 Thank you very much for organizing the conference, Erich and Karin and thank you also very much for the nice 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 Derg and my result I want to present here is a joint work with Harald Krossen and Weimar-Woltenhaar. Actually, this model is called Krossen-Woltenhaar model and the steps or some important steps which were found earlier actually were found on a conference organized by Derg and Spencer Bloch in Lesjuch two years ago, which Reimer already mentioned was a joint work with Erich. And we somehow then generalized it and found also the solution on the four dimensional moyale space, which I have to say is a 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. 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 Tomar and Reimer said. And then I want to make the connection between the moyale space, which is a non commutative space and how it's related to the quartic matrix model. And then I want to formulate and show the renormalized two point function. So if you do the continuum limit of 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. And I find really that we are some of directly on the edge where something is trivial 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. So let's start. So I define the quartic matrix model in the following way. So we have here the space of Hermitian matrices H n. There are n times n Hermitian matrices and E is one of the semi one Hermitian matrix with positive eigenvalues, capital E one up to capital E n. And this E should be understood as the Laplacian in momentum space. What doesn't 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. 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 emission matrices but 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 m matrices so we have to do here, exactly n squared integrals because the space of the degrees of freedom of Hermitian matrices is n squared. And we later take the limit to n goes going to infinity. So that means that we actually have here an infinite number of integrals make which make this not any sense for the partition function but actually the correlation function will make sense in this limit. I only want to talk today about the two point correlation function, which is defined as expectation value with Phi P Q Phi Q P, where you see that the eigenvalue of E at P and E Q are very important at this point. And yeah, we want to calculate that. And how is this related now to the more yard space. So, the definition of the quartic model on the non commutative more yard space and for dimension is the following you take the following action where Phi is now some Schwartz function you have here to Laplacian. And here you have some regulator here the mass and here you have the more yaw star product which is a non commutative product. And here, this regulator you had to introduce this was a work done by Reimer Wolkenhahn had across 1012 years ago 15 years ago maybe, and which avoids the infrared and ultraviolet mixing problem so this is a very important term, which will later somehow go to zero because this data is actually a metrics, which depends on the deformation parameter of this star product, and we will send this to infinity, such that this one over theta goes to zero. And I don't want to go into the details about the more yard star product. I only want to mention that it has one very important property that it has a metric spaces. So if you have some Schwartz function, you can approximate it or even more general functions you can approximate it. So this metric space, this is what you do so you write Phi of XS Phi FN Phi NM times this basis metric space and this metric place you can actually then perform this integral over DX and you end up with only some metrics equation. The ends are the eigenvalues here of this Laplacian. Here I have introduced additionally the renormalization field renormalization, then these are the coefficients in the metric spaces, and here are coupling constant renormalization and here is the, you know, interacting term. So and what also why am I did was that we want to have the eigenvalues that they admit some multiplicities. So we say E small E eyes are distinct eigenvalues where each of them has the multiplicity are I, and especially on the moya space we have this form of the eigenvalues. So we have the new bear so the bear mass. And here is some of the kinetic part and the eigenvalues or the multiplicity of the angry is is our end is equal to and which means the first time where you occurs once the second twice the third three times and so on and so on. So, further this already Rima have has shown this is the nonlinear equation of the two point function this is exactly the same what Rima showed before. We're 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 I mean you express the two point function by the two point function where you hear some odd one of the indices and 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 if M is equal to P makes somehow sense in a sense of a derivative at that point. And then we want to perform the continuum limit. This is now new Rima had done everything somehow discrete everything is fine in the discrete sense, and we have for discrete for the discrete version rational functions, but in the continuum limit actually something new happens. So, Rima mentioned it shortly that we will take two limits or one combined limit where we send n size of the metrics also the deformation parameter of the space of the more of the more space and the biggest eigenvalue, e d or the number of eigenvalues with the constant ratio in this limit, both to infinity, such that this constant rate, this ratio is exactly the cut off of your quantum field theory. So this is the biggest eigenvalue divided by the square root of n the square root comes from four dimension because you're D over two. So all your discrete numbers then converge to continuous variable. So you can think of that you have a lot of discrete numbers which all run together and then form a continuous interval between zero and the coupling and the cut off. So this function also converges to her. Yeah, it's not known at this point if it's continuous but to function depending on a continuous variables and if this functions continues we will see later that it's actually the point. And you end up with this integral equation. And the important point is here maybe that the two point function, the first term of the two point function is the free propagated goes was one over T. So you have DT times T times one over T. And this diverges with lambda squared cut off. And for that you need to renormalization. So in two dimension for instance you have here instead of DT times T only DT. And it was only logarithmically divergent and then you only need mass renormalization but in that case here you need also the field renormalization. And the coupling constant renormalize normalization actually is at the end trivial it's only finite because the beta functions actually zero in this model. And all this coupling, all this renormalization constant are of course depending on the cut off, 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. So the asymptote behavior of the eigenvalues of E in this scaling limit I have shown in this continuum limit will define the spectral dimension. And then we say that row of x dx is the spectral measure in this limit of the Laplacian 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. The integral is in four dimensions here, we have row 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's, yeah, it diverges with the logarithm and since you take the infimum, this is exactly four. And for in two dimension you have here one and here you have one over T which is again logarithmically dimension and since you take the infimum it's two. So these are the two spectra on the spectral measures on the mu yaw space and two and two and then four dimensions. Okay. And now we go back to the result of IMA 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 are which is a rational function for the district case and here again shortly the epsilon ends are defined by the difference which was the eigenvalues. And so are of epsilon and is he and which defines epsilon and here you have an implicit definition are depends on our prime. I even don't know if one can write this definition, because this is not allowed. So and the epsilon's actually the, yeah, are actually in the physical sheet, one would say if you use the notion of topological recursion guys. And if you send lambda to zero for epsilon and you end up with EN. So there are also the other epsilon heads with head Rima, which are different pre images because you if you multiply by the denominator everything you have a polynomial of degree D plus one that has D plus one solutions then and you one is very specific with you called the physical. 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 more yard space after taking all this renormalization stuff and so on to this linear integral equation so you have here something defined implicitly and here you have a linear integral 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 so this part here, it's exactly this part. And if you tell us subtract somehow twice, you get here. Yeah, square in T and the integration variable and here is that square in front. So that is then somehow defined by r of T. And the important point here is that mu is now coming from some boundary condition for some renormalization condition how you fix the renormalization and this is later we will see a quite universally or quite naturally found. And at this point I will have to mention that on the four dimensional more yard space. We see that the spectral measure is transformed into a effective spectrum measure depending on this our function. This comes from how we determine all this solution how we found this exact solution. I don't want to talk 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. And what I've used for that is the nice computer program of if like hyper and I mean if you want to try to compute the first orders and lambda you can go maybe to lambda square, maybe lambda to the three with Mathematica. And then it's the end. But if you use hyper and hyper and there's exactly this type of integrated 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 hyper and can go up to order lambda to the 10 that in the first 10 orders this type of serious satisfies this equation. So what is the interesting thing of the series. The interesting thing is that we have here alternating alternating letters so this h logs defined here below as iterated integral and here are only alternating letters and here also where the first letters the minus one and a lot of zeta to appear or that are for so even that are numbers being you can collect them in an arc sign. And this was very incredible when I found that that you can collect them in arc sign function. And the natural choice for the boundary condition from you is exactly by arc sign minus lambda arc sign square, and the constant here is C. And you can see maybe that this equation f of x satisfy the differential equation of second order if you derive this once you have something which has the first letter one, and all the rest is alternating. If you derive this you have a zero. Everything is alternating. So, we have shown that this is then later differential satisfies this differential equation of a second order. And this is solved by a hyper geometric function. So our of x is on the four dimensional model space within infinite deformation parameter which corresponds to this continuum limit. And this is given by this formula x times to have one alpha lambda one minus alpha lambda as parameters to minus x over mu squared where alpha lambda in this parameter you have the coupling constant by arc. Arc sign 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 know. Take this result and expand it and lambda is very, is very hard because you can it's very hard to expand something to expand hyper geometric function in the arguments. 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. Because I haven't maybe not so much I don't want to go into all the details. So hyper geometric function can even be generalized by something called Maya G function Maya G function is an analytic function defined by a contour integral separating sets of poles and for special poles this is actually a hyper geometric function and this my G function satisfies the convolution, which is exactly here. This type, you can plug in my G function for our and another my G function for the denominator here and then you can integrate that and you find again the my G function where you somehow then take this contour of the line 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 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 to convergent radius one over Pi because it depends on the arc sign of course that you everything can sum to the arc sign of lambda Pi. Good. And how is this result now, how's the R of that related to the two point function. So, here's the important thing that you divine define another function I of that as minus R of minus music at minus the inverse of our this is what I'm also tried to say that we need the inverse of this function. And the theory and then says that this party in red, exactly this part here is given by why plus this I function. And that's it. And then you know this I function and then you can solve the rest, because then it's not anymore nonlinear because you know what this part is, without knowing what the two point function is. And the important structure why the work of electric and lima was so important is because this I function has a very incredible structure hidden behind it because it's actually more or less an evolution so minus I of minus I of that is actually at the end that So this is some of 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, and then that this gives you exactly here the entity but then you plug it in this minus cancer plus this is then cancer, and here gets 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 is what I'm actually said was not so hard but you have to know this algebraic structure behind it. 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 Eric and why am I what they reached in the conference in the Jewish. So there the R function is X is actually very easy Z plus lambda log of one plus that and the inverse can be given in terms of the lambda 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 number function minus log of one minus a number function. And I have computed this two parts separately, but actually they are highly connected by this. I have said, and it's very easy if you know the structure here of I, and, and are, you can write it directly down and they, and I mean we have generalized it for any arbitrary spectral measure, and this is true there and the proof works out. 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, singular integral equation which is called of Kalaman type gives you the two point function and this is given by X, why in the continuum limit ultraviolet convergent, the X of n, and the n function is defined here so you have an integral for minus infinity plus infinity of log of your hyper geometric function from minus I infinity to plus infinity so in the, so some vertical integration in the complex plane. So basically to the branch point of the hyper geometric function and this parts you 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 hyper logarithms. So if you compute all the, or the Feynman diagrams use BPH that theorem to renormalize it, you find exactly the same result. So this is really a resumption of all Feynman integrals of all Feynman graphs for certain type of course for the two point function planar only, but you can resum them. And the important point is, I mean the sum of the number of Feynman graphs grows factorially, and you have additionally, if you look at perturbation theory the renormal problem in this model means that if you want to compute certain type of graph, the amplitude, if you then send the number of the loops to infinity, this type of graph grows also factorially. So there are the amplitude of the graphs grows. But nevertheless, you have this type of exactly miracle cancellation, which gives you the possibility to resummit at the end. So I want to end up with the spectral dimension of five to the four. So we have seen that the R function is given by this type of hyper geometric function. And a short lemma says that this behaves asymptotically was one over x to the A, if you have here just two coefficients, two parameters, and our function, it means for our function where you have x times the hyper geometric function behaves asymptotically with x to the one minus alpha lambda, where alpha lambda again was the arc sine of lambda pi over pi, which means that the spectral dimension by the definition of the spectral dimension behaves asymptotically was D over two minus one, which means that the effective spectral model is four minus two arc sine 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 need the inverse of our and the inverse of us some essential ingredient and it should be globally defined over our plus on all our plus you need the bijection. So 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 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 inverse globally. We are in the nice situation this are has a global inverse on our 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. 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 Weimar he he said of our work is in our limit no more algebraic at the end. So we go from some algebraic results to a limit where you have logarithms hyper logarithms or other functions. So, how is this continuum limit working. And there's also the block topological recursion structure true on the four dimensional space. There's also work in progress where we actually conjecture that this is the same structure. And the reason is that in the results of Weimar you have a lot of this ramification points beta he said he had and these betas the number of this ramification points depend on the number of eigenvalues and you send them to infinity. So you have infinitely many branch points but actually very interesting thing happens is all this ramification points come together and accumulate to one point. And that'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. And something quite recent. The recent idea came to me that one can maybe calculate the galore co action on our correlation function which means you assume the transcendental conjecture that all the motive iterated integrals are the iterated integrals so the period medicine isomorphism, and then you can compute maybe the galore co action on it and look if it's close. And because there's this paper of earlick and Oliver schnitz where they conjecture it for the fight to the forum model for the primitive log diversion graphs. And here you have exact results and can look if this is possible for a quantum or toy quantum field. Yeah, thank you. Thank you Alex. I hope I'm in time. You perfectly stuck for your time. Congratulations very well done. 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. So my question arises from the fact that there's an integer into halfway between four would naively want to modify your equations by changing your measure to square root of TDT. Yes. Is that a sensible thing to do. And what do you get between the conflict type of geometric function. The point is, maybe I can try to write something down the point is if you look how we computed the results. You can have any holder continuous measure. And we have a solution theory how you get for any holder continuous measure exact results. I mean, they are not. I mean you write them down in two implicitly defined functions, but this is valid for any holder continuous measure, you can imagine, and only on the D equal to and D equal for case, you have such nice equations. They break down to such equations. Thank you. And at the very end when you refer to the top logical recursion the block version in four dimension more in the speculation. Yeah. Now what is the integral structure in the two dimensional more. It's more or less. I mean also in two dimension you want to expect topological recursion. This is this important thing what I want to maybe I haven't said that that this underlying structure of topological recursion for this model, topological recursion is actually compatible with renormalization renormalization does not destroy the structure. We have seen that already for the conservative model where we also do such kind of continuum limit but it was too easy to easy to conserve each model. And in that case we have a lot of this ramification points but nevertheless the continuum limit and renormalization and renormalization does not destroy the structure. And this is some of the important case. Very interesting. Yeah. Yeah. Yeah, please. I have a little question. Your mechanism to avoid triviality. Is there anything generic about it. No, no, no. This is very special here for this model and it was somewhat surprised that we found it and this is really coming from the fact that this result results are defined implicitly in a system of equations. And I cannot expect to use it somewhere else. Let's thank Alex again.