Resurgent Trans-series Analysis of Hopf Algebraic Renormalization
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Dunne, Gérald
In the Kreimer-Connes Hopf algebraic approach to renormalization, for certain QFTs the Dyson-Schwinger equations can be reduced to nonlinear differential equations. I describe methods based on Ecalle's theory of resurgent trans-series to extract non-perturbative information from these Dyson-Schwinger equations. Even in the absence of exact results, there exist efficient methods to uncover non-perturbative information numerically from perturbative data.
 Alright, thanks Eric and thanks to the organizers for putting together this conference. It's been very good. So I want to talk about some recent work with Michi Burinski and my student Max Monig and I'll also be mentioning some work done recently with Abidu Kostin who's a mathematician, done a lot of work in the theory of research and transfer. So the motivation. I want to put together, try to put together two things. One we all know very well, this underlying Hopf-Algebra structure from perturbative quantum field theory renormalization. And one of the advantages of this from a practical point of view in quantum field theories that enables doing perturbative computations and understanding perturbative structures at very high order that's hard to access in other ways. I wanted to think about combining this with a Calv's theory of recidion and asymptotics, which is a very general theory, but we can think of it here as a way of extracting non-perturbative information from formal perturbative series. And the advantage from the physics point of view is that there is physics beyond perturbative expansions, which we call non-perturbative physics. And the question is whether that has any natural home in the world of Hopf-Algebras. I'm not able to. So the idea is to try and use these recidion transfer structures to decode some non-perturbative information about quantum field theories from their perturbative Hopf-Algebra structure. So just a few words because people may not be very familiar with recidion transfer is or a lot of people here are. The recidion functions can be described in many ways, but one way to think of them is that they're closed under essentially all operations in analysis. And the operations will be interested in things like rural transforms and little continuations, inverse rural transforms, things like that, monodromes. In applications so far in quantum mechanics and quantum field theories and in matrix models, the general structure looks something like this, that you have some formal series in some small parameter X, but you include also these beyond all orders, exponentially small things which we would call instantons or non-perturbative terms, and also powers of logarithms. And the essential result is that iterations of these three basic structures of X and e to the minus some constant over X and log X generate this huge class of so-called analyzable functions which is much more general than standard analytic functions. And we also know from quantum field theory calculations and various asymptotic expansions in large masses or large momenta or strong coupling, weak coupling, that these are exactly the objects out of which these asymptotic expansions are built. One of the advantages of this resurgent approach is that this trans series, this generalized series actually encodes analytic continuation information about the function you're trying to describe. Whereas if you just have some formal asymptotic series, it tells you something about the function, but it doesn't tell you everything about the function. And if this is achieved, if you can generate such a trend, in order to reinstate this analytic continuation property. So this was developed by Kyle in the context of dynamical systems and it's been explored in great detail in differential equation theory, difference equation theory, and it's been applied in quantum mechanics, matrix models, quantum field theory and strength theory. The things that are best understood are in the world of differential equations and difference equations. The word resurgence, if you're not familiar with this, here's an image to think of that there are some critical points in your problem, think of them as singularities of a Burrell transform of a formal interpretive series that's asymptotic. And so there are fluctuations around these singular points. And the idea is that the behavior near any given singular point encodes information about the behavior near other singular or critical points. And so these functions in some sense resurrect or surge up again, resurge near these other critical points. So that's the general perspective. And we can make it much more concrete once we're in a particular realm of talking about an integral equation or a difference equation or whatnot. But the basic advantage from the appeal of this formalism from the physics point of view is that it means that there should be some quantitative relationship between the fluctuations around different critical points, which if we think in terms of a functional integral would mean different saddle points of the functional integral. So what about going towards quantum field theory? So there are many examples at this point in matrix models and quantum mechanics, although having said that there is a lot of rigorous foundations to be solidified here. Once you go to quantum field theory, the fact that we have to deal with renormalization makes this whole story much more interesting because the expansion parameter is not just some small expansion parameter. It's actually running according to some well-defined renormalization group structure. There's been a lot of recent progress in the last five to 10 years for specifically regularized quantum field theories in some semi-classical regime and also in lattice quantum field theory. What I want to talk about here is something different. The idea is to try and see how this research instructor fits together with this Huff-Valjebre structure. So the starting point is to try and find the simplest possible example where you can explore what this looks like. So the main question I have in red is that if we think of the Dyson-Schringer equations for the quantum field theory, which we usually formulate in a formal, perturbative sense, one interesting question is, do those equations contain all of the information about the quantum field theory? If it happens to be a quantum field theory that has non-trivial, non-perturbative features? If the answer is yes, then you can ask further, how can you actually extract or decode that non-perturbative information from the perturbative formal series in a practical, possibly numerical way? And possibly the most interesting question is, given that this Huff-Valjebre structure is so natural and so elegant and so efficient at the perturbative level, is there some natural way that it fits into these so-called bridge equations of Eqal, which are the relations that relate all these different critical points? So these different critical points are associated with different non-perturbative features. So three papers from Dirk, at least in my thinking have been very influential in how to understand Dyson-Schringer equations. So these have some very beautiful features in them that we can take advantage of here to start thinking along these lines. So one way to phrase one of the conclusions of these papers is that for certain quantum field theories, when you combine the RG equations with this Huff-Valjebre structure, you can reduce some part of the problem to a set of nonlinear ordinary differential equations for the enormous dimensions of the problem. And this is the realm in which resurgence is best understood. There are a lot of rigorous results and practical results for understanding resurgence once you're in the world of talking about nonlinear differential equations. So this is a completely natural place to start. I emphasize that it's just the beginning. It's not the general picture, but I think it's worth having some very concrete examples to understand before trying to formulate the more general case, which would be aiming for something in the world of cage theories. So there are some well-studied examples, and I think we'll hopefully hear from Mark tomorrow, I guess. Maybe he'll say something about this beautiful work on the West Mino model. And recent paper with Michi, we studied the four-dimensional Yukawa model, and there'll be soon hopefully a paper about this six-dimensional Phi-Cube theory, which also Mark and Enrico have a recent paper about that. So just a summary of what's known for a nonlinear ODEs, and using the notation from a famous paper by Ovidu Kostin, you can write an n-thorder nonlinear ODE in first order form. So you make this an n-component vector. So y prime is just some function of the variable and y. And under certain conditions that are not too strenuous, you can form this into a norm form, where there's some formal ketobritz series. And then these are the eigenvalues of the first derivatives of f, and then one over z expansion. And the rest is suppressed at large z. Once you have it in this normal form, it's completely clear that the solution has the form of a formal series, an expansion here, here z is like one over h bar. So an expansion in inverse powers of z. But then because of these terms here, you get exponential contributions. And here you get inverse powers of z, depending on the eigenvalues. So you can diagonalize this with eigenvalues beta 1, beta 2, up to beta n. And then this nonlinear stuff generates some formal series here. So this is an example of a trend series. Here we have just a formal series in inverse powers of z, but it's complemented with these exponentially small beyond all orders terms with some pre-factor fluctuation terms, and also multiplied by formal for the formal series. And there are very rich bridge equations which relate the structure of these various fluctuation series to one another and to the original formal divergent series. And these are called the algebraic bridge equations. And if you form the singularity, the Braille transforms of these functions here, then the singularities of those Braille transforms label exactly the nonparticipative physics terms, which are these terms beyond all orders. These parameters here, sigma 1 up to sigma n, are the n independent parameters, given the boundary conditions of this nth order differential equation. And they're called trend series parameters. And here's a quote from this paper of Kostin. One of the interesting consequences of this structure is that given one formal solution, this thing here, you can actually generate from that a whole family of solutions beyond the formal asymptotics. So what I'm going to do is explore how this actually works in some particular examples of Dyson-Schringer equations. So the easiest one to start with is this example from this paper of David and Dirk of the four-dimensional massless Eucala theory. I don't think I need to spend terribly much time on this. There's a simple Dyson-Schringer equation iterating these self-energy terms. And the anomalous dimension in a momentum scheme defined this way in terms of the renormalized coupling leads to a RG equation that's this nonlinear first-order differential equation. So this is the result of this paper. And now given that, we can study the formal expansion of this. And this was done by David and Dirk. And it has some very interesting combinatorial structure and asymptotic structure. So just rescaled a little bit just to clean things up instead of function of alpha, divide alpha by 4. And here's the nonlinear equation. Again, you see it's quadratic in this function C. C is essentially the rescaled and almost dimension. But it's first-order in the derivative. And this rescaling has been done so that the expansion coefficients here are integers. And this is interesting combinatorially because these numbers correspond to the expansion of the generating function for connected chord diagrams. These numbers also have interesting large-order asymptotics. They diverge factorial E, gamma n plus 1 half. And there are subleading corrections going like gamma of n minus 1 half, gamma of n minus 3 halves. So this is the typical generic type of factorial divergence that is found very often in quantum mechanical and quantum field theory asymptotic expansions. But this can't be the full solution to this differential equation because there are no boundary condition parameters here. And this is a first-order equation. So there must be something else beyond this formal series. And this is a simple example of this idea of a card that we generalize as formal series to a trans series with one trans series parameter. It's a first-order equation. And the first power of this parameter, sigma, would just be this formal series. But the higher powers would correspond to these exponentially suppressed terms. So in practice, it's extremely simple to implement. You simply insert this type of ansatz into the differential equation, collect powers of sigma, and you see that you have a nonlinear equation for C0, the perturbative series. But all of these other ones satisfy linear but inhomogeneous equations. And you can solve them, it turns out, with a little bit of work. And here's the first term here. It's actually expressed as an exponentially small contribution involving C0. So this is an example of this feature of resurgence that this exponentially small contribution, non-trippinative contribution, corresponding to, say, the one instanton correction to this formal series, is explicitly expressed in terms of the formal series itself. Moreover, you can make an asymptotic expansion of this quantity here at small x. And there's this exponentially small instanton factor. And then it's multiplied by another formal series with certain coefficients that you can simply generate very easily since you know this. And now you look at these coefficients and go back and look at this large order behavior of the formal series coefficients. And you notice something very interesting that these numbers here are just identically equal to the coefficients of the subleading corrections to the large order behavior of the perturbative series coefficients. So this is a generic feature of resurgence that the large order behavior of the expansion around a given instanton sector, here the zero instanton sector, show up again in the expansion around another instanton sector. And these coefficients here in C1, this one, five halves, this here, have also some interesting common ethics associated with them, recent work of Carolyn Alley. So we can continue this. We can actually look at these coefficients of the C1 and look at its, sorry, their large order growth. And we generate another set of numbers here. And then we can go to the differential equation and look at the next exponentially suppressed term. And again, we see the same feature. These coefficients here show up in the low order expansion of the fluctuations around a higher exponential term. In fact, you can continue this to all orders in the expansion and generate and completely all orders summation for the full trans series. So just to remind you what we've done, we've generalized this formal, the turbotive series solution to this differential equation, this Deysen-Schringer equation, to a sum over what I'd be calling non-perturbative instanton terms. And each of these, it turns out, can be expressed in terms of C0, the lowest one. And not only that, not only can each of them be expressed in terms of C0. It can completely re-sum to all orders with this generating function here, little f, which is some function of two variables. And you'll notice that this is exactly the structure of this C1 solution here. So this itself is just exponentiated again. And that gives you the all orders complete non-perturbative solution of this Deysen-Schringer equation. So this is an illustration that we can indeed implement this resurgent asymptotics structure of a coil, this general structure. And since this is a first order differential equation in this case, even though it's not linear, we can carry out all of these steps in complete generality. I would also like to mention that another way to do this calculation is using this nice alien derivative structure that Mischi developed, where you use the fact that there's a functional relation satisfied by this function C of x. And applying this alien derivative, which is the thing that gives you this structure of bridge equations on all the singularities of the Braille transform, you can generate exactly the same all orders resummation. So just to summarize, here's the picture again. Imagine we started here with the formal divergent asymptotic expansion coming directly from the perturbative hot foundry structure. Using that nonlinear equation derived by David and Dirk, we can actually generate all of the other non-perturbative terms directly and show that they correspond exactly to the singularities of the Braille transform of the perturbative series and that they're all expressible in terms of the original formal series. So this is reassuring that one can actually do this in complete detail. So this hints the conclusion would be that indeed, these station-tringer equations have all of this non-perturbative information, you just have to do a little bit more work to extract it. So having done that, can turn now to a slightly more complicated and more interesting quantum field theory. Also studied by David and Dirk in their papers, which is Phi-cube theory in six dimensions. I'll use this expansion parameter alpha, the natural expansion parameter alpha. So this theory is interesting because it's asymptotically free. Six dimensions is the critical dimension of this theory. It has a known Lapatop style instant time, so that's a non-perturbative feature of the theory coming from a saddle point of the functional integral, not directly from some diagrammatic perturbative expansion. It's also interesting because it has a renormal chain structure from chains of these bubble diagrams. So all of these features here that I've listed hint at the possibility of interesting non-perturbative physics in this system. So in their work now, 10 years ago, David and Dirk uncovered a third order nonlinear differential equation for the anomalous dimension in this theory. So I've done a little bit of rescaling again to just get things in sort of natural units. And here's the third order equation. So you see there are three derivatives. It's fourth order in nonlinearity in C. It's very interesting, factorized type structure. And David and Dirk studied the perturbative solution to this equation. You just expand in small powers of x. And with some appropriate rescaling here, you have integer expansion coefficients. And I've listed the first few of them here. And as far as is known at this point, these interesting integers do not have any direct combinatorial interpretation. So that's an interesting open problem. Whereas in the Yukawa system, the corresponding sequence of integers had an interpretation in terms of these connected chord diagrams. So David and Dirk generated, I think, 30 or so of these coefficients and found, indeed, factorial divergence establishing that this was an asymptotic expansion. And they also studied the Borel summation of this. So with some more data, and we publicly thank David here for sending us a huge file of coefficients for us to play with. You can generate more refined asymptotics. So first of all, this is 12 at the end, just from the natural rescaling. But the factorial divergence is, in fact, not gamma of n plus 2. It's gamma of n plus 23 over 12. And you can actually generate these subleading corrections. And there are these particular coefficients corresponding to gamma of n plus 1112 and gamma of n plus n minus 112. So this has a very similar form compared to the asymptotics in the previous Yukawa case. However, this was a third order nonlinear differential equation. So from the theorem of the Ducosten, this means that there are actually three missing boundary condition parameters, not just one of the sigma trans series expansion parameters. There must be three of them working somewhere. And to find them is actually quite simple. Just insert an ansatz for the form of the expansion with this exponentially small term beyond all perturbative orders with some as yet undetermined pre-factor power multiplied by a formal series in powers of x. And just inserting that into the differential equation is very easy to see that there are only three possible solutions. And their combinations, lambda and beta, on lambda equals one, lambda equals two, and lambda equals three, with corresponding rational values of beta. And you should recognize this 23 over 12 here as this 23 over 12 here. And there's a very natural reason for that, which I'll explain in a minute. Another interesting feature of this is that these lambda parameters, which remember these are identified with the locations of the singularities of the barrel transform of the formal solution to this problem, they're actually resonant. They're multiples of one another. Whereas for a generic Thoder equation, these three eigenvalues lambda would be at any points in the complex plane. And this is related to the fact that this equation factorizes into this factorized form with one, two, and three here responsible for this one, two, and three. So what I'd like to talk about now is to explore this non-perturbative structure with these three different types of non-perturbative terms. So what it means is that the full trans series structure has the formal perturbative series. So remember, this is the thing coming directly from a formal expansion of this ophalgibra structure. But the full solution to the differential equation can be expanded to the form of a trans series with three trans series parameters, sigma one, sigma two, sigma three, these correspond to these three basic solutions from this trans series under us, and powers of the corresponding instanton factors here with these strange rational factors. And each of these is multiplied by some formal series. And then there are plus dot dot dot where you mix these various terms. But these are the basic series and seed non-perturbative terms. So given that you have an explicit differential equation and you have the form of what the trans series should look like, and if you have access to something like Mapl or Mathematica, you can simply insert this into the differential equation and start generating many terms in this series. So it's not something you want to do by hand, but it's fairly straightforward to do this using some mathematical manipulation program. So for example, you can look at the first few terms of this series here at the k equals one level. So the first instanton term with the lowest, so the dominant exponentially suppressed terms. And the first few coefficients are some rational numbers here, which you can get from the differential equation. But those numbers here should look familiar. They're actually the terms appearing in the subleading corrections to the large order behavior of the expansion of this formal series. So this is again an example of this generic resurgence structure that the large order behavior of the coefficients of this formal series research in a natural form in the low orders of the expansions here around these non-petrofit terms. So this is just one manifestation, one example of this infinite set of resurgence relations implied by a callous bridge equations. So there are all these relations between the different exponential contributions, which remember correspond to the different Braille singularities, they're all related by some intricate algebraic set of relations. And this is one of the examples where we can actually see it in full detail. To go beyond this, so this you can do just by things like studying the large order growth of the coefficients. So you can do ratio tests and then you can accelerate this with things like Richardson acceleration. But to go beyond this, since there are these three different Braille singularities to go beyond this, it's really necessary to invoke not just this sort of combinatorial treatment, but it seems to be much more powerful to go to the Braille plane and start using more analytic properties. So remember what I'm calling non-petrofit of information, these are directly related to these Braille singularities. So remember what a Braille transform does is it takes a formal asymptotic series such as would be produced by this Huff algebra, Dutch and Schwenger equations and converts it by factoring out the overall factorial growth converts it into a convergent Braille transform. But the Braille transform function is convergent, but it has singularities in the Braille plane. And those singularities correspond to these exponentially small corrections to the formal series when you do the inverse Braille transform, which is a Laplace transform. So what from the quantum field theory point of view, the locations of those Braille singularities correspond to what we would call instantons or renormal ones. And the nature of those singularities, whether they're branch cuts or if they're square branch cuts or logarithmic branch cuts or whatever type of branch cuts, the nature of those facts tell you something about the gas inflectuations around those non-petrofit transform objects. And the Stokes constants, which are the constants appearing in front of this overall factorial growth, the Stokes constants tell you something about the strength of these instanton terms and how they relate to one another through these bridge equations. So this is interesting and important information that you would like to be able to extract from the differential equation and, in fact, extract it from just this formal series solution to the differential equation. So to do this, you need to use, first of all, work in the Braille plane where you have some convergent series to deal with. And it turns out that to do this to the required and desired precision requires some new methods of Braille analysis. And this is the work I've been doing recently with the Vidyocustom. A very common method used by both mathematicians and physicists is so-called Braille-Pade, which is once you go to the Braille plane, you have a convergent series and you would like to understand the singularities. But, of course, in practice, you don't have necessarily all the terms of that convergent series. You only have 10 terms or 50 terms or 100 terms. So in order to study the singularities of that function, which is, in the end, just a polynomial, one possible approach, which is very powerful, is to make a Pade approximate to that truncated expansion. And the Pade approximate represents that truncated series as a ratio of polynomials. And so its singularities are necessarily only poles. And this is both good and bad. It's very easy to implement. And it gives you some hint of where the singularities are and what they might be like. But it doesn't give you much more detail beyond that. It gives you a good starting point approximate information. So for example, Pade struggles to represent the branch cut because it's inherently a representation in terms of rational functions. But what it does do is if it identifies the leading singularity in the Burrell plane, the T plane will be the Burrell plane, it would represent a branch cut by putting a whole sequence of poles along this axis accumulating to this branch point. And if you had detailed information about the distribution of these poles and also zeros, you would be able to tell what type of branch cut it is. But a problem with this is that in a nonlinear problem such as we're dealing with here, if there's a Burrell singularity here, there's necessarily a Burrell singularity repeated at all integer multiples of that. This is this generic truncated series structure. So it's hard to see these because these are further singularities, but they might just show up as another pole hidden under these line of poles that are trying to be a branch cut. So there's a way around this, which is to make a conformal map from this cut plane here, bring it within the unit disk. And a feature of this conformal map is that it resolves and separates these singularities. So the leading singularity is placed here and it has its Pade approximation will have a series of poles accumulating to that point indicating the nature of the branch cut. And the second singularity, which is hidden, is actually completely separated up here, also with the line of poles accumulating to it. And so on. So you can resolve this higher order research and structure. And in fact, a theorem, in fact, in this recent paper with Kostin, is that if you use instead of a conformal map, a uniformizing map, this is actually the optimal procedure to resolve and probe these structures. Because after all, what we want to know is not just where these things are, but what type of singularity they have and what is the corresponding Stokes constant. So when I say optimal, in practice, what that can mean is something like here. So this is an example of the Borel transform with a singularity of t equals minus one. So the behavior is known to be one plus t to the minus 35 over 12. And so if you're interested in the coefficient of that singularity, you simply multiply that by this and it should go to a constant. And so here we're approaching minus one and notice the scale here. So using a conformal map in blue, you can get to several digits, the Stokes constant multiplying this behavior. But if you use a uniformizing map, the precision is much, much greater. You get many, many more digits of precision. So that's useful in getting high precision in the Stokes constants. But it's actually more important in other examples or applications. So for example, if you look at the, not just the leading coefficient, leading singularity here, if you try to probe this subleading coefficient, so you're trying to probe this point here, you find, if you look at the asymptotics of the coefficients, it has a dominant contribution here going like gamma of n plus 35 over 12. And there are subleading corrections to that indicating that there's a branch cut at that point. But it turns out there's another singularity on the positive real Braille axis with exactly the same location, so the same distance from the origin. So the radius of convergence here of the Braille transform is say one, normalize it to be one, but both of these singularities are the same distance from the origin. Which means that if you just look at the asymptotics of these coefficients, you have a competition between these two terms, one of which is alternating in sign and the other of which is not. And simple things like ratio tests are not good enough to disentangle these two competing terms, let alone their subleading corrections. However, if you use one of these uniformizing maps, so you map from this doubly cut plane using some particular elliptic function, remember that you can use elliptic functions to map the three punctured sphere into one of these so called geodesic triangles here reflected to form a geodesic quadrilateral. This point here is mapped to here, this point is mapped to here, and these higher singularities are mapped to the boundary of this geodesic quadrilateral region. And once you're in this region here, that's the region in which you can optimally analyze the structure of the Braille transform. So with that information, you can identify all of these coefficients here of these leading and subleading corrections and just to illustrate what this means in the combinatorial language. If you just took the dominant singularity behavior, so this is the fastest factorial growth, which corresponds to this Braille singularity here, and you took the actual coefficients relative to that leading factorial growth, this is the type of behavior you would see, this oscillatory behavior in this wide band. However, if you include also this competing non-alternating growth from here and these subleading corrections, you see that you get this ratio tending to one much more rapidly and to much higher precision. And the reason that there are still these subleading oscillations is that the subleading corrections of this alternating behavior have a factorial growth which is extremely close to the strength of the factorial growth of the leading non-alternating term. It's only a difference of 23 over 12 versus 25 over 12. And working in the regime of just using the expansion coefficients, it's extremely difficult to disentangle these. But once you go to the Braille plane, you have access to more analytic tools for extracting these independent terms and their coefficients using these analytic continuation techniques based on conformal maps and uniformizing maps. So to summarize what we've learned about this scalar phi cube theory, it has a much, much richer non-pertorbitant structure than the Eucarla model, which you can trace back to the fact that the corresponding Dyson-Schwinger equation has a third order structure and a fourth order nonlinearity. There are three independent non-pertorbitant structures which nevertheless interact with one another because of this resonant structure of the Braille singularities lying on top of one another in integer multiples. Nevertheless, we still see evidence of these large order-loader resurgence relations, the deflectuations of the formal pertorbitant solution that was generated by David and Dirk. So if you look at the subleading corrections to the large order growth, those same coefficients show up as the lower order coefficients of the first non-pertorbitant correction to the formal series. And we also see just from the differential equation, this is just a concrete example of the result of Kostin's theorem, that all of the non-pertorbitant terms are generated from linear differential equations which have an inhomogeneous part which are all encoded in terms of the original formal pertorbitant series. So all of these non-pertorbitant terms can ultimately be generated from the formal pertorbitant series. All right, so that brings me to my conclusions. So there are more questions in the conclusions than outcomes, but let me just remind you what the goal of these two examples was. The idea is to ask whether it is indeed possible to learn something about the Klohmfeld theory starting from the pertorbitant Hock-Felgebra structure which is formulated in the language of formal and in-practice asymptotic expansions. And to use the techniques of recertent analysis, recertent asymptotic analysis, to extract from those formal expansions some non-pertorbitant information which when combined with the original formal expansions give you the complete non-pertorbitant solution of the Dyson-Tringer equations. So given you two examples, they're probably the simplest examples you can possibly think of, but the Phi-Cube theory is richer than the Yukar example. But we see that it's possible and it's actually fairly systematic and quite easy to implement. So that's sort of the good news. Now the more interesting question is, is there a more efficient way to do this? So this is a bit of a roundabout way to do things. So we've taken the natural Hock-Felgebra structure of the formal series, used the fact that there is this Hock-Felgebra structure which interacts nicely with the renormalization group equations as a means of generating this pertorbitant information, and then we apply, say, Borel analysis to this to extract the non-pertorbitant trend series structure. However, it would be much nicer if there are a way to directly understand this non-pertorbitant trend series structure in the same Hock-Felgebra context. And there are strong hints that this should be possible and it must be possible, just because in the end, Eichhahl's bridge equations, which relate, which are the resurgence relations, which tell you how to extract the non-pertorbitant information from the original pertorbitant series, those are inherently algebraic. And so I strongly believe that there should be some nice fit between that structure and the Hock-Felgebra structure. However, the bridge equations are extremely general and extremely formal, and it's hard to be very concrete about them unless you're in the formalism of dealing with a couple nonlinear differential equations or difference equations. And so that's why I would motivate these type of questions here as the right place to start because it's probably the simplest venue in which you'll be able to recognize how these two algebraic formalisms could fit together. Another possible direction to try and make this work is to use this very nice result of Michi from his thesis where by studying the ring of formal divergent series, he was able to find a very natural and extremely explicit implementation of these bridge equations and alien derivatives of Eichhahl. It's much, much more concrete than in the completely general resurgent case. And I think that is very likely to be a useful approach to this first question. So in the Eichhahl case, we're able to do this in complete detail, but even in this Phi-Q case, it's not completely clear yet how this works. Another interesting question is more from the quantum field theory side, relates to some of the very nice work that John Gracie, that these Phi-Q theories become a lot more interesting and especially in six dimensions when you're introduced more structure than just having one Phi field. And these types of generalizations would be extremely interesting from the quantum field theory point of view. Another question that's still open is how does this connect to these results of, say, LePatov and of Rinaldin? So these are other sources of non-petitutive physics. And it's not obvious yet how these other sources of non-petitutive physics fit with this half-algebraic picture. Instant times are usually understood as arising from the generic factorial growth of the number of diagrams, and Rinaldin's are usually understood in terms of the behavior of iterated structures of diagrams. So it feels like they should have some natural place in this discussion, but it's not fully understood yet. Other slightly more technical questions may be other Rinaldinization schemes beyond the momentum scheme might be more efficient. And then ultimately to look at some more complicated quantum field theories involving real gauge structure. And so that brings me to my real conclusions, which are the dope, happy birthday, looking forward to many more years of exciting physics and mathematics. And also thank you for your wonderful ideas that have been extremely rich and influenced many people in many fields. So thank you very much and happy birthday. Thanks, Jeroen. Thank you so much. Thanks, Deroen. In the chat there was a whole of crew dimensions that you want to ask a question. Yeah, okay. I didn't see that. Sorry. Hi, Olaf. Yeah, hi, Deroen. Hi, all. And unfortunately I had to come late to your talk, but I have a question to the very beginning of the talk. As I'm sure it was there, can you, could you go to the slide where we are starting ODE is for gamma? In which case, in the Yukawa case? Yes, in the Yukawa case. So here it is in terms of alpha, but let me show you the next slide. Wait, why? Oh, Jeroen, I don't think we can see your slide. Oh, sorry. Did I see the slide? Stop sharing. Okay. Hold on a second. Okay. Why did it stop sharing? Because it's real. Okay, do you see it now? Yes. Yes. Okay. Let me go to the next slide. But there it is. Yes. Yeah. Ah, I see. All right. Yeah, but can you go to the previous slide? Yes. Yeah, because the different. My question is regarding the differential equation for gamma for the anonymous dimension. And I'm asking what, I mean, this differential equation is obtained by David and Dirk in the paper called Exact Solutions. And my question is whether this, you have to put in more than just the Dyson-Schwinger equation to obtain this, right? So you need that it's scalar, so that's in six dimensions and that's a scalar. This is the Eucala. This is the Eucala. The Eucala, yeah, yeah. Okay. The Eucala. I can go to the scalar one if you like. No, no, I just want to know whether there's more than just the Dyson-Schwinger equation to obtain this nonlinear ODE. It's a conversation about the equation. Yeah. But no, no, the renonization group equation you get that the higher gamma, the gamma 2, gamma 3 are recursively given by gamma 1 by this, there's a recursive formula. But this is from the renonization group equation. But I can't see this. So this was a short talk. I didn't go into the fact that given this, you can generate the Green's functions and so on, but I try to keep it nice and explicit here. Yeah, I was talking to Michi yesterday during the evening and this morning I tried to obtain this equation using the renonization group equation and the Dyson-Schwinger equation and I've got the feeling that I'm missing something. And I think that something is about how the explicit integral is built. So there's this remarkable duality between these two and I think that's the key. So there's something missing in A. Yeah, just to, if I remember correctly, it is indeed true what you just mentioned is that this is particular for this insertion into this. So you write down this integral and then you compute the melanchor transform and that goes into this equation. And in the other case, you have a different melanchor transform in the six dimensional scalar case and you get this third order thing. So it is true. This particular equation really depends on the particular integration kernel. Yes, exactly. Okay. Yeah. Thank you, Eric. Yeah, thank you. We have also a question from David. Yes, thank you again, General, for this wonderful exposition. From my work in the 1990s and engage theories, I know that at least at large N, I had a much worse situation with no half plane analyticity. So I didn't know how to undo the Burrell transforms as Laplace transforms because there's intrinsic into ambiguity. Now in the Phi cubed example, you've pointed out in these higher instanton corrections that you don't have half plane analyticity. So is your result intrinsically ambiguous? If I give you the three boundary constants, is there still a unique answer? If you haven't got half plane analyticity? Yes, but that would take a long time to explain. That there is. The answer is yes. Yes. So this is in this 1998 paper of Causton. I can send you some more details about that. But there is a way to define a unique Burrell-Eccal solution to the differential equation. Thank you. Mark Belong, Mark, did you want to ask a question? I would just point to anchor that using the formalism of the Melling transform is quite easy to obtain the equation that the linear equation for gamma t3. Yeah, that's a very natural way to do it. We have one more question from Enrico also. Enrico. Can you hear me? Yes. No, sorry. It was just the same thing that Mark wanted to say. I wanted to answer to Olaf saying exactly that you have a natural way to write Tringer-Diesel equation. You use Melling transform and then you take the first derivative with respect to L, the log, and you have an equation for the anomalous dimension in a general way. Well, okay, that's it. Mark said. Yeah, thank you all. Thank you very much. All right, let's thank Gerald again. Thank you. Thank you. Thank you. Thank you. Thank you.
 Alright, thanks Eric and thanks to the organizers for putting together this conference.  It's been very good.  So I want to talk about some recent work with Michi Burinski and my student Max Monig and  I'll also be mentioning some work done recently with Abidu Kostin who's a mathematician,  done a lot of work in the theory of research and transfer.  So the motivation.  I want to put together, try to put together two things.  One we all know very well, this underlying Hopf-Algebra structure from perturbative  quantum field theory renormalization.  And one of the advantages of this from a practical point of view in quantum field theories that  enables doing perturbative computations and understanding perturbative structures at very  high order that's hard to access in other ways.  I wanted to think about combining this with a Calv's theory of recidion and asymptotics,  which is a very general theory, but we can think of it here as a way of extracting non-perturbative  information from formal perturbative series.  And the advantage from the physics point of view is that there is physics beyond perturbative  expansions, which we call non-perturbative physics.  And the question is whether that has any natural home in the world of Hopf-Algebras.  I'm not able to.  So the idea is to try and use these recidion transfer structures to decode some non-perturbative  information about quantum field theories from their perturbative Hopf-Algebra structure.  So just a few words because people may not be very familiar with recidion transfer is  or a lot of people here are.  The recidion functions can be described in many ways, but one way to think of them is  that they're closed under essentially all operations in analysis.  And the operations will be interested in things like rural transforms and little continuations,  inverse rural transforms, things like that, monodromes.  In applications so far in quantum mechanics and quantum field theories and in matrix models,  the general structure looks something like this, that you have some formal series in  some small parameter X, but you include also these beyond all orders, exponentially small  things which we would call instantons or non-perturbative terms, and also powers of  logarithms.  And the essential result is that iterations of these three basic structures of X and e  to the minus some constant over X and log X generate this huge class of so-called analyzable  functions which is much more general than standard analytic functions.  And we also know from quantum field theory calculations and various asymptotic expansions  in large masses or large momenta or strong coupling, weak coupling, that these are exactly  the objects out of which these asymptotic expansions are built.  One of the advantages of this resurgent approach is that this trans series, this generalized  series actually encodes analytic continuation information about the function you're trying  to describe.  Whereas if you just have some formal asymptotic series, it tells you something about the function,  but it doesn't tell you everything about the function.  And if this is achieved, if you can generate such a trend, in order to reinstate this analytic  continuation property.  So this was developed by Kyle in the context of dynamical systems and it's been explored  in great detail in differential equation theory, difference equation theory, and it's been  applied in quantum mechanics, matrix models, quantum field theory and strength theory.  The things that are best understood are in the world of differential equations and difference  equations.  The word resurgence, if you're not familiar with this, here's an image to think of that  there are some critical points in your problem, think of them as singularities of a Burrell  transform of a formal interpretive series that's asymptotic.  And so there are fluctuations around these singular points.  And the idea is that the behavior near any given singular point encodes information about  the behavior near other singular or critical points.  And so these functions in some sense resurrect or surge up again, resurge near these other  critical points.  So that's the general perspective.  And we can make it much more concrete once we're in a particular realm of talking about  an integral equation or a difference equation or whatnot.  But the basic advantage from the appeal of this formalism from the physics point of view  is that it means that there should be some quantitative relationship between the fluctuations  around different critical points, which if we think in terms of a functional integral  would mean different saddle points of the functional integral.  So what about going towards quantum field theory?  So there are many examples at this point in matrix models and quantum mechanics, although  having said that there is a lot of rigorous foundations to be solidified here.  Once you go to quantum field theory, the fact that we have to deal with renormalization  makes this whole story much more interesting because the expansion parameter is not just  some small expansion parameter.  It's actually running according to some well-defined renormalization group structure.  There's been a lot of recent progress in the last five to 10 years for specifically regularized  quantum field theories in some semi-classical regime and also in lattice quantum field theory.  What I want to talk about here is something different.  The idea is to try and see how this research instructor fits together with this Huff-Valjebre  structure.  So the starting point is to try and find the simplest possible example where you can explore  what this looks like.  So the main question I have in red is that if we think of the Dyson-Schringer equations  for the quantum field theory, which we usually formulate in a formal, perturbative sense,  one interesting question is, do those equations contain all of the information about the quantum  field theory?  If it happens to be a quantum field theory that has non-trivial, non-perturbative features?  If the answer is yes, then you can ask further, how can you actually extract or decode that  non-perturbative information from the perturbative formal series in a practical, possibly numerical  way?  And possibly the most interesting question is, given that this Huff-Valjebre structure  is so natural and so elegant and so efficient at the perturbative level, is there some natural  way that it fits into these so-called bridge equations of Eqal, which are the relations  that relate all these different critical points?  So these different critical points are associated with different non-perturbative features.  So three papers from Dirk, at least in my thinking have been very influential in how  to understand Dyson-Schringer equations.  So these have some very beautiful features in them that we can take advantage of here  to start thinking along these lines.  So one way to phrase one of the conclusions of these papers is that for certain quantum  field theories, when you combine the RG equations with this Huff-Valjebre structure, you can  reduce some part of the problem to a set of nonlinear ordinary differential equations  for the enormous dimensions of the problem.  And this is the realm in which resurgence is best understood.  There are a lot of rigorous results and practical results for understanding resurgence once  you're in the world of talking about nonlinear differential equations.  So this is a completely natural place to start.  I emphasize that it's just the beginning.  It's not the general picture, but I think it's worth having some very concrete examples  to understand before trying to formulate the more general case, which would be aiming for  something in the world of cage theories.  So there are some well-studied examples, and I think we'll hopefully hear from Mark tomorrow,  I guess.  Maybe he'll say something about this beautiful work on the West Mino model.  And recent paper with Michi, we studied the four-dimensional Yukawa model, and there'll  be soon hopefully a paper about this six-dimensional Phi-Cube theory, which also Mark and Enrico  have a recent paper about that.  So just a summary of what's known for a nonlinear ODEs, and using the notation from a famous  paper by Ovidu Kostin, you can write an n-thorder nonlinear ODE in first order form.  So you make this an n-component vector.  So y prime is just some function of the variable and y.  And under certain conditions that are not too strenuous, you can form this into a norm  form, where there's some formal ketobritz series.  And then these are the eigenvalues of the first derivatives of f, and then one over z  expansion.  And the rest is suppressed at large z.  Once you have it in this normal form, it's completely clear that the solution has the  form of a formal series, an expansion here, here z is like one over h bar.  So an expansion in inverse powers of z.  But then because of these terms here, you get exponential contributions.  And here you get inverse powers of z, depending on the eigenvalues.  So you can diagonalize this with eigenvalues beta 1, beta 2, up to beta n.  And then this nonlinear stuff generates some formal series here.  So this is an example of a trend series.  Here we have just a formal series in inverse powers of z, but it's complemented with these  exponentially small beyond all orders terms with some pre-factor fluctuation terms, and  also multiplied by formal for the formal series.  And there are very rich bridge equations which relate the structure of these various fluctuation  series to one another and to the original formal divergent series.  And these are called the algebraic bridge equations.  And if you form the singularity, the Braille transforms of these functions here, then the  singularities of those Braille transforms label exactly the nonparticipative physics  terms, which are these terms beyond all orders.  These parameters here, sigma 1 up to sigma n, are the n independent parameters, given  the boundary conditions of this nth order differential equation.  And they're called trend series parameters.  And here's a quote from this paper of Kostin.  One of the interesting consequences of this structure is that given one formal solution,  this thing here, you can actually generate from that a whole family of solutions beyond  the formal asymptotics.  So what I'm going to do is explore how this actually works in some particular examples  of Dyson-Schringer equations.  So the easiest one to start with is this example from this paper of David and Dirk of the four-dimensional  massless Eucala theory.  I don't think I need to spend terribly much time on this.  There's a simple Dyson-Schringer equation iterating these self-energy terms.  And the anomalous dimension in a momentum scheme defined this way in terms of the renormalized  coupling leads to a RG equation that's this nonlinear first-order differential equation.  So this is the result of this paper.  And now given that, we can study the formal expansion of this.  And this was done by David and Dirk.  And it has some very interesting combinatorial structure and asymptotic structure.  So just rescaled a little bit just to clean things up instead of function of alpha, divide  alpha by 4.  And here's the nonlinear equation.  Again, you see it's quadratic in this function C. C is essentially the rescaled and almost  dimension.  But it's first-order in the derivative.  And this rescaling has been done so that the expansion coefficients here are integers.  And this is interesting combinatorially because these numbers correspond to the expansion  of the generating function for connected chord diagrams.  These numbers also have interesting large-order asymptotics.  They diverge factorial E, gamma n plus 1 half.  And there are subleading corrections going like gamma of n minus 1 half, gamma of n minus  3 halves.  So this is the typical generic type of factorial divergence that is found very often in quantum  mechanical and quantum field theory asymptotic expansions.  But this can't be the full solution to this differential equation because there are no  boundary condition parameters here.  And this is a first-order equation.  So there must be something else beyond this formal series.  And this is a simple example of this idea of a card that we generalize as formal series  to a trans series with one trans series parameter.  It's a first-order equation.  And the first power of this parameter, sigma, would just be this formal series.  But the higher powers would correspond to these exponentially suppressed terms.  So in practice, it's extremely simple to implement.  You simply insert this type of ansatz into the differential equation, collect powers  of sigma, and you see that you have a nonlinear equation for C0, the perturbative series.  But all of these other ones satisfy linear but inhomogeneous equations.  And you can solve them, it turns out, with a little bit of work.  And here's the first term here.  It's actually expressed as an exponentially small contribution involving C0.  So this is an example of this feature of resurgence that this exponentially small contribution,  non-trippinative contribution, corresponding to, say, the one instanton correction to this  formal series, is explicitly expressed in terms of the formal series itself.  Moreover, you can make an asymptotic expansion of this quantity here at small x.  And there's this exponentially small instanton factor.  And then it's multiplied by another formal series with certain coefficients that you  can simply generate very easily since you know this.  And now you look at these coefficients and go back and look at this large order behavior  of the formal series coefficients.  And you notice something very interesting that these numbers here are just identically  equal to the coefficients of the subleading corrections to the large order behavior of  the perturbative series coefficients.  So this is a generic feature of resurgence that the large order behavior of the expansion  around a given instanton sector, here the zero instanton sector, show up again in the  expansion around another instanton sector.  And these coefficients here in C1, this one, five halves, this here, have also some interesting  common ethics associated with them, recent work of Carolyn Alley.  So we can continue this.  We can actually look at these coefficients of the C1 and look at its, sorry, their large  order growth.  And we generate another set of numbers here.  And then we can go to the differential equation and look at the next exponentially suppressed  term.  And again, we see the same feature.  These coefficients here show up in the low order expansion of the fluctuations around  a higher exponential term.  In fact, you can continue this to all orders in the expansion and generate and completely  all orders summation for the full trans series.  So just to remind you what we've done, we've generalized this formal, the turbotive series  solution to this differential equation, this Deysen-Schringer equation, to a sum over what  I'd be calling non-perturbative instanton terms.  And each of these, it turns out, can be expressed in terms of C0, the lowest one.  And not only that, not only can each of them be expressed in terms of C0.  It can completely re-sum to all orders with this generating function here, little f, which  is some function of two variables.  And you'll notice that this is exactly the structure of this C1 solution here.  So this itself is just exponentiated again.  And that gives you the all orders complete non-perturbative solution of this Deysen-Schringer  equation.  So this is an illustration that we can indeed implement this resurgent asymptotics structure  of a coil, this general structure.  And since this is a first order differential equation in this case, even though it's not  linear, we can carry out all of these steps in complete generality.  I would also like to mention that another way to do this calculation is using this nice  alien derivative structure that Mischi developed, where you use the fact that there's a functional  relation satisfied by this function C of x.  And applying this alien derivative, which is the thing that gives you this structure  of bridge equations on all the singularities of the Braille transform, you can generate  exactly the same all orders resummation.  So just to summarize, here's the picture again.  Imagine we started here with the formal divergent asymptotic expansion coming directly from  the perturbative hot foundry structure.  Using that nonlinear equation derived by David and Dirk, we can actually generate all of  the other non-perturbative terms directly and show that they correspond exactly to the  singularities of the Braille transform of the perturbative series and that they're all  expressible in terms of the original formal series.  So this is reassuring that one can actually do this in complete detail.  So this hints the conclusion would be that indeed, these station-tringer equations have  all of this non-perturbative information, you just have to do a little bit more work  to extract it.  So having done that, can turn now to a slightly more complicated and more interesting quantum  field theory.  Also studied by David and Dirk in their papers, which is Phi-cube theory in six dimensions.  I'll use this expansion parameter alpha, the natural expansion parameter alpha.  So this theory is interesting because it's asymptotically free.  Six dimensions is the critical dimension of this theory.  It has a known Lapatop style instant time, so that's a non-perturbative feature of the  theory coming from a saddle point of the functional integral, not directly from some diagrammatic  perturbative expansion.  It's also interesting because it has a renormal chain structure from chains of these bubble  diagrams.  So all of these features here that I've listed hint at the possibility of interesting non-perturbative  physics in this system.  So in their work now, 10 years ago, David and Dirk uncovered a third order nonlinear  differential equation for the anomalous dimension in this theory.  So I've done a little bit of rescaling again to just get things in sort of natural units.  And here's the third order equation.  So you see there are three derivatives.  It's fourth order in nonlinearity in C.  It's very interesting, factorized type structure.  And David and Dirk studied the perturbative solution to this equation.  You just expand in small powers of x.  And with some appropriate rescaling here, you have integer expansion coefficients.  And I've listed the first few of them here.  And as far as is known at this point, these interesting integers do not have any direct  combinatorial interpretation.  So that's an interesting open problem.  Whereas in the Yukawa system, the corresponding sequence of integers had an interpretation  in terms of these connected chord diagrams.  So David and Dirk generated, I think, 30 or so of these coefficients and found, indeed,  factorial divergence establishing that this was an asymptotic expansion.  And they also studied the Borel summation of this.  So with some more data, and we publicly thank David here for sending us a huge file of coefficients  for us to play with.  You can generate more refined asymptotics.  So first of all, this is 12 at the end, just from the natural rescaling.  But the factorial divergence is, in fact, not gamma of n plus 2.  It's gamma of n plus 23 over 12.  And you can actually generate these subleading corrections.  And there are these particular coefficients corresponding to gamma of n plus 1112 and  gamma of n plus n minus 112.  So this has a very similar form compared to the asymptotics in the previous Yukawa case.  However, this was a third order nonlinear differential equation.  So from the theorem of the Ducosten, this means that there are actually three missing  boundary condition parameters, not just one of the sigma trans series expansion parameters.  There must be three of them working somewhere.  And to find them is actually quite simple.  Just insert an ansatz for the form of the expansion with this exponentially small term  beyond all perturbative orders with some as yet undetermined pre-factor power multiplied  by a formal series in powers of x.  And just inserting that into the differential equation is very easy to see that there are  only three possible solutions.  And their combinations, lambda and beta, on lambda equals one, lambda equals two, and  lambda equals three, with corresponding rational values of beta.  And you should recognize this 23 over 12 here as this 23 over 12 here.  And there's a very natural reason for that, which I'll explain in a minute.  Another interesting feature of this is that these lambda parameters, which remember these  are identified with the locations of the singularities of the barrel transform of the formal solution  to this problem, they're actually resonant.  They're multiples of one another.  Whereas for a generic Thoder equation, these three eigenvalues lambda would be at any points  in the complex plane.  And this is related to the fact that this equation factorizes into this factorized form with  one, two, and three here responsible for this one, two, and three.  So what I'd like to talk about now is to explore this non-perturbative structure with these  three different types of non-perturbative terms.  So what it means is that the full trans series structure has the formal perturbative series.  So remember, this is the thing coming directly from a formal expansion of this ophalgibra  structure.  But the full solution to the differential equation can be expanded to the form of a trans  series with three trans series parameters, sigma one, sigma two, sigma three, these correspond  to these three basic solutions from this trans series under us, and powers of the corresponding  instanton factors here with these strange rational factors.  And each of these is multiplied by some formal series.  And then there are plus dot dot dot where you mix these various terms.  But these are the basic series and seed non-perturbative terms.  So given that you have an explicit differential equation and you have the form of what the  trans series should look like, and if you have access to something like Mapl or Mathematica,  you can simply insert this into the differential equation and start generating many terms in  this series.  So it's not something you want to do by hand, but it's fairly straightforward to do this  using some mathematical manipulation program.  So for example, you can look at the first few terms of this series here at the k equals  one level.  So the first instanton term with the lowest, so the dominant exponentially suppressed terms.  And the first few coefficients are some rational numbers here, which you can get from the differential  equation.  But those numbers here should look familiar.  They're actually the terms appearing in the subleading corrections to the large order  behavior of the expansion of this formal series.  So this is again an example of this generic resurgence structure that the large order behavior  of the coefficients of this formal series research in a natural form in the low orders  of the expansions here around these non-petrofit terms.  So this is just one manifestation, one example of this infinite set of resurgence relations  implied by a callous bridge equations.  So there are all these relations between the different exponential contributions, which  remember correspond to the different Braille singularities, they're all related by some  intricate algebraic set of relations.  And this is one of the examples where we can actually see it in full detail.  To go beyond this, so this you can do just by things like studying the large order growth  of the coefficients.  So you can do ratio tests and then you can accelerate this with things like Richardson  acceleration.  But to go beyond this, since there are these three different Braille singularities to go  beyond this, it's really necessary to invoke not just this sort of combinatorial treatment,  but it seems to be much more powerful to go to the Braille plane and start using more  analytic properties.  So remember what I'm calling non-petrofit of information, these are directly related  to these Braille singularities.  So remember what a Braille transform does is it takes a formal asymptotic series such  as would be produced by this Huff algebra, Dutch and Schwenger equations and converts  it by factoring out the overall factorial growth converts it into a convergent Braille  transform.  But the Braille transform function is convergent, but it has singularities in the Braille plane.  And those singularities correspond to these exponentially small corrections to the formal  series when you do the inverse Braille transform, which is a Laplace transform.  So what from the quantum field theory point of view, the locations of those Braille singularities  correspond to what we would call instantons or renormal ones.  And the nature of those singularities, whether they're branch cuts or if they're square  branch cuts or logarithmic branch cuts or whatever type of branch cuts, the nature of  those facts tell you something about the gas inflectuations around those non-petrofit  transform objects.  And the Stokes constants, which are the constants appearing in front of this overall factorial  growth, the Stokes constants tell you something about the strength of these instanton terms  and how they relate to one another through these bridge equations.  So this is interesting and important information that you would like to be able to extract from  the differential equation and, in fact, extract it from just this formal series solution to  the differential equation.  So to do this, you need to use, first of all, work in the Braille plane where you have some  convergent series to deal with.  And it turns out that to do this to the required and desired precision requires some new methods  of Braille analysis.  And this is the work I've been doing recently with the Vidyocustom.  A very common method used by both mathematicians and physicists is so-called Braille-Pade,  which is once you go to the Braille plane, you have a convergent series and you would  like to understand the singularities.  But, of course, in practice, you don't have necessarily all the terms of that convergent  series.  You only have 10 terms or 50 terms or 100 terms.  So in order to study the singularities of that function, which is, in the end, just a  polynomial, one possible approach, which is very powerful, is to make a Pade approximate  to that truncated expansion.  And the Pade approximate represents that truncated series as a ratio of polynomials.  And so its singularities are necessarily only poles.  And this is both good and bad.  It's very easy to implement.  And it gives you some hint of where the singularities are and what they might be like.  But it doesn't give you much more detail beyond that.  It gives you a good starting point approximate information.  So for example, Pade struggles to represent the branch cut because it's inherently a representation  in terms of rational functions.  But what it does do is if it identifies the leading singularity in the Burrell plane,  the T plane will be the Burrell plane, it would represent a branch cut by putting a whole  sequence of poles along this axis accumulating to this branch point.  And if you had detailed information about the distribution of these poles and also zeros,  you would be able to tell what type of branch cut it is.  But a problem with this is that in a nonlinear problem such as we're dealing with here, if  there's a Burrell singularity here, there's necessarily a Burrell singularity repeated  at all integer multiples of that.  This is this generic truncated series structure.  So it's hard to see these because these are further singularities, but they might just  show up as another pole hidden under these line of poles that are trying to be a branch  cut.  So there's a way around this, which is to make a conformal map from this cut plane here,  bring it within the unit disk.  And a feature of this conformal map is that it resolves and separates these singularities.  So the leading singularity is placed here and it has its Pade approximation will have  a series of poles accumulating to that point indicating the nature of the branch cut.  And the second singularity, which is hidden, is actually completely separated up here,  also with the line of poles accumulating to it.  And so on.  So you can resolve this higher order research and structure.  And in fact, a theorem, in fact, in this recent paper with Kostin, is that if you use instead  of a conformal map, a uniformizing map, this is actually the optimal procedure to resolve  and probe these structures.  Because after all, what we want to know is not just where these things are, but what  type of singularity they have and what is the corresponding Stokes constant.  So when I say optimal, in practice, what that can mean is something like here.  So this is an example of the Borel transform with a singularity of t equals minus one.  So the behavior is known to be one plus t to the minus 35 over 12.  And so if you're interested in the coefficient of that singularity, you simply multiply that  by this and it should go to a constant.  And so here we're approaching minus one and notice the scale here.  So using a conformal map in blue, you can get to several digits, the Stokes constant  multiplying this behavior.  But if you use a uniformizing map, the precision is much, much greater.  You get many, many more digits of precision.  So that's useful in getting high precision in the Stokes constants.  But it's actually more important in other examples or applications.  So for example, if you look at the, not just the leading coefficient, leading singularity  here, if you try to probe this subleading coefficient, so you're trying to probe this  point here, you find, if you look at the asymptotics of the coefficients, it has a dominant contribution  here going like gamma of n plus 35 over 12.  And there are subleading corrections to that indicating that there's a branch cut at that  point.  But it turns out there's another singularity on the positive real Braille axis with exactly  the same location, so the same distance from the origin.  So the radius of convergence here of the Braille transform is say one, normalize it to be one,  but both of these singularities are the same distance from the origin.  Which means that if you just look at the asymptotics of these coefficients, you have a competition  between these two terms, one of which is alternating in sign and the other of which is not.  And simple things like ratio tests are not good enough to disentangle these two competing  terms, let alone their subleading corrections.  However, if you use one of these uniformizing maps, so you map from this doubly cut plane  using some particular elliptic function, remember that you can use elliptic functions to map  the three punctured sphere into one of these so called geodesic triangles here reflected  to form a geodesic quadrilateral.  This point here is mapped to here, this point is mapped to here, and these higher singularities  are mapped to the boundary of this geodesic quadrilateral region.  And once you're in this region here, that's the region in which you can optimally analyze  the structure of the Braille transform.  So with that information, you can identify all of these coefficients here of these leading  and subleading corrections and just to illustrate what this means in the combinatorial language.  If you just took the dominant singularity behavior, so this is the fastest factorial  growth, which corresponds to this Braille singularity here, and you took the actual  coefficients relative to that leading factorial growth, this is the type of behavior you would  see, this oscillatory behavior in this wide band.  However, if you include also this competing non-alternating growth from here and these  subleading corrections, you see that you get this ratio tending to one much more rapidly  and to much higher precision.  And the reason that there are still these subleading oscillations is that the subleading  corrections of this alternating behavior have a factorial growth which is extremely close  to the strength of the factorial growth of the leading non-alternating term.  It's only a difference of 23 over 12 versus 25 over 12.  And working in the regime of just using the expansion coefficients, it's extremely difficult  to disentangle these.  But once you go to the Braille plane, you have access to more analytic tools for extracting  these independent terms and their coefficients using these analytic continuation techniques  based on conformal maps and uniformizing maps.  So to summarize what we've learned about this scalar phi cube theory, it has a much, much  richer non-pertorbitant structure than the Eucarla model, which you can trace back to  the fact that the corresponding Dyson-Schwinger equation has a third order structure and a  fourth order nonlinearity.  There are three independent non-pertorbitant structures which nevertheless interact with  one another because of this resonant structure of the Braille singularities lying on top  of one another in integer multiples.  Nevertheless, we still see evidence of these large order-loader resurgence relations, the  deflectuations of the formal pertorbitant solution that was generated by David and  Dirk.  So if you look at the subleading corrections to the large order growth, those same coefficients  show up as the lower order coefficients of the first non-pertorbitant correction to the  formal series.  And we also see just from the differential equation, this is just a concrete example  of the result of Kostin's theorem, that all of the non-pertorbitant terms are generated  from linear differential equations which have an inhomogeneous part which are all encoded  in terms of the original formal pertorbitant series.  So all of these non-pertorbitant terms can ultimately be generated from the formal pertorbitant  series.  All right, so that brings me to my conclusions.  So there are more questions in the conclusions than outcomes, but let me just remind you  what the goal of these two examples was.  The idea is to ask whether it is indeed possible to learn something about the Klohmfeld theory  starting from the pertorbitant Hock-Felgebra structure which is formulated in the language  of formal and in-practice asymptotic expansions.  And to use the techniques of recertent analysis, recertent asymptotic analysis, to extract from  those formal expansions some non-pertorbitant information which when combined with the original  formal expansions give you the complete non-pertorbitant solution of the Dyson-Tringer equations.  So given you two examples, they're probably the simplest examples you can possibly think  of, but the Phi-Cube theory is richer than the Yukar example.  But we see that it's possible and it's actually fairly systematic and quite easy to implement.  So that's sort of the good news.  Now the more interesting question is, is there a more efficient way to do this?  So this is a bit of a roundabout way to do things.  So we've taken the natural Hock-Felgebra structure of the formal series, used the fact that there  is this Hock-Felgebra structure which interacts nicely with the renormalization group equations  as a means of generating this pertorbitant information, and then we apply, say, Borel  analysis to this to extract the non-pertorbitant trend series structure.  However, it would be much nicer if there are a way to directly understand this non-pertorbitant  trend series structure in the same Hock-Felgebra context.  And there are strong hints that this should be possible and it must be possible, just  because in the end, Eichhahl's bridge equations, which relate, which are the resurgence relations,  which tell you how to extract the non-pertorbitant information from the original pertorbitant  series, those are inherently algebraic.  And so I strongly believe that there should be some nice fit between that structure and  the Hock-Felgebra structure.  However, the bridge equations are extremely general and extremely formal, and it's hard  to be very concrete about them unless you're in the formalism of dealing with a couple  nonlinear differential equations or difference equations.  And so that's why I would motivate these type of questions here as the right place to start  because it's probably the simplest venue in which you'll be able to recognize how these  two algebraic formalisms could fit together.  Another possible direction to try and make this work is to use this very nice result  of Michi from his thesis where by studying the ring of formal divergent series, he was  able to find a very natural and extremely explicit implementation of these bridge equations  and alien derivatives of Eichhahl.  It's much, much more concrete than in the completely general resurgent case.  And I think that is very likely to be a useful approach to this first question.  So in the Eichhahl case, we're able to do this in complete detail, but even in this  Phi-Q case, it's not completely clear yet how this works.  Another interesting question is more from the quantum field theory side, relates to some  of the very nice work that John Gracie, that these Phi-Q theories become a lot more interesting  and especially in six dimensions when you're introduced more structure than just having  one Phi field.  And these types of generalizations would be extremely interesting from the quantum field  theory point of view.  Another question that's still open is how does this connect to these results of, say,  LePatov and of Rinaldin?  So these are other sources of non-petitutive physics.  And it's not obvious yet how these other sources of non-petitutive physics fit with this half-algebraic  picture.  Instant times are usually understood as arising from the generic factorial growth of the number  of diagrams, and Rinaldin's are usually understood in terms of the behavior of iterated structures  of diagrams.  So it feels like they should have some natural place in this discussion, but it's not fully  understood yet.  Other slightly more technical questions may be other Rinaldinization schemes beyond the  momentum scheme might be more efficient.  And then ultimately to look at some more complicated quantum field theories involving real gauge  structure.  And so that brings me to my real conclusions, which are the dope, happy birthday, looking  forward to many more years of exciting physics and mathematics.  And also thank you for your wonderful ideas that have been extremely rich and influenced  many people in many fields.  So thank you very much and happy birthday.  Thanks, Jeroen.  Thank you so much.  Thanks, Deroen.  In the chat there was a whole of crew dimensions that you want to ask a question.  Yeah, okay.  I didn't see that.  Sorry.  Hi, Olaf.  Yeah, hi, Deroen.  Hi, all.  And unfortunately I had to come late to your talk, but I have a question to the very beginning  of the talk.  As I'm sure it was there, can you, could you go to the slide where we are starting ODE  is for gamma?  In which case, in the Yukawa case?  Yes, in the Yukawa case.  So here it is in terms of alpha, but let me show you the next slide.  Wait, why?  Oh, Jeroen, I don't think we can see your slide.  Oh, sorry.  Did I see the slide?  Stop sharing.  Okay.  Hold on a second.  Okay.  Why did it stop sharing?  Because it's real.  Okay, do you see it now?  Yes.  Yes.  Okay.  Let me go to the next slide.  But there it is.  Yes.  Yeah.  Ah, I see.  All right.  Yeah, but can you go to the previous slide?  Yes.  Yeah, because the different.  My question is regarding the differential equation for gamma for the anonymous dimension.  And I'm asking what, I mean, this differential equation is obtained by David and Dirk in  the paper called Exact Solutions.  And my question is whether this, you have to put in more than just the Dyson-Schwinger  equation to obtain this, right?  So you need that it's scalar, so that's in six dimensions and that's a scalar.  This is the Eucala.  This is the Eucala.  The Eucala, yeah, yeah.  Okay.  The Eucala.  I can go to the scalar one if you like.  No, no, I just want to know whether there's more than just the Dyson-Schwinger equation  to obtain this nonlinear ODE.  It's a conversation about the equation.  Yeah.  But no, no, the renonization group equation you get that the higher gamma, the gamma 2,  gamma 3 are recursively given by gamma 1 by this, there's a recursive formula.  But this is from the renonization group equation.  But I can't see this.  So this was a short talk.  I didn't go into the fact that given this, you can generate the Green's functions and  so on, but I try to keep it nice and explicit here.  Yeah, I was talking to Michi yesterday during the evening and this morning I tried to obtain  this equation using the renonization group equation and the Dyson-Schwinger equation  and I've got the feeling that I'm missing something.  And I think that something is about how the explicit integral is built.  So there's this remarkable duality between these two and I think that's the key.  So there's something missing in A.  Yeah, just to, if I remember correctly, it is indeed true what you just mentioned is  that this is particular for this insertion into this.  So you write down this integral and then you compute the melanchor transform and that goes  into this equation.  And in the other case, you have a different melanchor transform in the six dimensional  scalar case and you get this third order thing.  So it is true.  This particular equation really depends on the particular integration kernel.  Yes, exactly.  Okay.  Yeah.  Thank you, Eric.  Yeah, thank you.  We have also a question from David.  Yes, thank you again, General, for this wonderful exposition.  From my work in the 1990s and engage theories, I know that at least at large N, I had a much  worse situation with no half plane analyticity.  So I didn't know how to undo the Burrell transforms as Laplace transforms because there's intrinsic  into ambiguity.  Now in the Phi cubed example, you've pointed out in these higher instanton corrections that  you don't have half plane analyticity.  So is your result intrinsically ambiguous?  If I give you the three boundary constants, is there still a unique answer?  If you haven't got half plane analyticity?  Yes, but that would take a long time to explain.  That there is.  The answer is yes.  Yes.  So this is in this 1998 paper of Causton.  I can send you some more details about that.  But there is a way to define a unique Burrell-Eccal solution to the differential equation.  Thank you.  Mark Belong, Mark, did you want to ask a question?  I would just point to anchor that using the formalism of the Melling transform is quite  easy to obtain the equation that the linear equation for gamma t3.  Yeah, that's a very natural way to do it.  We have one more question from Enrico also.  Enrico.  Can you hear me?  Yes.  No, sorry.  It was just the same thing that Mark wanted to say.  I wanted to answer to Olaf saying exactly that you have a natural way to write Tringer-Diesel  equation.  You use Melling transform and then you take the first derivative with respect to L, the  log, and you have an equation for the anomalous dimension in a general way.  Well, okay, that's it.  Mark said.  Yeah, thank you all.  Thank you very much.  All right, let's thank Gerald again.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.
Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_0_47.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_48_119.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_120_395.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_396_1115.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_1116_6851.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_6852_8495.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_8496_11543.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_11544_11627.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_11628_11687.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_11688_11747.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_11748_11831.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_11832_12383.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_12384_14567.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_14568_18863.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_18864_19979.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_19980_22499.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_22500_24455.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_24456_24503.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_24504_25667.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_25668_26747.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_26748_26819.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_26820_26891.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_26892_27515.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_27516_27827.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_27828_27935.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_27936_29603.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_29604_30911.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_30912_34319.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_34320_38447.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_38448_38603.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_38604_38951.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_38952_39659.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_39660_39707.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_39708_44339.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_44340_46535.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_46536_46679.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_46680_46919.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_46920_47255.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_47256_53051.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_53052_54335.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_54336_54527.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_54528_54635.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_54636_54839.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_54840_54911.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_54912_56507.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_56508_57647.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_57648_58151.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_58152_60299.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_60300_62567.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_62568_70883.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_70884_71687.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_71688_71795.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_71796_71939.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_71940_72059.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72060_72107.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72108_72191.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72192_72251.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72252_72311.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72312_72383.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72384_72467.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72468_72539.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72540_72599.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72600_72839.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72840_72887.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72888_72935.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_72936_73115.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_73116_73283.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_73284_78107.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_78108_78155.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_78156_78203.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_78204_78251.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_78252_78551.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_78552_78611.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_78612_78695.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_78696_79451.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_79452_79511.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_79512_79559.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_79560_79631.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_79632_81071.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_81072_81119.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_81120_81179.png Gerald_DUNNE_-_Resurgent_Trans-series_Analysis_of__29ENkKqUDvI_81180_81259.png