Solvable Dyson-Schwinger Equations
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00:00
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Transcript: English(auto-generated)
00:16
You know, first, many thanks to Kevin and Eric for organizing
00:23
this event in three different versions. And I think it's a good compromise. It really works well. Then thanks for the invitation to speak here. It's a great honor for me. And as many other speakers, I will go back to history, to March 98.
00:43
I just arrived at CPT Marseille at the time as a postdoc. And then a larger group of us went to a conference on non-confrontist geometry in Wiederseldnauer in south Italy. And of course, there was a contribution by Anaconda.
01:02
And I remember reported on a really fascinating groundbreaking work by a physicist, Joe Cramer, who discovered that the realization of quantum field theory, which was a mystery for many mathematicians, is actually encoded in Hopf algebra.
01:21
So this was absolutely remarkable. And even more remarkably is that the same Hopf algebra in a slightly different paper appears in another work, what Alain was doing with Morimoto Scavici, on the local index formula for certain hyper-lytic operators
01:42
on space affiliations. So of course, we didn't understand immediately all the details. But it was clear to everyone that this is absolutely great development. And we have to understand it as well. So it had enormous impact to our group in Marseille.
02:02
So we had seminars on it. We stopped all other work and wanted to understand it. So OK, I worked with Thomas Kroski, our next speaker, on this. And we soon understood the generic structure.
02:22
But we had problems with overlapping divergences. And OK, I think it was Bruno Jukum who invited Jukum to Marseille for the end of May. And as preparation for discussion, Thomas and I made our notes on archive.
02:44
And OK, here that's more or less the first part of our paper. And I want to highlight one sentence, which is to recite already of discussion with Jukum. So here we describe how Jukum treats
03:04
overlapping divergences. And there is a story behind this little sentence. The story is more or less this. So I found on an old computer my email correspondence
03:20
with Jukum. So this Jukum's message from May 1998. And so OK, he explains what to do. And what I want to emphasize is here that, so he points out that to understand overlapping divergences,
03:40
it's assumed that the reader digested the use of Schwinger-Dyson infection. And he refers then to the amputation thesis where OK, details can be found. All right, so I stop history here at this point because I discontinued working on top of H1.
04:01
There are other speakers who can do this much better than I can. But so for me, these diastronic equations, they are of absolute central importance. Back in 1998, it was new for me. But in the meantime, they really become I would say my most important tool.
04:25
So then I would like to show this impressive page. So this impressive number of 11 papers authored by Jukum, which have diastronic equations in the title. So just in title.
04:41
So I'm sure there are many, many more where diastronic equations are the main, one main subject. But already in title, there are 11 papers. So three with Karen, but many, many more. So OK, therefore, I think contributing something on diastronic equations is something
05:02
well-suited for this nice version. OK, one can say really a lot about diastronic equations. We can have lectures here and so on. So I will be very short and give some general idea. And then we go to a specific instance.
05:24
So diastronic equations are some equations of motions for Green function in the 23 theory. One can understand them graphically as the collection of Feynman graph series of same external structure. So they form blobs, and the equations are of that type.
05:43
So for instance, one is interested in connected two-point function, and then one follows here the first line on the left, so what can happen? So either the line goes through, nothing happens. That's a free theory. Or it hits somewhere a vertex. So then you write the vertex, and then you ask,
06:00
what else can there be? So in this case, it will be a connected four-point function, which then would happen. So yeah, so OK, this picture is very easy
06:25
to understand when collecting Feynman graphs. But the point is somehow this assumes that the series of Feynman graphs converges, which is absolutely not clear. But there is another remark. So such an equation can be completely
06:43
rigorously derived without any reference to penbavian theory. So that's an exact equation, which can be solved by penbavian theory, but it's not needed. So therefore, in fact, these di-Sicher equations are non-perturbed equations.
07:01
The problem is it's always sometimes often hard to solve it. And the problem you see already here in the structure. So you want to know something on the two-point function. That's what you want to study. But this equation involves here the four-point function.
07:22
So that's a typical feature that somehow you need information which you don't have yet. So either one has to solve all of these equations simultaneously, which of course is difficult, or one has to work. And I'm working in particular on concrete series
07:44
on finite dimension approximations, not with geometries. Another word is made, Tricks model. So in this setting, it happens that these equations somehow are disentangled. So we get equations where this relation sign is reversed.
08:04
So we are interested in n-point function, and then we only need functions where the number of legs is smaller, not larger than n. So the price to pay is that this equation will become non-linear, and that's also difficult.
08:25
So OK, now I come to the subject. One feet series of a particular type on matrices. So all right. So a starting point is fundamental theorem
08:43
in harmonic analysis to Bogner, who proved that whenever we have an inner product on a real vector space, so finite dimension for simplicity, for instance, the space of separate joint n by n matrices. OK, so it's fine.
09:02
So OK, the theorem is whenever you have an inner product on such space, then there is a unique probability measure on the dual space whose Fourier transform is exp of minus 1 half the inner product, completely general.
09:21
So we take a very specific inner product on space of matrices. Namely, we give ourselves n cross different numbers, e1 to en. So this somehow stands for the energies on the spectrum of Laplace operator truncated to n levels.
09:41
So when we built a scalar product, well, this one. And kl and prime k divided by ek plus el. So that gives an inner product. And OK, we take this one. So as I said, it has something to do with Laplacians or number geometries.
10:01
And OK, here, that's a Gaussian measure, and everything is now, of course. OK, we will not stop here for the next step. And also, next step will explain why we are interested in this particular inner product. So OK, the reason for choosing this inner product
10:22
is the Consevich model. So the Consevich model is a deformation of the measure I just introduced, what free theory, by a cubic potential. So e to the minus lambda n. We've discovered that if you do computations
10:46
with this particular measure, so vacuum graphs, and you understand them as function here of the spectral values e1 to n, then this gives rise to the numbers which you get,
11:01
rational numbers. They are, in fact, intersection numbers of topological crevice classes on the moody space of complex graphs. So it's a very deep result. And so there are many, many developments connected to this result, among other sorts. Incubate model related to the KDP hierarchy.
11:21
So in fact, all these relations were already suggested by Whitten shortly before. And Consevich gave the proof by inventing this particular measure. But actually, it's, of course, not a real measure. As you know, phi cubed is unbounded below. So you better have lambda purely imaginary,
11:41
and then it's not a measure. But formally, it is. So OK, and it's connected to something which is very professional now, topological recursion, also talked about tomorrow, deal with it, and so on. OK, so and OK, everything is known.
12:00
So we are interested in the deformation by aquatic potential. So phi to the 4 instead of phi cubed. This is a very small step from point of view of concrete theory, a very natural step. Lambda phi to the 4 is much more natural than lambda phi cubed. But the mathematics is completely different.
12:21
So there is a clear logic by Consevich chose phi cubed, and namely phi cubed has something to do with simple zeros of Schreiber differential on Biemann surface. And you cannot replace it by phi 4, which would be quadratic
12:41
zeros, but these are exceptional. So the generic case is really phi cubed, and there is absolutely no relation, mathematically, between phi cubed and phi to the 4. Also, perturbation theory is completely different. It turns out, however, that that's a recent result of the previous year,
13:02
that nevertheless, we find structures which are completely similar, mathematical structures, as something to do with topological recursion. And well, this is the subject of the talk. But one remark, we get something for concrete theory.
13:24
So it's a toy model, all right? But it's something where we have nearly exact solution, exact in the coupling constant. So OK, that's the contents. OK, so let's start.
13:41
A starting point, well, is we really derive disjoint equations. So we consider the full transform of the measure, and that's a full measure. So depending on the spectral values, you want to do that again, and the coupling constant.
14:03
So that's the 40 differential. So then it's a complete elementary exercise to derive these equations of motions. For the full transform. So and these are important. Because if you remember, I try to go back to the picture.
14:24
So OK, these external x in this formulation preside by differentiating the full transform with respect to the source matrix. So if I have n derivatives, then I get n external x. So a derivative with respect to m brings the phi downstairs,
14:47
and then graphically, this is the endpoint function. All right, so the problem is that in this equation, we have too many external x. And these equation help us, because they
15:03
allow to express here a second derivative, which gives two additional x, second derivative of the full transform by something which has only one leg, or in this case, no leg at all. So OK, we can express these derivatives
15:20
by something which has less derivatives, or maybe the same number. But OK, that's the key point, which is not available in digital conflict theory. That's special for matrix models. And that's the key why we can go much further than for normal conflict theories. All right, so I will show some of these equations
15:46
then which you can decide. So the idea is you differentiate some more times, and then you get identities. And these identities are the Dyson string iterations. OK, so these Dyson string iterations, they have then,
16:02
they inherit these matrix indices. So these are matrix indices, of course. And then we have a finite number distinguished by a finite number of matrix indices of these equations. So there's a key step. Namely, it turns out that one can complexify the resulting
16:22
Dyson string equations so that they become equations between meromorphic functions in many complex variables. Of course, complexifying finite number of, they are given by a finite number of points, and complexifying them is not unique.
16:42
But there's a natural way to do it. So we stress that there is a very natural way to obtain Dyson string equations for meromorphic functions. And then we can use the whole power of complex analysis to tackle and to solve them. So in doing this, so after complexification,
17:03
we can relax a little bit the conditions that these e1 up to n were previously degenerate. So we can omit multiplicities. So our different eigenvalues, spectral values, are little e1 up to ed, and they come with a certain multiplicity.
17:23
It's not really important, but it gives us a lot of flexibility. So here is the simplest, or the first equation which one gets. So we consider kilosecond moment of the measure with a two-point function.
17:42
All right, there are certain factors of n. We give it a name, g, with two indices. OK, if b is different from q, no, that's OK. But if here is something else, then it's 0 at the measure. So that's the interesting object to study.
18:02
So we're interested in large n expansion. So this is a formal power series in inverse powers of n. Turns out, it goes in even steps. And the little number g, the parameter, is or will be related to the genus of a Riemann surface.
18:24
So that's it. But OK, here, it's just a name. But yes, it's formal expansion. So as I said, we complexify. So here, everything is discrete. We complexify it. So we arrange that our g, pq, is actually
18:42
the evaluation of a meromorphic function in two complex variables, sine eta, no, zeta, zeta eta, where zeta is ep. So no, this was fast. And eta is eq.
19:00
So these initial spectral values, if you evaluate our meromorphic function on them, we get back this, OK, the expansion coefficients. So the first equation is this one. So it has a long history of playing with Hart's Gauss many, many years ago.
19:22
So the important thing is that's an equation which only involves this two-point function, complex two-point function. So that's nice to have. But as I said, the prize is it's a nonlinear equation. So here is the nonlinearity, which
19:40
was really hard to understand and well to solve. So it's a very general setting. So if we go to the large n limit, then we could have divergence, so better have bare mass and wave function optimization there to have a lot of freedom.
20:03
Well, so OK. Well, for me, the breakthrough was in joint work with Eric, which we achieved during a school in Lisieux organized by Duke two years ago, where we
20:23
were able to solve this equation, learning equation in a special case, namely that the multiplicity is 1. And all these spectral values are equally spaced. So this is a particular choice, which corresponds exactly to the lambda for the 4-model
20:43
or two-dimensional model space. So there is particular interest studying this choice in large n limit. So it turned out that doing this, so all these sums become integrals and so on. And then, OK, you do perturbation theory,
21:03
not for this equation, but for some axial equation. And remarkably, using Eric's hyperin program, pushing far enough, we understood the first coefficients. So we did a pre-market pattern,
21:22
and we understood then that they are the first terms of a series, which is the series of the Lambert function, Lambert-W function. OK, so well, that's something. Solving such an equation to Lambert-W was unexpected,
21:44
and this opened the way. So it opened the door to tackle the general case. So the Lambert function has special properties, and after understanding them, which took some time, we understood the general case.
22:01
So take general case, general r case, finite or infinite, whatever, we can solve in any case this equation. So of particular interest is the Froude-Majim-Moyali space, where we get, instead of Lambert function, we get the inverse function of the hypergeometric function.
22:23
So this will be covered in the overnight talk by Alex Hock, 1550 in Paris time. So he will present more details on this approach and very nice consequences of this example.
22:41
So I continue with the discrete case because for finite d or finite n, there are extremely nice structures which appear, which has something to do with complex analysis and algebraic geometry.
23:01
All right, so it's also a long history, but I will just give the final theory. So I will describe the solution of this equation in complete generality in case of finite n, finite d. So you have to do something.
23:20
OK, remember, we are given the spectral velocity k, multiplicities, and coupling constant. So define new parameters, epsilon k, rho k, which somehow deform from this data in the following way. So build such a function of z, z minus lambda divided by n.
23:41
OK, let's find some. So here you should see in a certain regime, this will produce a logarithm. And that's z plus lambda log z is then, OK, the inverse function of this lambda function. So there's a logic behind this, which can be guessed from the lambda function.
24:04
All right, so take this problem, function r. And OK, here you have epsilon k, rho k. And they are implicitly defined to satisfy r of epsilon k is ek. And r prime epsilon k rho k is rk. So that's implicitly defined problem.
24:24
So then, so remember, this was the function we want to have, solve this equation, hopefully. So you go to the pre-image, so zeta eta given. And then you go to the pre-image via r, so z and w.
24:43
And you transform variables. And after transformation, the result is script g. So for this function script g, we have a formula. So here it is. So it's a, well, it's a rational function. That's important.
25:01
And it involves the definition of r. And so r of z can be replaced by zeta. That's clear. But the point is we also have the negative, r of minus w, r of minus z, where we cannot do it directly. So there's no direct relation. So we have to go via r.
25:20
So r is important. Then a notation, there's a little hat. So a hat means, OK, first go back here. It's, well, you see this is a rational function, which is the ratio of polynomial of degree d plus 1
25:40
by polynomial degree d. So there are d plus 1 pre-images of every value. And the other pre-images, except of the principal one, they carry a hat. So this epsilon k hat are then the pre-images of these deformed energies. OK, so that's the solution.
26:01
And so proving it when one knows what to do is very, very easy. So you can give it as an exercise in a complex analysis course. But it really took us 10 years to come from this equation to the solution. So OK, remark, symmetry, although it is not manifest,
26:21
it is automatic. It can be checked. And yeah, OK, that's remark. And that's a pointer. Good. More remarks. So again, stress, that's a solution of nonlinear Dieneschian equation. We found it first with Eric by, well, brute force, I would say,
26:44
perturbation theory. And a lot of luck understood what we were computing there. But then we proved it. So we got an idea. And then the proof is beautiful. It's beautiful complex analysis. So then, of course, you ask, how is this possible that you solve a nonlinear problem?
27:01
And it's more or less clear there must be something behind, some deep edge-back structure, which we don't have it. But we are confident to find it. And this involves the affine equation, which I will then talk next.
27:22
OK, one remark. It was very important to deform the data. So originally, we had ek, spas, multiplicity, lambda. And in these variables, you don't see anything. It's absolutely important to transform
27:40
by the inverse of such a relatively simple function. Then we see a structure. And OK, it's an empty statement. But nevertheless, maybe one should think about it. Maybe something similar could be true also in familiar confidence, that we have to transform in a way which we do not know
28:01
to other variables. And in these other variables, we see more structure. OK, but I have no idea how to do it in nice examples. OK, so now I come to the affine equations.
28:20
So well, by general theory analysis, it's clear they are solvable. But there doesn't seem to exist any theory which could give us this equation explicitly. So the key idea was due to Alex. So he suggested to look at auxiliary functions.
28:44
So namely, so here, these are our two-point functions. And so well, they somehow depend implicitly on the initial data, the energies, and the coupling constant. And so you differentiate these two-point functions
29:01
with respect to the energies. Of course, you can do this in practice because we don't control the measure. But OK, let's introduce them and call them omega with indices a1 up to n, which, OK, one a1 special and the others are derivatives. And then again, apply these genus expansions.
29:23
OK, so the key step is we don't need to know here the two-point function. We can derive this equation directly for these omega n's. Well, not directly. They are coupled to other functions which I will introduce. And then we can try to solve it. So what we need are the initial data, our rational
29:44
function r, and the planar two-point function, but nothing else. Everything else is contained in the initial equations. So for me, at least, complete and unexpected result was that these auxiliary functions unexpectedly
30:03
translate to something which has a clear mathematical meaning, namely they are object of block topological closure proposed by Burrow and Chaterin five years ago. So I will say a little bit on this. OK, here that's more or less the overview
30:23
what one has to do. So it is not that we are interested in these omega's. So from them, we later get, then, the moments here, these, which we're going to have. So OK, they were auxiliary functions, but we have to introduce more auxiliary functions, namely t's, two sorts of them.
30:42
And then one has to go through such a triangular pattern. So here, this first box is what we know, the two-point function, planar two-point function. And then, OK, for genus zero, it's enough to compute this chain in the upper line. But if you want to increase the genus,
31:01
then we also need here this strange other function. And then we won't go here. Then, OK, this is the way to go. So OK, for all these functions, we have downstream equations. And so evaluating the t's is relatively easy.
31:24
But the result is horribly complicated. And you insert this horrible result into the equation for the omega's. And then many, many cancellations arise. And finally, the outcome for the omega's is extremely easy. So I will only show the results, not the equation
31:42
and not the way to compute it. So the first result is quite marked points, omega 0, 2. So of interest is this expression here in parenthesis, especially the red one.
32:01
So this is, OK, up to differentials, it's the Birkin kernel of topological excursion. So having found this, we knew that we are on the right way. So and there are always these prefactors. So the idea is we multiply by the differential of R
32:21
and any of these variables. Also, lambda goes away. And then we come to differential forms, the one form in each variable. And so OK, well, that's so we know it's true in the planar case, otherwise it's contracted.
32:41
We are sure that these omega GNs have a very simple post structure. So they are more monomorphic. And we think we know where the ports are. So we are sure in planar case, but otherwise it's contracted. So they have only ports at 0 on the opposite diagonal.
33:04
And on special points, there are two of them, where the derivative of R vanishes. So these are the so-called ramification points of the ramified cover encoded in this rational function R.
33:21
Yeah, OK. So we'll mark these T functions. They are much more complicated. They have many, many more ports. OK, so here's the few solutions. OK, looks also complicated, but in fact, there is a key logic.
33:45
And in fact, these are relatively simple functions. I wanted to show, maybe show first this. OK, so these are known perturbative results.
34:03
So that's important. And in principle, you can amuse yourselves in finding which Feynman graphs, use the Feynman graphs, contribute to such an omega GM. Of course, there are many of them. And so the sum is convergent.
34:23
And so there are certain graphs of a certain external structure, certain topology, and well, all they sum to such an expression. And of course, the dependence on the coupling constant is a little bit hidden. So remember, we had the system of equations.
34:41
The epsilon case and the rho case, they are implicitly defined. And then we have the betas, the solution of r prime beta plus 0. And well, this system has to be solved somehow by the implicit function theory. So abstractly, it's completely clear. We have a holomorphic solution.
35:01
And also, as you know, we have good convergence results if one wants to find a solution numerically. But one can also approach them via a Taylor expansion to the implicit function theory. And then this Taylor approach will coincide
35:22
with the Feynman graph expansion of a certain topology and external structure. So our idea is that this is something which we would like to contribute to this sigma volume for Dirk's birthday. It's still nice to compare the Feynman graph series with the exact result, going in particular
35:44
through the simplification points. So a few remarks on color code. On the next two pages, I will show that the terms in blue have a very clear mathematical meaning. So there's something very, very known. So in particular, here, you see these rational numbers
36:03
appearing. At least some of them have a meaning as intersection numbers of characters, classes on the modal space of graphs. Not all of them, but many of them. So then here, this term in black
36:21
with ports on the opposite diagonal, we understand them at least for all planar functions, so genus zero, projected, but it's clear it's true. And so we are confident that very soon we will understand them completely. So my feeling is that they are just there for confusory,
36:43
but maybe they will not have a meaning in mathematics. But these terms here in red, ports at zero, because zero is special. So you remember, we had always this reflection that goes to minus z, so here, eta minus beta. This is very special for this particular model.
37:03
And zero is the fixed point of this reflection, and these numbers, one-eighth, one-sixteenths, and so in particular, these combinations of ratios of derivatives of r and zero. So my feeling is they have a certain meaning. So there's something which will extend
37:22
the result of conception to something, but I have no idea to what. That's for the future. So, okay, now what are the blue terms? So here, that's the general picture of the ramification points. Of course, I can only draw a real picture.
37:42
So if they are simple zeros, then the real behavior is just a z-square function. And then you have z, and the reflection is called sigma offset. So it means r of z is r of sigma, i of z, and at beta i, the two come together.
38:04
Good. So this reflection is the local Galois rule issue. Good. So, okay, now you do the following. Omega zero one is special. So you define it as minus r of minus z, r prime of z dz. So in all other omega gn, as I said before,
38:22
and then, well, surprisingly, but there will be some origin, we have so-called loop equations. So, okay, these omega gns, they have ports very high order at beta, but if we sum the contribution at z, and at, sorry,
38:41
and at sigma of z, the reflect value, then the ports cancel, and you get something, also constant term vanishes. So it's something which starts linearly about the ramification point. And then there's quadratic loop equation that, okay,
39:00
it's a very strange combination. So you take omega, okay, one genus lower, and so at z and sigma of z, and then you share it in product quadratic, and you share z and sigma of z in both variables, and otherwise you distribute the other variables,
39:20
and the genus. And then, so again, these are ports of high orders, but in this particular combination, all the ports cancel, also linear term cancels, and you get some things quadratic form in z, which starts quadratically about the point. So, okay, this is true,
39:41
and planar case up to five, and also four, one, one, and of course we can check chart, this can be an accident, that all these ports cancel, so this should be true in general. So then there's a general theorem of block double recursion, that whenever you have a family of differential forms,
40:00
neuromorphic, which satisfy these epsilon equations, then you know what is the polar part, so polar part of this omega GM with poles at the ramification points, and this part is given by such a formula, okay, it looks complicated, but anyway, but this is familiar, so that's the famous formula of topological recursion,
40:22
and so in the original version of topological recursion, was just this, so that's the formula how you build from, so here on the right, everything is known, so you want to compute omega GM, and all the others are already known, you have them, and all you have to do is to evaluate just this residue,
40:44
and to create a certain recursion kernel, or the Birkin kernel, integrated, omega zero, one are given, and so on. So this is something easy, and the remarkable thing is that many, many structures in mathematics follow exactly this formula.
41:02
So in large class of examples, PZ omega is equal to omega, so you can remove this P, there are no other poles, but as proved by Boren and Chardin, that's not the general case, so when you only require these absolute equations,
41:22
then you can afford more, and then these formulae describe the polar part, and then there's something else, H, the holomorphic part, holomorphic at ramification points, and they are formed by these blobs,
41:40
so additional input, so here's a picture, so that's the picture of topological recursion, that, so okay, we live here on space of lines, compactified complex lines, two mark points on a Riemann surface, and so you want to build such a particular form
42:01
on that space, and the idea is you distribute lines and genera in all possible ways, and then you glue everything with recursion product, so that's the natural recursion, and in the block case, there is an additional input at every recursion step. So the important point is,
42:22
that's an infinite space, but our omega GN select one particular point in this space, and question is, what's the significance? Okay, good. Yeah, it's a sort of outlook, whatever,
42:43
so it is known from the paper that whenever we have these absolute equations, then these omega GM encode intersection numbers on the modal space of stable complex graphs, that's a general effect.
43:01
So, and they give formula, so most ideas we get in general, several copies of the same intersections of the psi and kappa classes, so psi and echelon classes, kappa and Manford classes, as in the conservative model, and, but they are not independent, they are coupled somehow,
43:21
or described, well, they give a precise description, via the blobs. It's completely generic. so any case, one gets something, but it's not clear that this of interest. So here, but in our case,
43:40
we have this global involution z to minus z, which places special order, so that, let's hope, it's not proved, but I hope that at least these points about zero, could have some meaning in mathematics. We do not know what, but it could be something. So this global involution as an additional structure
44:02
on the modal space of curves. So another thing is, so we have endpoint functions here, forms. We don't have the partition function itself. At some point, we would like to identify it. And then the interesting question is,
44:21
whether this partition function, is a tau function for a hero type equation. So if so, this would be a precise formulation of incurability. So in general, topological recursion, there's always this relation, there's always a tau function. For block topological recursion, this is not known.
44:43
So it could be, could also be not the case. So there's something which one should decide. Okay, I think I'm on time. All right, so someone. Yeah, first talk on disjoint equations.
45:01
And so it arches from the proper fiber graphs, where it's the tool to disentangle the problem of open divergences. Two things are presented. di-shwinger equations are central to DURB and to me.
45:24
And yes, so at least I tried to convince you that di-shwinger equations are really, I would say the best approach to study from field theory non-perturbatively. So in this particular case of matrix models,
45:40
confused on a basis, they produce a complete understanding. And I'm sure, but I can prove it, that also other concrete theories should be treated in this way. So it brings me to an end, to Dirk. So I wish you a lot of pleasure and success
46:02
with your work on downstream tracing or also everything else, and happy birthday. Thank you very much, Reimer. So I have a question for Reimer. Yeah, all right. So what would be your advice,
46:21
how to use topological or blob topological recurve when I want to go from solvable to true quantum field theory in the finger equations? Well, I would say this relation is very special. But, so I'm sure there's some integrability
46:44
and you can expect this for normal quantum field theories. But I'm sure that the idea to look for, to disentangle the problem when you have a quantum field theory that you split, you somehow understand it
47:00
as the convolution of function with the inverse of another function. Part of the difficulty you throw in this problem of determining the inverse. So we have seen for the Lambert function, this expanded gives all the logarithms, integrals, then the Niesen polygon and so on.
47:25
And it's inverse of a very, very simple function. So I'm sure something in this direction should be there also in other quantum fields. Topological recursion, I'm not sure. I have Garth and Bobo now sitting next to me
47:42
so I can actually try to get something for him. Garth thinks that you hired him. Reminds me, yep, thanks. If there are further questions, feel free to raise your hand if you're a viewer or just speak up as a speaker.
48:00
I had a question when you showed your formula for the, I mean, these omega G and everything's rational, right? There's no transcendental numbers in these things. This one? Yeah. Yeah, so I was just confused because if I remember at some point, as you just mentioned, there are transcendental things, right? There's logarithms and maybe some zeta values
48:21
or polylogarithms. Oh, good point, yeah. And you said you can somehow match some classes of Feynman diagrams to these individual omegas. Then I just wonder where does the transcendental score? Well, so to do this,
48:43
well, that's a trial model. In concrete theory, there's no organization. Everything is, it's a regularization. So everything is fine dimensional. For concrete theory, you have to go to the large N limit into infinity, D to infinity in a certain way, and this produces here integrals.
49:00
And in the case which we had, this was the logarithm. You almost see it here, that this converges to logarithm. In other cases, so that will be the presentation by Alex in one hour or so, this would be a hypergeometric function. And then you have to produce the inverse. You see it somewhere here after this transformation.
49:23
And even if you have a very simple integral here, for instance, the inverse of the hypergeometric function is not clear to me that this, well, of course it exists. It's a monotonous function. So it must have an inverse, but what it is, it's not clear. Expand it and you get all the hyperlogarithms.
49:45
So there's a step which is hidden here. So I described the fine dimensional case for concrete theory, you want to have the large N limit. Yeah. Thank you very much. That's very clear. All right. I suggest we thank Reimer again for his nice talk.