Solution of ϕ44 on the Moyal Space
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Hock, Alexander
We show the exact solution of the self-dual ϕ4-model on the 4-dimensional Moyal space. Using the results explained in Raimar's talk, an implicitly defined function converges to a Fredholm integral, which is solved, for any coupling constant λ more than −1π, in terms of a hypergeometric function. We prove that the interacting model has spectral dimension 4−2arcsin(λπ)π for |λ| less than 1π. It is this dimension drop which for λ more than 0 avoids the triviality problem of the ϕ44 model on the Moyal space.
 Thank you very much for organizing the conference, Ileka and Karen. 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 Wultenhaar. Actually, this model is called Krosser-Wultenhaar model. And the steps, some important steps which were found earlier, actually were found on a conference organized by Derg and Spencer Bloch in Lejus two years ago, which Reimer already mentioned was a joint work with Ileka. And we somehow then generalized it and found also the solution on the four-dimensional moyale 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 a 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. 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. So we find really that we are somewhat 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 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. And this E should be understood as the Laplacian in momentum space. But 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. 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 OK, I like more this type of definition because it's a little bit old school like in the 80s or 70s. OK. And OK, this integral over all Hermitian n times n matrices. So we have to do 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 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 with phi pq phi qp, where you see that the eigenvalue of E at p and Eq are very important at this point. And yeah, we want to calculate that and how is this related now to the moyale space. So the definition of the quartic model on the non-commutative moyale 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 and here you have the moyale star product, which is a non-commutative product. And here this regulator you had to introduce. This was worked on by Weimar-Woltenhauer and Haddad-Klosse 10, 12 years ago, 15 years ago maybe, which avoids the infrared and ultraviolet mixing problem. 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. And we will send this to infinity such that this 1 over theta goes to zero. But I don't want to go into the details about the moyale 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 Schwarz function, you can approximate it or even more general functions. You can approximate it inside this metric space. This is what you do. So you write phi of x as phi fn, phi nm times this basis metric space. In this metric space, you can actually then perform this integral over dx and you end up with only some matrix equation. So where the e ends are the eigenvalues here of this Laplacian. Here I have introduced additionally the field renormalization. Then these are the coefficients in the metric spaces. And here are coupling constant renormalization and here is the interacting term. So what also Leimer did was that we want to have the eigenvalues that they admit some multiplicities. So we say small eis are distinct eigenvalues where each of them has the multiplicity ri. And especially on the moyale 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. And the eigenvalues or the multiplicity of the eigenvalues is rn 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 Leimer has shown. This is the nonlinear equation of the two point function. This is exactly the same what Leimer showed before. We 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 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 if m is equal to p makes somehow sense in a sense of a derivative at that point. So and then we want to perform the continuum limit. This is now new. Leimer had done everything somehow discreet. Everything is fine in the discreet sense. And we have for the discrete version rational functions. But in the continuum limit, actually something new happens. So Leimer mentioned it 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 molybdenum space and the biggest eigenvalue, e d 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 dimension because you're d over two and 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 cutoff and your function also converges to a, yeah, it's not known at this point if it's continuous but to a function depending on your continuous variables. 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. So in two dimension for instance, you have here instead of dt times t only dt and it was only logarithmic divergent and then you only need mass 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 renormalization constant are of course depending on the 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. 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 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. So one example is in four dimensions here we have rho of x is equal to x so you insert here 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 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 Muyaal space in two and in two and in four dimensions. 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 R which is a rational function for the discrete case and here again shortly the epsilon ends are defined by the e ends which was the eigenvalues and so R of epsilon and is E n which defines epsilon and here you have an implicit definition R depends on our prime. I don't know if you can write this definition because this is not allowed. So and the epsilon's are actually in the physical sheet one would say if you used a notion of topological recursion guys which means that if you send lambda to zero for epsilon n you end up with E n. So there are also the other epsilon heads with RIMA which are different pre-images because you multiply by this denominated everything you have a polynomial of degree d plus one that has d plus one solutions then and one is very specific with you called the physical. So and what can you do in the continuum limit? So actually this is a theory and I don't want to go into the technical details but this implicit equation converges in the continuum limit for the four dimensional Muyaal 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 as a Fredholm 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 is exactly this part and if you Taylor subtract somehow twice you get here 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 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 a quite universally or quite naturally found and at this point I have to mention that on the four dimensional Muyaal 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 and how we found this exact solution and I don't want to talk about the details but here you see an effective measure depending on the coupling constant. And now solve this integral equation so again here is the integral equation I want to solve and what I have used for that is the nice computer program of E of lich hyperend 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 but if you use hyperend in hyperend there is 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 rid of this Mu and then you find if you go with hyperend you can go up to order lambda to the ten that in the first ten 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 an integrated integral and here only alternating letters and here also where the first letters are 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 sign. This was very incredible when I found that you can collect them in an arc sign function and the natural choice for the boundary condition for Mu 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 satisfies the differential equation of second order if you derive these ones 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 a differential satisfies the differential equation of the second order and this is solved by a hyper geometric function 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 in this parameter you have the coupling constant by arc sign 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 now take this result and expand it in lambda is very it's very hard because you can it's very hard to expand something to expand a hyper geometric 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 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 two sets of poles and for special poles this is actually a hyper geometric function and this Maya g function satisfies the convolution which is exactly here this type you can plug in a 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 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 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 define define another function I of that s minus r of minus mu squared minus the inverse of R this is what weim 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 and then you know this I function and then you can solve the rest because then it's not any more non-linear because you know what this part is without knowing what the two point function is and the important structure why the work of Euler can rhyme over some port and is because this 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 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 this gives you exactly here the entity but then you plug it in this minus cancer this must be a 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 and yes and what you're left with is with a 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 Weimar what they reached in the conference in Lejus so there the R function is X is 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 and they have computed this 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 and R 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 and 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 Y 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 plus infinity of log of your hyper geometric function from minus I infinity to plus I infinity so in the so some vertical integration in the complex plane closely to the branch point of the hyper geometric function and this part 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 and if you compute all the all 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 plane are 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 which means that if you want to compute certain type of graph the amplitude if you then send the number of of the loops to to to infinity this type of graph grows also factorially so there the the amplitude of the graph grows but nevertheless you have this type of exactly miracle cancellation which gives you the possibility to resum 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 five to the four so we have seen that the r function is given by this type of hyper geometric function and 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 hypergeometry 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 with d over two minus one which means that the effective spectral dimension of this model is four minus two arcs and 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 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 uh not yeah defined uh 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 upper bound and you cannot write down the 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 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 it's also the block topological recursion structure true on the four-dimensional moya space there's also work in progress where we actually conjecture that it is the same structure the reason is that in the results of Weimar you have a lot of this ramification points beta he said he had and this these beta's 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 these ramification points come together and accumulate to one point and it's very incredible how this works and we have not understood 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 at the end something quite recent the recent idea came to me that one can maybe calculate the Galois co-action on our correlation function which means you assume the transcendental conjecture that all the motivic iterated integrals are the iterated integrals so the period metas and 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 Erlich and Oliver Schnetz where they conjectured for the fight to the form 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 theory yeah thank you. Thank you Alex. I hope I'm in time. You perfectly stuck to 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. Thank you. So my question arises from the fact that there's an integer halfway between two and four. There are what naively want to modify your equations by changing your measure to square root of t dt. Is that a sensible thing to do and what do 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 you can have any held continuous measure and we have a solution theory how you get for any held a continuous measure exact results. So I mean there are not I mean you write them down in two implicitly defined functions but this is valid for any held a continuous measure you can imagine and only on the d equal two and d equal four case you have such nice equations. They break down to such equations. Thank you. I had a question at the very end when you refer to the topological recursion the blocked version in four-dimensional moyao in the speculation. Now what is the integral structure in the two-dimensional moyao? 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 or blocked 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 to easily the conservative model and in that case we have a lot of these ramification points but nevertheless the continuum limit and renormalization does not destroy the structure and this is somewhat important case. It's very interesting. 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 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. Okay. All right let's thank Alex again.
 Thank you very much for organizing the conference, Ileka and Karen.  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  Wultenhaar.  Actually, this model is called Krosser-Wultenhaar model.  And the steps, some important steps which were found earlier, actually were found on  a conference organized by Derg and Spencer Bloch in Lejus two years ago, which Reimer  already mentioned was a joint work with Ileka.  And we somehow then generalized it and found also the solution on the four-dimensional  moyale 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 a 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.  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.  So we find really that we are somewhat 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 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.  And this E should be understood as the Laplacian in momentum space.  But 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.  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 OK, I like more this type of definition because it's a little bit old school like in  the 80s or 70s.  OK.  And OK, this integral over all Hermitian n times n matrices.  So we have to do 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 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 with phi pq phi qp, where you see that the eigenvalue of E at p and  Eq are very important at this point.  And yeah, we want to calculate that and how is this related now to the moyale space.  So the definition of the quartic model on the non-commutative moyale 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 and here you have the moyale star product, which is a non-commutative product.  And here this regulator you had to introduce.  This was worked on by Weimar-Woltenhauer and Haddad-Klosse 10, 12 years ago, 15 years  ago maybe, which avoids the infrared and ultraviolet mixing problem.  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.  And we will send this to infinity such that this 1 over theta goes to zero.  But I don't want to go into the details about the moyale 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 Schwarz function, you can approximate it or even more general functions.  You can approximate it inside this metric space.  This is what you do.  So you write phi of x as phi fn, phi nm times this basis metric space.  In this metric space, you can actually then perform this integral over dx and you end  up with only some matrix equation.  So where the e ends are the eigenvalues here of this Laplacian.  Here I have introduced additionally the field renormalization.  Then these are the coefficients in the metric spaces.  And here are coupling constant renormalization and here is the interacting term.  So what also Leimer did was that we want to have the eigenvalues that they admit some  multiplicities.  So we say small eis are distinct eigenvalues where each of them has the multiplicity ri.  And especially on the moyale 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.  And the eigenvalues or the multiplicity of the eigenvalues is rn 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 Leimer has shown.  This is the nonlinear equation of the two point function.  This is exactly the same what Leimer showed before.  We 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 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 if m is equal to p makes somehow sense in a sense of a derivative at that point.  So and then we want to perform the continuum limit.  This is now new.  Leimer had done everything somehow discreet.  Everything is fine in the discreet sense.  And we have for the discrete version rational functions.  But in the continuum limit, actually something new happens.  So Leimer mentioned it 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  molybdenum space and the biggest eigenvalue, e d 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 dimension because you're d over two and 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 cutoff and your function  also converges to a, yeah, it's not known at this point if it's continuous but to a  function depending on your continuous variables.  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.  So in two dimension for instance, you have here instead of dt times t only dt and it  was only logarithmic divergent and then you only need mass 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 renormalization constant are of course depending on the 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. 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 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.  So one example is in four dimensions here we have rho of x is equal to x so you insert  here 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 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 Muyaal space in two and in  two and in four dimensions.  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 R which is a rational function for the discrete case  and here again shortly the epsilon ends are defined by the e ends which was the eigenvalues  and so R of epsilon and is E n which defines epsilon and here you have an implicit definition  R depends on our prime.  I don't know if you can write this definition because this is not allowed.  So and the epsilon's are actually in the physical sheet one would say if you used a notion of  topological recursion guys which means that if you send lambda to zero for epsilon n you  end up with E n.  So there are also the other epsilon heads with RIMA which are different pre-images because  you multiply by this denominated everything you have a polynomial of degree d plus one  that has d plus one solutions then and one is very specific with you called the physical.  So and what can you do in the continuum limit?  So actually this is a theory and I don't want to go into the technical details but this  implicit equation converges in the continuum limit for the four dimensional Muyaal 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  as a Fredholm 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 is exactly this part and if you Taylor subtract somehow twice you get  here 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 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 a quite universally or quite naturally found and at this point I have to  mention that on the four dimensional Muyaal 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 and how we found this exact solution  and I don't want to talk about the details but here you see an effective measure depending  on the coupling constant.  And now solve this integral equation so again here is the integral equation I want to solve  and what I have used for that is the nice computer program of E of lich hyperend 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 but if you use hyperend  in hyperend there is 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 rid of this Mu and then you  find if you go with hyperend you can go up to order lambda to the ten that in the first  ten 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 an integrated integral and here  only alternating letters and here also where the first letters are 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 sign.  This was very incredible when I found that you can collect them in an arc sign function  and the natural choice for the boundary condition for Mu 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 satisfies the differential equation of second order if you derive these ones  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 a differential satisfies the differential equation of the second order  and this is solved by a hyper geometric function 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 in this parameter you have the coupling constant  by arc sign 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 now take this result and  expand it in lambda is very it's very hard because you can it's very hard to expand something  to expand a hyper geometric 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 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 two sets of poles and  for special poles this is actually a hyper geometric function and this Maya g function  satisfies the convolution which is exactly here this type you can plug in a 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 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 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 define define another function  I of that s minus r of minus mu squared minus the inverse of R this is what weim 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 and then  you know this I function and then you can solve the rest because then it's not any more non-linear  because you know what this part is without knowing what the two point function is and  the important structure why the work of Euler can rhyme over some port and is because this  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 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 this  gives you exactly here the entity but then you plug it in this minus cancer this must  be a 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 and yes and what you're left with is with a 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 Weimar what they reached in the conference in Lejus so there the R function is  X is 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 and they have computed this 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 and R 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 and 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 Y 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 plus infinity of log  of your hyper geometric function from minus I infinity to plus I infinity so in the so some  vertical integration in the complex plane closely to the branch point of the hyper geometric function  and this part 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 and if you compute all the all 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 plane are 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 which means that if you want to compute certain type of graph the amplitude if you  then send the number of of the loops to to to infinity this type of graph grows also  factorially so there the the amplitude of the graph grows but nevertheless you have this type  of exactly miracle cancellation which gives you the possibility to resum 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 five to the four so we have seen that the r function is given by this type of hyper geometric  function and 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 hypergeometry 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 with d over two  minus one which means that the effective spectral dimension of this model is four minus two arcs  and 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 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 uh  not yeah defined uh 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 upper bound and you cannot write down the 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  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 it's  also the block topological recursion structure true on the four-dimensional moya space there's  also work in progress where we actually conjecture that it is the same structure the reason is that  in the results of Weimar you have a lot of this ramification points beta he said he had and this  these beta's 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 these ramification points come together and accumulate to one point and it's  very incredible how this works and we have not understood 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 at the end something quite recent the recent idea came to me that  one can maybe calculate the Galois co-action on our correlation function which means you assume  the transcendental conjecture that all the motivic iterated integrals are the iterated integrals so  the period metas and 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 Erlich and Oliver Schnetz where they conjectured  for the fight to the form 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 theory yeah thank you.  Thank you Alex. I hope I'm in time. You perfectly stuck to 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. Thank you. So my  question arises from the fact that there's an integer halfway between two and four.  There are what naively want to modify your equations by changing your measure to square root of t dt.  Is that a sensible thing to do and what do 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 you can have any held continuous measure and we have a solution theory how you get  for any held a continuous measure exact results. So I mean there are not I mean you write them down  in two implicitly defined functions but this is valid for any held a continuous measure  you can imagine and only on the d equal two and d equal four case you have such nice equations.  They break down to such equations. Thank you.  I had a question at the very end when you refer to the topological recursion the blocked version in  four-dimensional moyao in the speculation. Now what is the integral structure in the two-dimensional  moyao? 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 or blocked 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  to easily the conservative model and in that case we have a lot of these ramification points but  nevertheless the continuum limit and renormalization does not destroy the structure and this is  somewhat important case. It's very interesting. 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  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. Okay. All right let's thank Alex again.
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