Solution of ϕ44 on the Moyal Space
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Transcript: English(auto-generated)
00:16
Thank you very much for organizing the conference Eric and Karen and thank you also very much for the nice
00:24
talks yesterday all the days before it is really a pleasure for me to talk here in front of you all. And of course, congratulations to Dirk and my result I want to present here is a joint work with Harald Gross and Reimer Wülkner.
00:41
Actually, this model is called Gross-Wülkner model and the steps or some important steps which were found earlier actually were found on a conference organized by Dirk and Spencer Bloch in Le Juche two years ago, which Reimer already mentioned was a joint work with Eric. And we somehow then generalized it and found
01:09
also the solution on the four dimensional Moya space, which I have to say is just renormalizable model. And I guess this is very interesting to have an exact solution which I want to show you for just renormalizable model in four dimensions.
01:29
Okay, let's start. So the outline of the talk is the following. I want to somehow start again with a matrix model which is mainly built on what already Thomas and Reimer said.
01:43
And then I want to make the connection between the Moya space, which is a non-commutative space, and how it's related to the quartic matrix model. Then I want to formulate and show the renormalized two-point function. So if you do the continuum limit,
02:00
as Reimer said already in his talk, we have to renormalize, of course, everything, field renormalization, mass renormalization, etc. And a final result is that actually this model is non-trivial. So we find really that we are somehow directly on the edge where something is trivial
02:24
or not trivial, and I want to explain or somehow argue why this is not trivial. And this is a very interesting thing and only visible if you have the exact result. So not perturbatively visible.
02:43
So let's start. So I define the quartic matrix model in the following way. So we have here the space of Hermitian matrices HN. There are N times N Hermitian matrices, and E is one of these Hermitian, one Hermitian matrix with positive eigenvalues, capital E1 up to capital EN.
03:06
And this E should be understood as the Laplacian in momentum space. What does it mean? It means that the eigenvalues are essentially the spectrum of the Laplacian, which is now discrete, but we want to perform a continuum limit where it's a continuous spectrum.
03:24
And the partition function is defined as an integral over all Hermitian matrices, where we take the exponential of the action, where E, the Laplacian, think of the Laplacian, stands in front of the quadratic term, plus some quartic interaction. And Reimer's definition is exactly equivalent to this definition here, the integration over the new space of Hermitian matrices. But
03:48
okay, I like more this type of definition because it's a little bit old school, like in the 80s or 70s. Okay, and okay, this integral over all Hermitian N times N matrices, so we have to do
04:04
here exactly N squared integrals because the space of degrees of freedom of Hermitian matrices is N squared. And we later take the limit to N going to infinity. So that means that we actually have here an infinite number
04:25
of integrals, which make this not any sense for the partition function, but actually the correlation function will make sense in this limit. So, and I only want to talk today about the two-point correlation function, which is defined as expectation value
04:41
with phi PQ, phi QP, where you see that the eigenvalue of P and EQ are very important at this point. And yeah, we want to calculate that. And how is this related now to the Muehl space?
05:05
So the definition of the quartic model on the non-commutative Muehl space in four dimension is the following. You take the following action, where phi is now some Schwarz function. You have here the Laplacian, and here you have some regulator, here the mass,
05:23
and here you have the Muehl star product, which is a non-commutative product. And here, this regulator you had to introduce, this was a work done by Reimer, Wülkenhe, and Harald Kloss, 10, 12 years ago, 15 years ago, maybe, which avoids the infrared and ultraviolet mixing problem.
05:47
So this is a very important term, which will later somehow go to zero, because this theta is actually a matrix, which depends on the deformation parameter of this star product.
06:02
And we will send this to infinity, such that this one over theta goes to zero. But I don't want to go into the details about the Muehl star product. I only want to mention that it has one very important property, that it has a matrix basis.
06:21
So if you have some Schwarz function, you can approximate it, or even more general functions, you can approximate it inside this matrix space. This is what you do, so you write phi of x as phi f n, phi n m, times this matrix space, and this matrix space, you can actually then perform this integral over dx, and you end up with only some matrix equation.
06:48
So where the ens are the eigenvalues here of this Laplacian, here I have introduced additionally the field renormalization, then these are the coefficients in the matrix basis, and here are coupling constant renormalization, and here is the interacting term.
07:11
So what also Leimer did was that we want to have the eigenvalues that they admit some multiplicities. So we say small e i's are distinct eigenvalues, where each of them has the multiplicity r i,
07:26
and especially on the Moyer space, we have this form of the eigenvalues, so we have the mu bear, so the bear mass, and here is some of the kinetic part.
07:42
And the eigenvalues, or the multiplicity of the eigenvalues, is r n is equal to n, which means the first eigenvalue occurs once, the second twice, the third three times, and so on and so on. So then let's go further, this already Reimer has shown, this is the nonlinear equation of the two-point function.
08:04
This is exactly the same what Reimer showed before, where only had to plug in the eigenvalues. And the important point here is that this is, again, I will mention that this is a nonlinear equation, it is a closed equation.
08:21
I mean, you express the two-point function by the two-point function, where you here sum out one of the indices, m comes from the multiplicity. And here you have some difference quotient of the two-point function, and here m comes also from the multiplicity. And if you, let's say you want to approximate it by continuous function, you see that also here,
08:43
if m is equal to p, makes somehow sense, in the sense of a derivative at that point. So then we want to perform the continuum limit. This is now new. Reimer had done everything somehow discrete, everything is fine in the discrete sense. And we have for discrete version rational functions, but in the continuum limit, actually something new happens.
09:07
So Reimer mentioned shortly that we will take two limits, or one combined limit, where we send n, the size of the matrix, also the deformation parameter of the space of the Moyard space.
09:22
And the biggest eigenvalue, ed, or the number of eigenvalues, with a constant ratio in this limit, both to infinity, such that this constant, this ratio, is exactly the cutoff of your quantum field theory. So this is the biggest eigenvalue divided by the square root of n. The square root comes from four dimensions, because you have d over two.
09:47
And all your discrete numbers then converge to a continuous variable. So you can think of that you have a lot of discrete numbers which all
10:02
run together and then form a continuous interval between zero and the coupling and the cutoff. And your function also converges to, it's not known at this point if it's continuous, but to a function depending on continuous variables.
10:21
And if this function is continuous, we will see later that it's actually the point. And you end up with this integral equation. So the important point is here maybe that the two -point function, the first term of the two-point function is the free propagated goes with one over t. So you have dt times t times one over t. And this diverges with lambda squared, the cutoff, and for that you need renormalization.
10:51
So in two dimensions, for instance, you have here instead of dt times t, only dt. And it was only logarithmically divergent, and then you only need mass
11:01
renormalization, but in that case here you need also the field renormalization. And the coupling constant renormalization actually is at the end trivial, it's only finite because the beta function is actually zero in this model. And all this coupling, all this renormalization constant are of course depending on the
11:24
cutoff, and then you want to send it to infinity such that everything is finite. And before doing that, I want to talk about the spectral dimension. So we want to say something about the dimension of this model coming from the spectrum of the eigenvalues of E.
11:44
So the asymptotic behavior of the eigenvalues of E in this scaling limit I have shown in this continuum limit will define the spectral dimension. So we say that rho of x dx is the spectral measure in this limit of the Laplacian.
12:03
And then you define the spectral dimension as follows. So d is defined by the infimum of p such that this integral over the spectral measure converges.
12:21
So one example is in four dimensions here, we have rho of x is equal x, so you insert your t, t over, and you say four dimension one, plus t to the four over two, so t squared, so t over t squared. It diverges with the logarithm, and since you take the infimum, this is exactly four, and in two dimensions you have
12:46
here one, and here you have one over t, which is again logarithmically divergent, and since you take the infimum, it's two. So these are the two spectral measures on the Muhyal space in two and in four dimensions.
13:05
Okay, and now we go back to the result Reimer has shown. So here, I don't want to say all the details again, but he has said that the most important thing is to define this function r, which is the rational function for the discrete case.
13:26
And here, again, shortly the epsilon n's are defined by the e n's, which was the eigenvalues. And so r of epsilon n is e n, which defines epsilon, and here you have an implicit definition, r depends on r prime.
13:43
I even don't know if one can write this definition, because this is not allowed. So, and the epsilons are actually the, yeah, are actually in the physical sheet, one would say, if you use the notion of topological recursion, guys, which means that if you send lambda to zero for epsilon n, you end up with e n.
14:05
So there are also the other epsilon heads with Reimer, which are different pre-images because you multiply by this denominator, everything, you have a polynomial of degree d plus one. It has d plus one solutions then, and one is very specific, which you call the physical.
14:26
So what can you do in the continuum limit? So actually, this is a theorem, and I don't want to go into the technical details, but this implicit equation converges in the continuum limit for the four-dimensional Moya space, after taking all this renormalization stuff and so on, to this linear integral equation.
14:47
So you have here something defined implicitly, and here you have a linear integral equation. It's a Fredhorn equation, and you can maybe see that here that comes somehow from something like Taylor subtraction, which is essentially what you can do by renormalization, right?
15:04
So this part here is exactly this part, and if you Taylor subtract somehow twice, you get here a square in t and the integration variable, and here z squared in front, and r of z is then somehow defined by r of t.
15:24
And the important point here is that mu is now coming from some boundary condition, some renormalization condition, how you fix the renormalization, and this is later, we will see quite universally or quite naturally found.
15:42
And at this point, I have to mention that on the four-dimensional Moya space, we see that the spectral measure is transformed into an effective spectral measure, depending on this r function. This comes from how we determine all this solution, how we found this exact solution, and I don't
16:05
want to talk about the details, but here you see an effective measure depending on the coupling constant. Okay, and now solve this integral equation. So again, here's the integral equation I want to solve.
16:21
And what I have used for that is the nice computer program of ESIC, HyperInt. I mean, if you want to try to compute the first orders in lambda, you can go maybe to lambda square, maybe lambda to the three with Mathematica, but then it's the end.
16:41
But if you use HyperInt, in HyperInt, there's exactly this type of iterated integrals implemented that give you the possibility to compute that. So we define this function f of x in this way, so to get a little bit rid of this mu. And then you find, if you go with HyperInt, you can go up to order lambda
17:03
to the 10, that in the first 10 orders, this type of series satisfies this equation. So what is the interesting thing of the series? The interesting thing is that we have here alternating letters, so this h log is defined here below as iterated integral,
17:27
and here only alternating letters, and here also, where the first letter is a minus one, and a lot of zeta two appear, or zeta four, so even zeta numbers appear, and you can collect them in an arc sine.
17:41
This was very incredible when I found that you can collect them in an arc sine function. And the natural choice for the boundary condition for mu is exactly by arc sine minus lambda arc sine squared. And the constant here is c. And you can see maybe that this equation f of x satisfy the differential equation of second order.
18:03
If you derive this once, you have something which has the first letter one, and all the rest is alternating, and if you derive this, you have a zero, everything is alternating. So we have shown that this is then later differential, satisfies the differential equation of a second order, and this is solved by a hypergeometric function.
18:23
So r of x is on the four-dimensional moial space with an infinite deformation parameter which corresponds to this continuum limit given by this formula x times two f one alpha lambda, one minus alpha lambda as parameters, two minus x over mu squared, where alpha lambda
18:43
in this parameter, you have the coupling constant by arc sine, arc sine of lambda pi. So this is really the important thing, that the coupling constant is hidden here in the parameters. And if you want to now take this result and expand it in lambda, it's very
19:03
hard, because it's very hard to expand something, to expand a hypergeometric function in the arguments. But I mean, this is the solution, and the proof of that is actually not by guessing this thing here, but by knowing that this is the right solution, inserting it in the equation, and then finding the right answer,
19:26
because I have maybe not so much, I don't want to go into all the details. So a hypergeometric function can even be generalized by something called Maya-G function. Maya-G function is an analytic function defined by a contour integral separating two sets of poles.
19:48
And for special poles, this is actually a hypergeometric function, and this Maya-G function satisfies the convolution, which is exactly here, this type, you can plug in the Maya-G function for R, and then the other Maya-G function for the denominator here.
20:06
And then you can integrate that, and you find again the Maya-G function, where you somehow then take this contour of the definition, and move through one of the poles, you pick up a residue, and find exactly that, again, the Maya-G function where you have started with
20:22
comes out, where you actually use also for two gamma functions, the Euler reflection identity. So this is very interesting and nice calculation. And what you see here is that our result, our model has the convergent radius one over pi, because
20:41
it depends on the arc sine, of course, that everything can sum to the arc sine of lambda pi. Good. And how is this result now, how is the R of z related to the two-point function? So here's the important thing, that you define another function, I of z, as minus R of minus mu squared minus the inverse of R.
21:03
This is what Reimer also tried to say, that we need the inverse of this function. And the theorem then says that this part here in red, exactly this part here, is given by y plus this i function, and that's it.
21:22
And then you know this i function, and then you can solve the rest, because then it's not any more nonlinear, because you know what this part is, without knowing what the two-point function is. And the important structure why the work of Ehrlich and Reimer was so important is because this
21:40
i function has a very incredible structure hidden behind it, because it's actually more or less an involution. So minus i of minus i of z is actually at the end z. So this is somewhat formal, because you have to make sure that you take the right inverses, because you have a lot of them in the different sheets or in the different branches.
22:01
And then this gives you exactly here the identity, but then you plug it in, this minus cancel, this minus to a plus, this is then cancelled, and here gets a plus, and this is then the identity. And this observation, this structure i, gave us the possibility to guess the results, and then to prove what Reimer actually said was not so hard, but you have to know this algebraic structure behind it.
22:27
And yes, and what you're left with is with the singular integral equation for the two-point function, where the solution theory is known. And let's go back to the result of Ehrlich and Reimer, what they reached in the conference in Les Juches.
22:44
So there the r function is x, it's actually very easy, z plus lambda log of one plus z, and the inverse can be given in terms of the Lambert wave function. And if you plug it in this i, so minus r of minus one minus r to the minus one, you end up with this function, with the Lambert function, minus log of one minus the Lambert function.
23:04
And they have computed these two parts separately, but actually they are highly connected by this i of z. And it's very easy, if you know the structure here of i and r, you can write it directly down, and I mean we have generalized it for any arbitrary spectral measure, and this is true there, and the proof works out.
23:27
So let's go back to the four-dimensional case, so I want to show you the exact result of the two-point function. So solving this singular integral equation, which is called a Carlemann type, gives you the two-point function.
23:42
And this is given by xy in the continuum limit, ultraviolet convergent, given by x of n, and the n function is defined here. So you have an integral from minus infinity to plus infinity of log of your hypergeometric function from minus i infinity to plus i infinity.
24:04
So some vertical integration in the complex plane, closely to the branch point of the hypergeometric function, and these parts come from renormalization to make everything convergent, and so on. And the incredible thing is now, if you expand this result in a small coupling constant, you find hyperlogarithms,
24:27
and if you compute all the Feynman diagrams, use BPHZ theorem to renormalize it, you find exactly the same result. So this is really a re-summation of all Feynman integrals, of all Feynman graphs for a
24:47
certain type, of course, for the two-point function, planar only, but you can re-sum them. And the important point is, I mean, the number of Feynman graphs grows factorial, and you have additionally, if you look
25:02
at perturbation theory, the renormal problem in this model, which means that if you want to compute a certain type of graph, the amplitude, if you then send the number of the loops to infinity, this type of graph grows also factorial. So the amplitude of
25:22
the graphs grows, but nevertheless, you have this type of exactly miracle cancellation, which gives you the possibility to re-sum it at the end. And now I want to end up, do I have a few minutes? Yes, I guess so, with the spectral dimension of
25:41
phi to the four, so we have seen that the R function is given by this type of hypergeometric function, and a short lemma says that this behaves asymptotically with one over x to the a, if you have here just two coefficients, two parameters, and our R function, it means for our R function where you have x times the hypergeometric function behaves asymptotically
26:08
with x to the one minus alpha lambda, where alpha lambda again was the arcsin of lambda pi over pi, which means that the spectral dimension by the definition of the spectral dimension behaves asymptotically with D over two minus
26:24
one, which means that the effective spectral dimension of this model is four minus two arcsin lambda pi over pi, which means for positive lambda, you are effectively in a lower dimensional setting. And why does it avoid the triviality problem? Because we
26:44
need the inverse of R, and the inverse of R is some essential ingredient, and it should be globally defined over R plus. On all R plus, you need the bijection, and if you instead would have here another R function, let's say with not defined by this linear integral equation, but instead defined by the usual
27:09
measure, then you would have something which has an upper bound, which behaves with one over lambda. So I mean for finite lambda, you have an upper bound, and you cannot write down the
27:21
inverse globally, but we are in the nice situation. This R has a global inverse on R plus, and which means we have an effective dimension drop, which is only visible by having this exact solution. So if you would look at perturbation theory, you would never see that this type of effective dimensional drop would come out.
27:45
And yeah, I want to finish with some open question for the future. So I want to understand how this algebraic structure from the finite results from Mavaima, he said of our work, is in our limit, no more algebraic at the end.
28:02
So we go from some algebraic results to a limit where you have logarithms, hyperlogarithms, or other functions. So how is this continuum limit working? And there's also the block topological recursion structure true on the four-dimensional Moya space.
28:22
There's also work in progress, where we actually conjecture that it is the same structure. The reason is that in the results of Mavaima, you have a lot of these ramification points, beta, he said, he had. And these betas, the number of these ramification points, depend on the number of eigenvalues.
28:41
And you send them to infinity, so you have infinitely many branch points. But actually, a very interesting thing happens is all these ramification points come together and accumulate to one point. And it's very incredible how this works, and we have not understand it yet. Then I want to understand the structure of this generating series. I mean, every correlation function is a generating series of iterated integrals.
29:06
And at the end, something quite recent, a recent idea came to me that one can maybe calculate the Galois co-action on our correlation function, which means you assume the transcendental
29:21
conjecture that all the motivic iterated integrals are the iterated integrals, so the period medicine isomorphism. And then you can compute maybe the Galois co-action on it and look if it's close. And because there's this paper of Ehrich and Oliver Schnetz, where they conjectured for the fight to the fore model for the primitive log-diversion graphs.
29:44
And here you have exact results and can look if this is possible for a quantum or toy quantum theory. Yeah, thank you. Thank you, Alex. I hope I'm in time. You perfectly stuck with your time. Congratulations, very well done.
30:04
We have a question straight away from David Broadhurst, David. Yeah, really big congratulations on this very beautiful jump from d equals two to d equals four. Thank you. So my question arises from the fact that there's an integer halfway between two and four. I would naively want to modify your equations by changing your measure to square root of TDT.
30:28
Yes. Is that a sensible thing to do and what you get between the conflict hypergeometric function? The point is, maybe I can try to write something down. The point is, if you look how we computed the results,
30:45
you can have any Hölder continuous measure and we have a solution theory how you get for any Hölder continuous measure exact results. So, I mean, there are not, I mean, you write them down in two implicitly defined functions,
31:01
but this is valid for any Hölder continuous measure, you can imagine. And only on the d equals two and d equals four case, you have such nice equations. They break down to such equations. Thank you.
31:21
I had a question at the very end when you refer to the topological recursion, the block version in four-dimensional Moyau in the speculation. Now, what is the integral structure in the two-dimensional Moyau? It's more or less, I mean, also in two-dimension, you want to expect topological recursion.
31:40
This is the important thing, what I want to, maybe I haven't said that, that this underlying structure of topological recursion for this model of block topological recursion is actually compatible with renormalization. Renormalization does not destroy the structure. We have seen that already for the conservation model, where we also do such kind of continuum limit, but it was too easy to conservation model.
32:05
And in that case, we have a lot of this ramification points, but nevertheless, the continuum limit and renormalization does not destroy the structure. And this is some of the important case. Very interesting.
32:21
Yeah, yeah. Dirk, please. I have a little question. Your mechanism to avoid triviality, is there anything generic about it? No, no, no, no. This is very special here for this model, and it was somehow a surprise that we found it and this is really coming
32:45
from the fact that this result results are defined implicitly in a system of equations, and I cannot expect to use it somewhere else. Okay. Alright, let's thank Alex again.