
Yeah.
So the problem is that you see this guy, well, I mean,
the problem is that my domain of integration
intersects the my graph polynomial.
And this guy has boundaries, actually given
by this normal crossing divisor.
So in a sense, I mean, it's not a proper period integral.
OK?
So I want to see that normally a period integral.
We're integrating a top form on a cycle.
So if you want, I mean, to make contact with what
was said in the previous talk, and I will mention that as well
for the case of the elliptic curve,
you can write this thing as a residue.
So you can write, if you want, you can push things.
I mean, essentially, the singularity
is at most logarithmic.
Well, it doesn't work.
But anyway, so essentially, what happens
is that you can write this thing as, if you prefer,
as a residue form evaluated on x gamma,
and then you integrate over a cycle in the homology.
So because of that, what I want in that language
is to look at something that is a relative homology setup,
where I want to avoid the place where my graph polynomial
intersects with my domain of integration.
So for the sunset, I was telling you,
and I will be more precise, explicit in a moment,
the graph polynomial defines an elliptic curve.
And elliptic curve goes exactly through just the corner
of the triangle of my domain of integration.
So I just blow it up by putting p1s.
And now I'm going to look at my integral as a period
integral for this homology, where I just
do the appropriate blow up.
So as I mentioned before, the Feynman integral
tend to have ultraviolet and UV divergences.
But in the expression that I give here,
I can work, if my integral has divergences, it's not finite.
If it has divergences in four dimensions, for example,
I can do an expansion.
I can take d to be 4 minus 2 epsilon
and do an epsilon expansion.
And then when you do that, of course, you get poles.
And then you get, of course, and the expansion doesn't stop.
So even when it is finite, it's interesting
to study what is the epsilon expansion in that context.
And the ci are still period integrals,
in the sense of the one that was studied in this work.
And in particular, they fit the definition
of conservations again.
So now the question for me, which
is to understand the master's integrals that
are needed for evaluating my amplitude, scattering
amplitude in quantum field theory,
is just a question of understanding what
are the periods of my motive.
So this is my motive.
And of course, quantum field theory
is not just at fixed value.
So I will vary the energy at the time of the incoming
and outgoing particle.
So in particular here, I want to vary the value of p square,
keeping the mass mi.
So we'll have a family of, oops, I'm sorry.
Yes.
So what's going on?
OK, yes, I see.
OK, very good.
OK, so I see.
So I will have a family of actually odd structure.
And so that means I will have to look
at variation of odd structure.
And so this is where, by understanding
the variation of the odd structure,
and in particular, looking at the Gauss-Manin connection,
I will be able to derive the Pickard-Fuchs equation.
And a generic fact about all the Feynman integrals
is that there is an inhomogeneous terms.
So in a sense, there is always an extension
that you have to deal with.
And the extension can be complicated or not,
depending on the setup.
And I will describe that.
So in a sense, to make contact with the previous talk about
why it was the Riemann-Hilbert problem was mentioned,
I made a little list.
So these items are what are the mathematical questions.
And this is what the corresponding quantum field
phase, so I put this little graph.
So compute the, I mean, the period.
So you have no setup.
I want to compute the period.
I want to know what are the period of my motives.
So essentially, this is what we want to do in quantum field
phase.
We want to evaluate them numerically efficiently.
We want to know what kind of function they are.
So we want to understand the local monodromy.
And actually, the problem with the local monodromy
is exactly the one that you have when you look at unitarity,
when you look at the discontinuities of your scattering
amplitude and particle production.
It's not just words.
I mean, it's really the same.
Then you want to consider the complex system
of differential operators.
So a very, very common method, a popular method
in quantum field phase, what's so-called integration
by part identity that is used in QCD,
which essentially is a way to build a Gauss-Mannin connection.
But I want to use a different method to build that,
because it's not very practical in a lot of cases,
but you can consider the PicoFux operator in a way
I will describe in a minute.
And then, what are the special function you need?
So in particular, this is where the Motivica approach
tells you whether you know what kind of obstruction
of being polylogged, multiple polylogged, elliptic polylogged,
or obstruction or not.
So let me look at the sunset family.
So what I'm going to do is I'm going to look at this family,
special family of graphs, where the masses are fixed
and piecewise vine.
So as I will tell you, I mean, actually,
that leads to an interesting family of Motiv
that is not mixed state and has a lot of non-trivial extension.
And then, we will discover mirror symmetry
at some point in it.
So if you want, I mean, if you write
it in this little factorized way, little phi,
what characterizes the graph is this homogeneous polynomial here.
And this guy has a reflexive Newton polynomial,
his polar part has only integer point, and actually, it's
So essentially, all the sunset family
is a one-parameter family of Calibrio hyper surface
in phano type.
So let me now look at the, let me
start with the first non-trivial case, which is the Janus 2 case.
The Janus 2 case, my cubic defines an elliptic curve.
So you want to compute this over this domain of integration
that is positive x.
So of course, I can write it, and this is the way, actually,
we write it in the papers with Spencer and Matt,
as a residue, you take the residue of that guy,
evaluate it on the elliptic curve, and you push it to h1.
But anyway, I prefer to write it this way in this talk.
So in general, when the value of the masses are different here,
this is an open elliptic curve.
When all the masses are the same,
it's actually a nice modular curve, x1, 6,
that is in the Bovell classification.
So already, you see something nice.
So what I want to do is that I want
to derive the Picker-Fux operator.
So instead of having a differential operator
to which I want to find a period,
I want to avoid, actually, a GZK.
So I know that I have a period that
is given to me by physics.
A natural period is the period where I integrate the same object,
but I change the domain of integration,
I integrate over the torus.
Why do I do that?
Because this is exactly what the maximal cut,
this is what unitarity tells you in quantum filtering.
So you cut all the lines, and you compute that integral
with delta functions.
That means you have replaced all your Feynman propagators
by delta functions.
And you know this guy, pi0, is going
to be a solution of your Picker-Fux operator with a 0.
It's going to be a pure period.
So in a sense, I know one period,
and I will construct the Picker-Fux operator.
If I cut only two lines, I get the other period,
the non-allomorphic one.
So quantum filtering tells you what
parts do you understand the period of your elliptical.
The nice thing is that when you do the expansion,
when pi square is infinite, so naturally,
you want to do an expansion when pi square is small.
This is what physics wants normally.
But actually, if you look at pi square go to infinity,
it tells that this period is completely analytic
and has the good taste for the banana family
to involve the appearing numbers.
So and the system has maximal unipotent monodromy
for this case one.
So essentially, we fit in the context of this paper by Yao.
And I will comment about how we can extend our work at the end.
But from that period, it's just trivial to just derive
the Picker-Fux operator.
I mean, you have either the recurrence on the coefficient,
and then you find the second order differential operator.
So now, what happens is that if now you do it for, I'm sorry,
if you do it for i or loops, you get
to sum with more integers here.
So you have the generalized Aperi.
And again, you build up the Picker-Fux operator
using standard method, just looking
at the differential equation associated
to a series expansion that you know very well.
So now, I have constructed my Picker-Fux operator.
And it acts on my integral.
So by generic statement, the action
of the Picker-Fux operator on my form is not 0.
It's exact.
But so that means I can integrate by part.
But because of the problem I mentioned before
that my domain of integration is not in the homology,
I get to boundary term.
And this is what gives me the inhomogeneous term.
The inhomogeneous term has to do with the extension
of the edge structure.
And instead of going into explicit construction
of this, I want to tell you, actually physically,
you can understand this for this class of graph.
This inhomogeneous term has two pieces, y and some piece
that is in your log in the masses here.
The y part is easy.
Because actually, when you think about the system,
it's a simple demodule.
And the y piece actually comes from the Gauss-Manning
connection, and it immediately gives you the U-cover coupling
for the elliptic curve, which is exactly,
you take the form, actually, the residue.
The actually, I should have right here.
I mean the residue form.
I differentiate it with respect to my parameter.
I wedge it, and I compute the integral over my elliptic curve.
And that gives this.
Another nice property of the system
is that actually, this U-cover coupling is just the Wronskian.
So it's just given to you from the Picker-Fux operator.
If you want, this is constructing the connection,
and y is part of your connection.
The log terms are more complicated, because they
have to do with the extension.
But still, if you think geometrically, in fact,
that, OK, so what you have, the geometry of your graph,
it's a filtering graph.
But nevertheless, it has an elliptic curve.
You have an elliptic curve, that means you have a torus,
and it has punctures.
So when you have a torus with punctures, what is very natural
is to look at the integral of your form between, I mean,
on divisors.
So essentially, all the guys that come in
are actually obtained by computing
this integral of the form between the puncture
that you have on your elliptic curve,
and the punctures are given explicitly
in terms of the mass and the p squared that
is given to you by physics.
And by construction, this object, actually, the new y
gives you the ci's, they are just rational function.
So now, we're almost done.
So in a sense, I mean, I have constructed
the differential operator just by essentially understanding
the geometry.
And I can do it, if you want, at your loop.
But now, once you have the differential operator,
you can try to solve it, right?
OK, well, what you do is that actually, as I told you,
I mean, my integral is going to be a period of my motives.
So in a sense, what we essentially,
what you have to prove, actually,
is that, actually, your sunset integral
is a regulator period associated with your motive.
And this is for all the families of bananas.
All the funnels, this is the case.
So essentially, what you have to do
is that you know that your integral divided
by your period of your elliptic curve
is essentially going to be your regulator.
And everything is fine.
The symbol is tame, and so you are very happy.
So one way to compute the regulator,
instead of going to abstract math,
I want to tell you the intuitive and physics way to do it,
is to have a representation of the coordinate on your cubic
in term of something that is very natural when
you come from string phase propagators, theta functions.
So these theta functions have the property of P1, Q1
are the various points that are on my elliptic curve.
And I'm just having a function that
are the right poles and zeros on my elliptic curve.
And the theta 1 function is this one, is the standard one.
Now, the regulator is just the integral
of the log of x over z dy.
So now, you see, if you integrate the log of theta x,
so I'm forgetting the piece.
OK, there is a constant piece that doesn't depend on x.
But you do the sales expansion of these guys.
So the sales expansion of these guys
gives you the Li1, the log, which
integrate over the measure that is dx over x.
And that gives you exactly Li2.
So this is the way you find that actually your sunset
integral and the regulator is given by this elliptic delog
evaluated at the various punctures on your elliptic curve.
And it takes exactly that form.
So I like that form because it's explicitly
convergent because I'm assuming over positive ends.
You can use the functional equation of the dialogue
to write it as a sum of all n, positive and negative.
And then you can get an expression
that is close to the one used by Brown and Levin.
But then you need a regulator.
You need to regulate it to explain
what happens when n is negative.
So this one is perfectly convergent.
It's the one that comes from the computation.
So now, let me go to the free loop banana,
so the free loop amplitude.
So the free loop amplitude, then I'm
going to have one more variable, and the degree
is going to increase by one.
So now, the graph polynomial is this.
So when all the masses are the same, I'm sorry.
So the statement here is when all the masses are the same.
When all the masses, in the general case,
that when t varies, this is a family of K-free surfaces.
But when all the masses are the same,
this is a very special K-free.
It is this K-free that is actually
is obtained by the same product of the elliptic curve
I described before.
And this was already studied in the past by Elena Verill.
And you can derive the Picker-Fux operator
by just looking at the period that I described to you.
So you just have the period.
Then you have the apiary number for the K-free problem.
And then, bingo, you get it.
So I write it for all the equal mass case.
Now, in the all equal mass case, the regulator,
so you have to look at the K-free of the K-free.
Sometimes, this is so complicated
that sometimes you want to make stupid jokes,
that it's essentially given by, so essentially,
you want to think that this three-loop guy is
one more integration on the top of the two-loop guy.
So I add an elliptic Li2, so one more integration
will give me an elliptic Li3.
And this is indeed what you find.
So you find this is elliptic free log.
And this is exactly on the nose the one that was considered
in these references.
So in a sense, I mean, this setup of this banana graph
is exactly a case where the Feynman
integral is a regulator.
So of course, this regulator has a special property
of being a multivalued function.
So in particular, all the subject
when you do a modular transform on Q, they are not invariant.
But when you change Q, you change the value
of the energy, p squared.
So that means you explore in the complex energy plane
how the integral behaves.
And then you will have to look at the branch cut
and the monodromies.
So it's very natural.
In a sense, I mean, we describe that in the paper.
And we show that how the modular transformation is exactly
compatible with what you expect from an eternity
and the monodromies of the function you want.
OK, so now, in a sense, I mean, here,
what I have in that description is something that is essentially
using, OK, so let me just go back.
So the Q here, the Q here that I have
has to do with the complex structure of the torus
because I have explained, I'm sorry, here.
So this is how I defined my elliptical.
It's C star divided by Qz, right?
So Q is exponential 2i pi tau.
And the tau is the period ratio.
And this is an expansion that is nice and good
when you want to study your integral when, actually,
the value of the momentum, k squared, is small.
So now I'm going to do another expansion.
I'm going to look at the expansion when k square or p
square is infinite.
So I told you that, actually, this is a good place
because normally, this is where my period that I call pi 0
were at this very nice analytic structure
and my system at maximal unipotent monodromy.
And I could construct on my period using Frobenius method.
So if you do that and you realize that, actually,
the expansion near p square equal infinity
is better organized this way.
So what you have is that you have here parameters,
the mass parameters, come here.
And you have the exponential of r0 to the power l.
l is the sum of three integers that are defined here.
And r0, so if you want, the derivative of r0
is this analytic period I described.
Or another way to characterize what is r0
is to say this is the logarithmic Miller measure.
So this is the same domain of integration as pi 0.
And pi 0 was this integral when I have 1 over phi.
This one is just the log of phi.
So people have already studied that kind of object
in mathematics.
But I know the exponential of r0 comes actually
at your killer parameter.
So here, I have three masses.
So in a sense, if I put the mass with my r0,
I have naturally three killer parameters.
So what you get here is that you get something
that is your period times the log of your killer parameter,
so cube, plus this sum here.
And then you have numbers.
You compute them.
And you discover this.
You discover that this number are actually rational numbers.
And they exactly satisfy what you
want for being a genius 0 degree d number.
And you can introduce the virtual numbers here.
So yeah, I give a table.
And actually, the way to extract these numbers,
so there is a way to extract them just from,
I mean, just mathematically from the graph,
I mean, you don't have to think about something
very complicated to do that.
And if all the masses are the same,
then you can even write a nice form for the mirror mark
between your big Q, which is your killer parameter,
to your little Q that is your complex parameter, which
is this way.
And then, yes, I'm going to come to it.
And then you have this.
So first, I mean, that was discovered by accident.
But then, of course, we understood why.
So essentially, what's happening is that the mirror symmetry
of an elliptic curve is not, this is not a mirror symmetry.
These Gromov return environments are not
associated with the mirror symmetry of the elliptic curve.
So what you have to do is that you have to,
and this is true for all these banana families,
you have to construct a non-compact callebio.
So essentially, you construct a local freefall by this way,
elliptic vibration.
And then you use the fact that actually the graph polynomial
that you get from the sunset is the Del Pedro surface.
And the non-trivial statement that we proved
is that actually, so the first part of my talk
that has to be with the motif that was the B-modal language.
So we used the fact that actually the Feynman integral
is a regulator period in the motific cohomology.
Then we had to prove that why it is identified
to the local Gromov return pre-potential for the freefall.
And so we had to look at what is the cohomology on the M-modal
side, and in particular, I used some result
by Ratani on the quantum cohomology, which actually
is very difficult to understand, honestly, to me.
But anyway, you can do it.
But to give you a sense of what's happening is that,
so what happens is that you have your callebio freefall that
is on the top.
So as everybody knows from string saying,
when you have a callebio freefall,
there's a natural object, the ukawakoupling,
that is given by this.
But it's a firm derivative.
But when you go to this non-degenerated
and you go to this limit, essentially,
you just keep the ukawakoupling for your local non-compact
version of it.
And this guy gives on the nose the ukawakoupling
that I've constructed in a very elementary way
in the first part of my talk.
Now, the prepotential for callebio free,
and in particular for this elliptically-fibered callebio
free base on the alpezo has been, especially,
computed, for example, by Albrecht Klem.
So there is a generic form that is given like that.
This is the callebio class.
And here, you have all instant on some.
So then there is a map that the qi are exponential of ti.
And as Albrecht explained to us a few weeks ago that
came in time for that conference,
that actually worked out the identification
of the parameters between our parameters and its parameters.
And if you take the formula that it gives in this paper,
you just apply this mapping, you
can reproduce exactly what we have computed on the other side.
So that's pretty remarkable.
That's pretty remarkable because first, I mean,
that still works for the next one, which
is the free banana that is based on the K3.
Everything is fine.
So the guy after that, the four-loop sunset,
actually, this one is a bit more tricky.
I mean, normally, everything should work,
but this is more conjectural.
So this one, there is a sun here because everything
is under control.
And I'm sure it works, but still needs to be understood,
for example.
The interesting thing that happens for the four-loop one
is that this graph polynomial defines a Calabi-O freefold.
And then you can already see, even
when all the mass are the same, that you are not
going to get an elliptic four-log.
Essentially, what you get when all the mass are the same,
the model is a Bart-Nietro model.
And then you already see obstruction
to get elliptic polylogs.
And now, if you want to construct the differential operator
and look at the structure at infinity, well,
you have to pay attention to the smoothness of your construction.
And so normally, that was some kind
of a requirement in the work by Yau and his collaborator.
But essentially, you can forget about it, and it still works.
The very difficult part, when you go to higher loops,
that mean higher degrees of this polynomial I described,
is to really extend the construction of the motivation
class and the regulator period.
In a sense, the machine seems to work,
but then you have to check many things, in particular, the symbol.
And it's complicated to have a proof that will be valid to all orders in n.
Although, by physics, using unitarity,
I don't see any reason why the story should stop.
So in a sense, what I'm saying is that, by looking at this graph,
I think that there is maybe a way to go over some of the difficulties that
were actually expressed in that paper for going to higher degrees anyway.
So essentially, this is the thing I wanted to tell you,
is that as a physicist, I think it gives a very efficient way
of deriving the differential equation that people are looking for.
It is a completely alternative way of constructing the differential operator
compared to the integration by part identity method.
The integration by part identity method looks a bit more systematic,
but it has the same problem.
It has a problem that, OK, you can get a system differential operator
that you have to check whether it is reducible or not.
So it has some of the defect of the JZK model, the JZK approach.
The Dwork-Griffitz way of constructing the differential operator
is what we used so far.
But I mean, at some point, whoever is systematic,
it still needs to be, depending on the complexity of your Feynman integral,
that may not be the optimal way.
So the other thing is that what Broderst has noticed
is that for special values of the momentum p square and the masses,
this Feynman integral evaluates to value of L functions.
And what we showed, actually, with Spencer and Matt,
is that actually when all the mass are the same and p square equal 1,
the graph polynomial is very symmetric.
The period is not anymore a mixed period.
It is a pure period.
And essentially proving the relations that Broderst has discovered
to the order to which we could do
was essentially proving the linear conjectures
that relate period to value of L functions.
So at two-loop and three-loop, the elliptic curve and the K3 case,
that's kind of easy because this is what we did.
So at higher loop orders, I mean, this is as hard as that.
But this is of great interest for understanding, as well,
all these values conjecture that Broderst has been put forward
in relation between quantum field theory and number theory.
And there is, as well, some nice connection.
And I hope that sometime I will have to say something interesting
about the relation to the work of the gamma class by Goloshev and Zagye.
But you see, I think that by physics intuition,
you can then approach very, very abstract maths
and get a very concrete way of trying to get around
some of the obstruction that people will think you could get.
And this is all I wanted to say.
Thank you.