 Okay, thank you very much for the invitation to come to this conference. I'm very happy to be here and I'm going to be telling you about some work that I did with my student Thomas Reyes, trying to apply the techniques of resurgence to physical systems superconductors. So what is the motivation for this? Well as you have seen in the talks today, precision resurgence, I mean using resurgence really in its full glory has been applied very much into quantum mechanics, matrix models, and so on. But these are in a sense toy models, especially now that we have an engagement with the European taxpayers with our ERC grant, we should try to move to something a little bit more serious, and more... I know, I know, but the course can be destroyed as you know very well, and also more from a point of view of mathematics, I think it's very interesting to find new sources for trans-series, and I think I will try to present you here a really very concrete source for trans-series. Now one possibility you want to enlarge the examples where resurgence can be applied is to look at quantum many-body systems in low dimensions, because I think they offer a very good balance between analytic control and physical relevance, and sometimes they are simpler than quantum field theories. We have seen the talk by Mithat where he has been already addressing issues of resurgence in quantum field theories, but in a sense many quantum many-body systems are even simpler than quantum field theories, although they share the same flavor of quantum field theories. Now there is some more concrete motivation and it's the following. Now in resurgence, we search for non-pertreative effects, so we really want to correct perturbative expansions with exponentially small effects in coupling constants, but actually one of the most important non-pertreative effects in quantum physics, and one effect which is actually something that people have measured, etc., is the superconducting energy gap. And this is a robust, I will explain you a little bit more what it is. I understand that most of the audience here is not an expert in quantum many-body theory. Mithat, am I, by the way, so this is also a disclaimer, not an expert in quantum many-body theory, so I'm going to tell you what is the superconducting energy gap, but this is a non-protreative effect in quantum physics, exponentially small in the coupling constant, and it's a robust feature of Fermi monibody systems with an attractive interaction. You have a weakly coupled Fermi system with an attractive interaction, there will be a superconducting energy gap. And this actually, this effect feature very importantly in the famous paper by Bardin, Cooper and Schroeder in 1957, where the theory, the modern theory of standard superconductors was built. Now, once you see a non-protreative effect, something which is exponentially small in the coupling constant, if you are doing resurgence, another question is, can you incorporate this effect in some way or another in the theory of resurgence? Is, for example, to make the question more concrete, can you relate this exponentially small effect, for example, to the larger revivigelo of the perturbative series? So, this is the kind of question that I want to explore in this talk. So, let me then make a short summary what is the superconducting energy gap. So, you might have a system of the spin-halve thermions, which have the standard non-relativistic kinetic term, and then you have like an interaction, and this is a two-body interaction, so I'm using the language of quantum field theory to describe this thermosystem. So, this is a two-body interaction, and for simplicity, this is a delta function interaction, okay? In one dimension, the delta function interaction actually is very well behaved. In two dimensions, higher dimensions have to do a little bit of work, but you can also consider it. And for simplicity, I will consider this kind of system. So, you have a coupling constant here, g, m is the mass of the fermions, sigma is the spin of the fermions, and so on. So, this is a system of weakly interactive fermions which interact with this delta function interaction, and in mind that you want just to compute the energy of the ground state of this system of fermions. Now, of course, this is a basic question in quantum-many-body theory, and you can answer these questions with Feynman diagrams. Now, you have to use the Feynman diagrams of many multi-body theory. The energy, ground state energy of this system is going to be the energy of a system of free fermions, and then you will start adding corrections all of Feynman. So, the first correction is what is called the Hartree term. Then, you have the Fock term, and you have more complicated diagrams, and you have many series of diagrams. And function of one variable. Yeah, yeah, this is a function of m and g. So, g, you can fix m, of course, you have to, this is going to... No, no, no, no, no, psi itself is thermo... So, psi is the thermo field, okay? It's a... It's a non-special variable. What do you mean? No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, Oh this is a Hamiltonian density, okay? So you integrate over this. This is, this is a field. This is a two-dimension? No, this is any dimension, okay? Then, this is a Hamiltonian density, you integrate over the space, whatever it is, and you get the energy, okay? Now, this is... I'm going to focus later on one spatial dimension, but this is impr harmful gene. And, again, you know, this expansion is generic for any dimension. Now, if you change dimension, you will have to change the type of integral as you do, but the Feynman diagram structure is the same for any dimension. Sorry, but just to make sense, should we make sure it depends on what? If we make sure it depends on the language you're using. You're using the grand canonical ensemble, then this depends on the special directions, and then on the tau, on the Euclidean tan direction, which is related to the temperature and so on. And the sum is always the same. The sum here is over spins. Here, already the spins are already fixed. This is not a standard Hamiltonian formula for thermions. It's been helped. You do have three up and one down. Okay, sorry, this is a misprint. You have this, this will be down, okay? Okay. Now, this is really a standard textbook thing apart from my misprint, which is also standard in some textbooks. Now, if you want to, if you have a superconductor, many condensed matter theories will tell you that actually summing these five-man diagrams is not a good point of view. What they will tell you is that you have to use the point of view of BCS theory, and they will tell you that the ground state is not really a perturbed free-firm system, but really a condensate of Cooper pairs. And they will actually write for you a nice formula for the energy of the system, which is essentially going to be the free energy part minus delta square. And delta, a weak coupling when G is very small. And delta is exponentialismo, it's an exponentialismo effect, and is what is called the superconducting energy gap. And this is the binding energy of a Cooper pair. So what happens is that fermions with opposite spin pair together, and they form some sort of condensate, and if you want to excite the system, what you have to do is to break these Cooper pairs, you know, have less energy than independent fermions, then you want to excite the system, you have to break one Cooper pair, and this is the energy of one single Cooper pair, so you break one Cooper pair, you excite the system from the ground state to the excited state. So you see the energy has this dependence on delta square, and if you just want to excite the system, you just have a gap of delta. So delta is actually the energy gap between the ground state and the first excited state. Now, one modern point of view of VCS theory of this point of view is that what you are doing is really a large approximation to the system. So you could imagine that you have n flavors of fermions, so you have an additional index for the fermions. And then, when you do this analysis of the CLR chain, the path integral has two different saddle points. You have the standard free system in which the fermions are free, and then you have a more stable state in which the fermions have the energy E3 minus this, no exponentially small non-analytic term. So this is what people sometimes call, talk about the Cooper instability, because this condense state is supposed to be more stable than the standard state. Now, this is not an analysis based on ordinary perturbation theory. This is not the analysis that you would get based on the Feynman diagram. This is a different point of view which you can interpret as a large chain point of view. One question that I would like to address as well is how this instability is actually reflected in ordinary perturbation theory. Now, in a sense, the energy gap, this quantity here, sets the non-pertuative scale of the theory. So this is a very important quantity of the system. So you must expect that anything which is exponentially small in G is going to depend on this energy gap. So I'm going to state for you two conjectures that I will try to illustrate later, so let me give you the conjectures here. Now, the conjecture is that, the first conjecture is that the perturbative series of the ground state, the one that you can compute with the standard Feynman diagrams, forget about BCS theory, just start computing diagrams. The perturbative series of the ground state energy of a superconductor is factorially divergent and non-bottled sum of. That's my first conjecture. And the second conjecture is that the first singularity in the Borel plane for this number of volts series is determined precisely by the energy gap. So the energy gap sets the scale for the first Borel singularity. And let me tell you that this, my look trivial, must look simple. I mean, you have a non-pertuative scale, which is this energy gap, and then, you know, this should be the non-pertuative scale in the problem and then it should also determine the position of the Borel singularity. But people have not always been of this advice. People, for example, have thought for a long time that in theories with fermions, you wouldn't even have factorial growth in the perturbative series, because fermions, you know, when you have fermions, things can have plus or minus signs and there will be cancellations that will make the growth of perturbative series smaller than factorial, okay? So these have been previous works arguing that you don't have such a factorial growth. So my estimate is that, yes, you do have a factorial growth, and moreover, this factorial growth is not related to a semi-classical configuration, and this is related to the talk by Mithath, but it's more like a renormal type of singularity. So we will have, so what I claim here is that the superconductive energy gap is like a renormal, I would, an infrared renormal for this many body system, okay? So these are the conjectures. I could give you, I can give you an additional argument maybe at the end of the talk, but when you have a conjecture like this, what is good is to have a simple model where you can do explicit calculations and eventually test the conjecture. So what is the simple model of a superconductor that you have in the market? Well, this is probably not the simplest model, but this one of the most beautiful models is called the Godin-Yam model. So Godin was working in Sacle, if I understand correctly. So this is also like a product of this South area of Paris. This is a one-dimensional spin-hal fermion with a delta-fansion interaction. So it's the one-dimensional case of the Hamiltonian nest that I showed you at the very beginning. Now this model was introduced and solved in 1967 and it's a close crossing of another fermions model in condensed matter which is the leaf-linear model which describes not fermions but bosons also interacting with a delta-fansion interaction, with a delta-fansion interaction. And why these models are interesting to study? Well, if you want analytic control, it's good to have models which can be solved exactly and these both models can be solved exactly with the better answer equations. So how do I do, let me just summarize very quickly for you how this better answer equation looks like. Now the effective coupling of the model is the dimensionless coupling is the coupling constant g and now in one dimension and you have to divide by the density of particles. I will work all the time in the thermodynamic limit. The number of particles is infinite, the number, the dimension, the volume or the length of the one dimensional space is infinite but the quotient is finitely fixed to be n. Then there is an integral equation which describes the density of better roots. You don't know what is a better root, think about fx as a sort of auxiliary function which gives a result for the problem. It's a very simple integral equation with a very simple kernel. There is a parameter b here which appears in this integral equation and from the solution of this integral equation for a given b, you can find the coupling, one over gamma is the integral of this function over b, so if you want this is a relation which gives, this relation gives you the way in which b is related to the effective coupling constant and then by integrating the second moment of this distribution, you get actually the ground state energy divided by n cubed, which is again the dimensionless quantity. So this is really a complete solution of the problem in the sense that if you just solve these two equations numerically, for example, here is the solution, for example, this is gamma and this is the energy. Here, when gamma is zero, you get pi square over 12 which is the energy, the density energy of a free system of thermos in one dimension and then as the interaction increases you can see the energy becomes negative and this is the energy. This is just the solution of this integral equation. You can do it numerically. Now, you say, well, you have solved the problem. What's the point? Well, the point is that this is just a number, is this, this integral equation cannot be solved analytically. This is just a number that you get and what you can now try to do is, as Ricardo was mentioning this morning, you can try to do a semi-classical decoding of this quantity. So can we write this quantity as some generalized Borel-Ecal resumation of a trans-series? This is the first question that you have to do and when you want to write a trans-series, the first thing you have to do is to find the perturbative series, just the perturbative expansion in gamma. Of course, we could do this with finite diagrams but the next question you should ask is, can I extract from these sets of integral equations the perturbative expansion of the problem? Now, before doing this, let me remind you that the physics of this Godinian model is actually the physics of a superconductor. The grand state of this model is a BCS-like state made out of copper pairs and you can compute the energy gap, an important calculation you can do and it's exponentially small in defective coupling constant with a very precise weight. And you can see from the Betyans' equations that fermions here actually pair up. For copper pairs, you have a given number of spin-up fermions and spin-down fermions, they will pair up and if the number of spin-up fermions and spin-down fermions is different there will be some isolated fermions of the kind of couple too but the absolute ground state of the system occurs when there is equal number of spin-up fermions and equal number of spin-down fermions and all of them are paired up forming copper pairs. And this is the case I will consider, this is the case what is called zero polarization, equal number of spin-up and spin-down fermions, everybody's in a copper pair. So this is really a toy model for a superconductor in one dimension. It's not such toy models because people are actually striving to implement experimentally these systems and they are quite close to do it, they have not done it yet. So let's come back to this question of extracting the perturbed system from the Betyans'. Now it turns out that this is a very difficult problem. And if you read the paper of Liv Linniger which is the paper which essentially introduced the Betyans' in condensed matter systems, you find that extracting the perturbed system was something that they could not do. And this is just because zero coupling is a singular limit of these Godan's integral equations. So it's a problem that you find both in this Godanian model and in the Liv Linniger's model. Now after 50 years, there are only the first two coefficients of these series are known analytically. And this was done by Idan Guadatti, Tracy and Whedon wrote various papers in the last four years trying to do this. As I said, it's the same problem in the Linniger model. So the first thing you have to do if you want to do resurgence is try to extract the perturbed disease from these Betyans'. It turns out that you can actually solve this problem. There is a very nice old paper by Popov, the same one of Father Jeff Popov, where he actually already found the right strategy to do based on previous works by mathematicians. And there was a very nice technical breakthrough by Dimitri Volling in 2009 in the context of ADSEFT that you can adapt to this problem. Yes. In terms of all this, I can take this as a one-dimensional integrable system. Yes. And because your rules precisely come from the corresponding Betyans' way. Yes. And the question is, can I address this question from the quantum spectral formalism like Dimitri? Well, I don't know if you can do it from the quantum spectral curve. I mean, I don't... You know, the question is you want to extract this perturbed disease and that's the problem you want to solve. The strategy that I'm going to show you is not based on the quantum spectral curve. It's just like different. Maybe it turns out to be the same thing, but I don't know any other way of doing this. And as you see, it's not I. I mean, people have even done numerical fitting of these coefficients. The kind of things that Ricardo was showing today, you compute a lot of numbers and then you try to compute these coefficients numerical and you try to fit into exact expressions. So let me give you a very short summary. This is in a sense the kind of technical, the kernel, technical kernel of our method was actually to solve this problem. So the strategy is very simple. You find, you write two answers for the integral equation. The integral equation describes a density of roots, okay? So it's a function like this and it ends up in B and minus B and then you write two answers for this solution. One is around the origin, another is around the boundary and you try to match these things. This is an old idea which goes back to the world of Hadso in the 50s. Popovri discovered this idea in the 70s. But doing this doesn't take you very far. With a lot of work, you can get two coefficients. This is what people did until very recently. Now, there is a key improvement which is based on the idea of Ivan Kostov, Cervan and Wolling, again, Sackler and Worker and Dimitri Wolling to use the resolvent of this density. So this is the function that you want to solve and it turns out that it's much better to use the resolvent of this function which is defined in this way. Now, once you find this resolvent, it turns out that the inverse Laplace transform of this resolvent near the boundary and in the limit of infinite B can be obtained in closed form and this is in a sense the key ingredient that was missing. And this, the plus transform, has a very closed form which is this form here and this function encodes all the information of the perturbative series in a kind of very convoluted way. You have now to extract it from this function but this function which is the, as I said, the inverse Laplace transform of this resolvent near the boundary and in the limit of infinite B has all the information. It looks like one of these anarchist functions that we were seeing this morning, but I don't know if. This is still in form, right? This is still in form. Yeah, yeah, yeah. This is gamma of, yeah, yeah. But as you will see, it has a lot of information in it. Now once you do this fitting, you can actually write the perturbative series and we were able to break this bottleneck of two coefficients and now we can essentially calculate any number of coefficients if we have enough computer program, computer power. We have completed 35 coefficients but we are now getting 40 or whatever. So you see that this is the energy of the free theory. Then you have these two contributions that were computed, were known. And then starting with the order, cubic order, you have the zeta functions evaluated as odd numbers. They do not even occur linearly. You can have problems and squares and so on. And this is the answer. This is the answer for the perturbative series. And we have the first, these first coefficients can be also checked with Feynman diagram techniques. You can actually do the Feynman diagram technique and check that you actually get this number. So this is the exact formal power series for the perturbative energy in the ground state obtained from the beta ansatz. But as I said, this agrees to all the orders we know with Feynman diagram calculation and we expect it's the right answer. Okay, so now, if you want to do resonances for this, in a sense, the easiest way to work, the easiest way to go is to actually look at the larger degree here of the series. How does it divert the series? So you can do a numerical study of the series. So let me define this coefficient Ck and then you can see numerically very easily that first of all, they diverge factorially. So these cancellations in fermion systems is not actually taking place. There is no cancellation between different contributions. And actually, you can see that the divergence is controlled by this pi square. So you can see numerically, after doing some research extrapolation that this quotient actually converges to pi square very fast. And you can actually guess that the non-pertorative, you can actually deduce from this article here that the non-pertorative ambiguity that we have here is presently of this form into the minus pi square over gamma. And this is exactly twice the value of the energy gap or the superconductive energy gap. So this has the exponential dependence of the square energy gap in agreement with the conjecture we were making. Now the further energy gap appears square, I don't have a full justification for it, but you can see that in BCS theory, the correction to the energy involves delta square, not just delta. So this is in sort of heuristic agreement with this. Now of course, this is a non-trivial test. As I said, we have to generate all these series. And it's also good to have other models, but we actually did another test of this conjecture in the Haber model. The Haber model is a sort of discretized version of this model on a lattice. And it has essentially qualitatively the same properties. And you can study this model a whole feeling where the ground state energy series can be calculated analytically. So you have a closed formula expression for this coefficient Ck. And you can check again that the larger the bigger of the series is exactly governed by the superconducting energy gap. So for one of your resurgence, this means that remember that these things can be computed. I mean, this is a number. When you solve the integral equation, you get a number here. This number has to be decoded as a trans-series. The first part of this trans-series is the perturbative series that I just showed you. And then there should be exponentially small terms of this type. This is the simplest answer you can get. And actually, from the numerical results we have, this coefficient, C1, is there just to cancel the imaginary part of the lateral resumations. You can think about this as a sto constant. And its value is plus minus i, from our numerical results. So it's just purely imaginary. Now, when you look at a perturbative series in a quantum theory and you find that the larger behavior is controlled by this polar singularity, the first question you have to ask is, what is the physical meaning of this singularity? What is this singularity representing? What are the sources of Borel's singularities in a quantum theory? Well, essentially, as far as I know, there are two essential sources for this. And this recaps a little bit the discussion that Mithat did in his previous talk. The first sources are instantons. Now, one way that series can diverge factorial is that because at each order, the number, the total number of Feynman diagrams contributing at each order grows factorially. But each diagram has only exponentially growing growth. So exponentially, only growth exponentially. So for example, if you go to this perturbative series that I was showing you here, you can see that this is order 1, this is order 2. So at order n, the number of Feynman diagrams will increase factorially. And this factorial source of growth of the series is what is supposed to be encoded by instanton, by semi-classical configurations of the path integral. No, we provided the definition of normalization. No, no, here everything is finite. You don't have to do renormalization. And this is also interesting regarding what I'm going to do. So this source of factorial growth has a semi-classic interpretation. Now, another source of factorial growth in quantum theory are precisely these renormalons that Mitard was mentioning. So what are the renormalons? The renormalons are special types of diagrams that after you integrate over the Feynman momentum, the Feynman integral, will lead to factorial growth. So it's not the number of diagrams that is increasing. But at a given order, there's a very bad diagram that is giving you these factorial growth when you integrate over momentum. So I'm going to give you some examples. And in general, so you take a random theory, they do not have a semi-classical description. Also, there is the work of Mitard, Gérald, and other people who have found such semi-classical descriptions in many models after this adiabatic compactification. And I have to say that one of the motivations for this work was to try to find scenarios where these ideas of theirs could be tested, maybe in a simpler way. So maybe this is going to provide a simple example of this behavior. Now, renormalon diagrams are absent in many theories. For example, in quantum mechanics or insurance and most theory, there are no renormalon diagrams. Now, in the presence of renormalon diagrams, as Mitard was explaining, instanton effects might become secondary. They give you singularities in the word replay, but are very far from the origin. And then, renormalon singularities are going to be more important than instanton singularities. And this is believed to be the case in QCD, for example. Now, if you do a pure diagrammatic analysis, it's very hard to know exactly where is the location of the renormalon singularities. Now, typically, what you do is you try to argue that some classes of diagrams are going to dominate. They are going to be more important than instantons. And in renormalizable theories, you can use additional tools like renormalization group, operator product expansions to locate the singularities. But in general, if you find that there are special diagrams that lead to factorial divergence after integration over momentum, you can suspect that these are going to be the responsible persons for the singularity that you are obtaining. So what happens in the case of superconductors? What happens in the case of this Godanja model? Now, it has been pointed out sometimes, although this is not universally a knowledge from what I have seen in the condensed matter literature, that there is a special type of diagrams in Fermi system, which give you these divergence. So these are called ladder diagrams. So these are diagrams which are circular like this, and they have ladders. They can have as many ladders as you want. And you can see that at every given order, there is only one single diagram like this. But this diagram, once integrated over momentum, is going to diverge factorial. And this, by the way, are also the diagrams that people use to exhibit the Cooper instability. You can use these diagrams to argue that there is some pole in the 2-point function or in the 4-point function, and these poles will be interpreted as a boundless state of resonance and so on. Now, on the other hand, you use instant underestimates for the Godanja model. As I said, you don't even expect full factorial growth. So there are grounds to believe that this Borel singularity that you observe clearly in the Godanja model is of a renormal type. It is due to some diagrams which are divergent too fast. And we have identified actually two classes of diagrams with factorial growth. Ones are ladder diagrams, the ones that I was showing here. You can actually consider a subset of these diagrams, which are the ones that are called with one pair of whole lines. This is a technicality. If you are not familiar with quantum-many-body theory, it's OK. I was not familiar myself with these things six months ago. You can actually compute the growth of these diagrams and they grow factorially with a factor 2 pi square minus k. So this is twice the singularity that we found. But it turns out that in one dimension, you have another class of diagrams that also diverge factorial. And this actually has not been really appreciated in the condensed matter literature that on top of ladder diagrams, in one dimension, also the standard ring diagrams also diverge factorial. Ring diagrams are very important in condensed matter because these are the diagrams that, for example, people use to actually calculate the energy of a system of interacting electrons. There is this Femios-Gell-Mann-Brugner formula, which gives you the contribution of these ring diagrams to the energy of a gas of electrons or so. It turns out that in one dimension, ring diagrams grow factorially and precisely with the growth that we have observed empirically in the total answer. So I think this shows, this gives a strong indication, this is not a proof, but this is a strong indication that these diagrams, these divergent diagrams, are the ones which are responsible for the Borel-Singrate that you observe in the total energy. Now, it's very interesting if you compare this to standard renormals. Now, in renormals, there are infrared renormals, as Mithal was mentioning, but there are also ultraviolet renormals. And they are called ultraviolet or infrared renormals because the divergence, the factorial growth that they give come from the region of very large momentum or very small momentum. Here, the dangerous region is not the momenta very large or the momenta very small, but are the momenta who are close to the Fermi surface of the Fermi system. So when the momentum of this part of this Cooper-Perser are close to the Fermi surface, then you have a logarithmic singularity in the integrand from these Feynman integrals and you have this factorial growth. So I guess this gives a strong indication that these renormal type of diagrams are actually responsible for this singularity. And actually, analyzing these diagrams is easy in higher dimensions. For example, in three dimensions, you can consider now an actual superconducting system, the kind of superconducting system that people observe, for example, in gas of cold atoms and so on. Those are these systems in three dimensions which are described as a Hamiltonian density. And then, of course, you don't have a better answer that allows you to extract the exapertorative series. But you can analyze ladder diagrams and see how they diverge. And what you find is that the singularity that you find, this ladder has diverged factorially and they lead to a Borel singularity which is twice what is expected from our conjecture. But as I said, this is a restricted type of diagram. So it could happen in three dimensions, this conjecture is not yet settled, but it could happen that if you add more diagrams, you could actually enhance the singularity to what it should be. So the analysis of diagrams is not so classic, but it gives you strong suspicions that first the singularity is not semi-classical and second that it really comes from some families of diagrams. I mean, it's unfortunately here we don't have these additional tools yet, like in quantum theory, to locate the singularity more precisely. Yes. You mean like if I consider this three-dimensional model, there might be contributions from non-letter or other type diagrams? Well, the problem is that the ladder diagrams, yeah, absolutely. No, your question is very good. The question is that when I consider these ladder diagrams, I only consider technically what are called ladder diagrams, one pair of whole lines. So these are not even all the possible ladder diagrams. So if you consider all other diagrams with more than one pair of whole lines, yeah. It's a restricted set of ladder diagrams, absolutely. Yeah, sorry for the imprecision. It's a restricted set of ladder diagrams. But also, you know, we know that ladder diagrams diverse factor, but it's not, I don't think anybody knows for sure that there are no other diagrams that come diverse factorial. I mean, we know, for example, that ring diagrams in three-dimensions do not diverse factorial. They only grow exponentially. But who knows what is there, right? I mean, this is a space of many, many diagrams. It could have something, there could be other-than-genius diagrams, okay. And now, you know, I think in the worst of the cases, if this is not enhanced, this just means that our conjecture has to be slightly changing three-dimensions to allow for a multiple of the singularity. Notice that these are integer multiples of the singularity that I'm telling you. I mean, this is a multiple, this is twice the singularity that our conjecture produced. This restricts the set of ladder diagrams. So, it's not so far off, no? Okay, now, let me now mention another model. This is not a superconductor, but it's very similar to what I have been discussing, which is now the model of bosons, the leaf-linger model. Now, with the same techniques, you can analyze now this model, which is one of the most important models. This model can be implemented in the laboratory. This is really a model where people actually compare almost in a routine way, experiment to theory. And it can describe bosons with repositive data function interaction and can be so with the data answers. Now, the integral equation that appears in this model is exactly the same as the Gotan's equation, but with a different sign here, instead of a plus or a minus. And you can solve with the similar techniques. Now, you can actually also compute the inverse or plus transform of this resolvent, and you will find this function here. So, you see, it's very similar to what I have before. Instead of having a one-half, you have a one. And again, this encodes the full solution to the problem. So, you can use all this information to actually calculate the perturative expansion of the energy. And this transparency is especially dedicated to Jan, who studied the capacitor, the problem of the... So, the same integral equation appears in the old problem of calculating the capacity of circular play condenser at a small separation. So, this is a problem in electrostatics that maybe you have been tortured with this problem when you were a student in physics. And, Jan, actually, when I was working on this project, I discovered that Jan had written many papers on this. You forgot about them, but the web keeps everything. So, one of the things that Jan was studying was the formula, you know, how to calculate the asymptotic expansion of this capacitor, and again, only these terms were known in the literature, okay? This is everything that people knew before we developed these methods, and now you can actually compute many of these terms. And I think Jan was relating this to invariance in Riemannian geometry, right? Now, you can actually compute all these numbers, you see in the involved data functions as well and so on. Okay, so this is, in a sense, a bonus of this project. Okay? Now, coming back to the perturative series of the ground state energy of this linear model, you find these series. So, these first two terms, you can compute very easily using Bogolubov's theory of interacting bosons, and again, you have all these coefficients here. You can notice that there are fractional coefficients. This is the typical signature that you have a condensed state of bosons. The series is, again, factorial divergent. It's number is summable, and you can also deduce the larger behavior. Numerically, you find that, with high probability, this is 8 pi. And, this is a little bit what happened with this Bender-Buch calculation. You know, Bender and Bu were studying the quartic oscillator, and they just computed many coefficients of the perturative series, and they found a number that they identified by numerical fitting. So, this is what a little bit what we are doing here, but we don't really know what is this number. Now, you can ask now the same question. What is this number giving you? I mean, every time you have a single, Borel singularity in a quantum theory, there should be something behind. There should be some phenomenon. Some instanton, some renormal behind. Now, for the Liefling-Niger model, I don't really know what is this 8 pi. We know that this would correspond to a transidious which would have a exponentially dependence of the 4 e to the minus 8 pi over gamma 1 to the 1 half, and what I suspect, but this is something that I would like to make sure in the future, is that this is really due to an instanton. And, this is an instanton that nobody has studied. It should be the instanton of a non-relativistic 2D field theory describing the Liefling-Niger model. The Liefling-Niger model, in terms of quantum field theory, is a two-dimensional quantum theory for a complex scalar field with a non-relativistic kinetic term and with a 5-4 interaction. And, if this phenomenon is related to this, to an instanton of this theory, you should be able to find a finite action solution of this, of the question of motion of this action principle that should reproduce this 8 pi. So, I think this is really very interesting because the Liefling-Niger model is an experimental model. So, this could be a prediction of resurgence for something that you might eventually see someday in the lab. I mean, and this really, nobody knows about this. I talked to the expert, one of the world experts on one-dimensional condensed matter systems in the world, who happens to be in my department, and he doesn't know what kind of effect could be exponentially small in the Liefling-Niger model. But resurgence tells you that such an effect should exist because this is what is controlling the larger behavior of this perturative series. It's there. So, what is this? And maybe there is something interesting here. Yes. There is no way to relate this to the BCS. I mean, because the Cooper pair is also, they can represent those opposites. Yeah, yeah, yeah, no, I understand what you mean. There is even this idea that you can, in a sense, interpolate between the superconducting state and the condensate state. Yeah, this could be the case, but when you do this, no, this is something that people have. This is called the BCS-BC crossover, the crossover between BCS superconductivity and Bose-Einstein condensation. But still, when you do this, you essentially glue the Goddanya model to a Lively-Niger model. This has been studied. But still, this doesn't have the dependence of that you would, I mean, it's not the same kind of effect that you have. And I suppose I can extract this from the leather construction. I mean, I extract this from... Well, the problem is that the perturative series, the perturative series here gets completely reorganized. I mean, the perturative series for this model, the Bosone-Perturative series is very different from the Fermin-Perturative series, precisely because you have the condensate. So you have to do perturative series in the presence of a condensate. And you don't... I mean, the diagrams get completely reorganized, even if you think that this is a continuation of the Goddanya model. I think I would say that the most straightforward possibility is that there is a finite action solution of this model. I think you're sure about the... I don't have any extra special set of points like in previous case, say, the normal one on top of nature or instant one. Well, it could be that there are no normals in this model. But you see, there is a difference between fermion-existing and bosonic systems. In fermion-existing, instantons, you don't even expect that they give you factorial behavior. Here, instantons could give you factorial behavior. So maybe we can try first to see if there are some instanton solutions, no? I mean, after all, we know, for example, that the relative... Let me actually tell you something else. We know that the relativistic version of this theory has not finite action of instantons, but almost finite action of instantons. These are the famous vortex that people like Polyakov studied. So it's not so... It wouldn't be... But of course, it could happen, as you say, that there are no instantons that lead to this, and then one should look again to some sort of renormal on diagrams that are responsible for this. We know, for example, that in two-dimensional field theories, this is a non-relativistic two-dimensional field theory. Relativistic two-dimensional field theories with five foreign behavior. And I think once that we've studied recently by Marco Ceroni and collaborators, again, instantons are really the responsible for that sort of behavior. So I would say, based on this, that this is the first place where I would look at. But as you are saying, maybe this is a renormal once again. So, but I think it's very interesting that resurgence is telling us about an effect that we didn't know about in a very central model to condense matter theory. Okay, let me already reach my conclusions. But I think that the main idea is that quantum many-body theory is a very important arena where resurgence ideas could and maybe should be applied. I'm not a moralistic person, so I wouldn't say it should be applied, but it can be applied and it's a pleasure to apply them and it's a very interesting new place where you can think about resurgence. And as I said, the larger the behavior of this, the main technical point of our work was to actually generate this prototype series. Already using resurgence to this series, you see interesting new insights on the prototype physics. For example, the Cooper instability, nobody has understood this instability as a singularity in the overall plane. I mean, this is just because in condense matter it's very hard to generate this prototype series. So I think this gives really a new perspective on the Cooper instability. So this energy gap, which is one of the most important effects in quantum physics now you can really understand it as an exponentially small correction in a trans series. And it controls, this is also an open question. I mean, if you read all these papers on larger, there is very few results on larger the behavior in condense matter theory, very few results. And you read all these papers, there's not even a hint of what is the object driving the larger the behavior of the prototype series in many fermions systems. Also, when you look at these, the answer is obvious. Is the Cooper instability? Is the exponentially small number of effects discovery in this series? This is the effect which is controlling the larger the behavior, probably generically, in any weak interactive thermosystem. And it's also very interesting that when you start looking at the physics behind this, just find very similar physics to the normal physics. And I think this was also not really pointed out in the literature in a clear way. So, of course, when you see these things, you can start seeing many things which are similar. You see that in both cases, there is something like a lambda or pole and so on. But it's just more, it's really something that you can see very clearly. I mean, the physics of these superconducting energy gap as it manifests in the larger behavior of the prototype series is a renormal problem. It's a renormal phenomenon. It's the same one. It's surprising that people have not realized this before. Now, in the Livlin-Negre model, the larger the analysis also unveils this new non-protective effect, which I think should be understood better. And I think there is a very interesting possibility of finding new physics there, in a sense. Now, from the point of view of mathematics, I think this opens also a very interesting area where we should use our resurgence tools. Because, again, this idea of semi-classical decoding is that it's not enough to have a Gooth numeric or Gooth explicit solution. I mean, having this better answer solution, in a sense, doesn't tell you much about what is going on. I mean, as Bigner said once, it's good to know that the computer understands the problem, but I would also like to understand the problem. So resurgence is a way to understand these numbers that your computer speeds up in a more conceptual way. So what I think is a very interesting arena of research is to take all these better answers solutions, these all these exact solutions in terms of the better answers, and try to extract the trans-series which is behind, which has to be there. I mean, you take the Godaña model, there is this perturative series, the perturative series is non-borel sumo, there is a borel singularity in the positive real axis, so for sure there has to be something exponentially small that has to cure this, such that by doing lateral borel resumption, you recover the exatancer. That has to be the case. So how could you compute such a trans-series? I mean, it's not a simple problem because we have a lot of trouble just to compute the perturative series. Now computing analytically, these exponentially small corrections from the better answers, I don't think it's going to be very easy, but I think it's a very interesting question. And I think, again, coming back to these ideas of MITAT and GERAL, I think also these thermosystems give you, in a sense, a surprising realization of renormal on physics, which has all the ingredients of renormals, I mean, factorial, average, and diagnosis, and so on, but in a sense, in a much simpler setting, as Maxingua said, here you don't have to worry about renormalization in the standard sense of regulating. I mean, all diagrams are finite. At least, more precisely, at a given order, the sum of all diagrams can be trivially regulated in such a way that they are finite, but you don't have to use an explicit RG group or renormalization group or anything. So in a sense, this is maybe an opportunity to understand there is renormal efficiency in a nicer way, in a nice way, and we also know that these renormal on effects, these, the exponentially small contributions which correspond to these diagrams are also in the beta ansatz equations. So in a sense, the beta ansatz equations is telling you how you should solve the renormal on problem in these theories. So again, I think this opens a very interesting door to understand these problems, and thank you very much for your attention. Questions and remarks? Okay, yeah. What happens to the latter diagrams in 2D? Oh, they also diverge factorially. Yeah, they also diverge factorially. That's a very interesting case. The singularity you can actually compute numerically is involves a three somewhere, something like three pi or something like that, but I haven't looked at yet at it because in 2D, the way that, in 2D, the delta interaction potential you have to renormalize it, and then it's a little bit more subtle to understand, you know, the relation to the coupling and so on, but this is something that we plan to do. So it's clear that ring diagrams do not diverge factorially, so you have to look at the diagrams, but this is one of the things that we're really planning to do. But there's no clear relation to a thermo surface approach? No, no, the renormal on singularity is also related to a, is due to a singularity near a thermo surface as well. Yeah, yeah, yeah, I think so, yeah. Yeah, the question, in the very beginning, you have a series and besides the first two terms, the rest of the polynomials in odd digits, yeah? Yes, yeah. In fact, I heard something similar in X, X, Y models. Well, somebody told me. Sure, sure, sure. It might be the case because these are some models solved with the answer, so I don't know if you can extract, I don't know if these techniques have been applied to these models, but of course there is a better answer than the right. Of course, the appearance of these zeta-3 and zeta-5 is... No, the miracles are not multiple zeta-settles, it's only polynomials, it's odd zetas. Right, right, yeah, yeah, yeah, yeah, yeah, yeah. You don't have any explanation? No, no, I don't have any number three explanation. Actually, people thought at the very beginning that it was linear in the zetas, so there were no polynomial structures, it was just linear. But actually, the first ones are linear, but you see it already at order five or six, you have a problem, so yeah, I don't have any clue about this. But these are analytic results, I mean, because people have derived some of these coefficients by numerical fitting, they're so really analytic, I mean. Yes? And in fact, we're on some models where you can have a phenomenon from the Schrodinger-Eisen equation. Do you think that you could use something like a self-consistent equation to plug a modified propagator in your... Ah, yeah, yeah, yeah. No, I see what you mean, yeah, yeah, yeah, yeah. No, people, you know, that's a very interesting question as well, is that in some cases, you can resound the ladder diagrams. And in a sense, you hide these normal singularities. You can do that, for example, in three dimensions. In three dimensions, you can actually resound partially at least one of these ladder diagrams, to define your vertex in such a way that includes these ladders. And then, you know, you can do perturbation theory with some sort of renormalized vertex, and then the degree of divergence changes when you do this, okay, because you have resound some partial diaries. But it's not always the case that you can do this, because this, actually doing this is, in a sense, a partial resumption, a partial prescrision for resumming these divergent series. For example, in one dimension, you cannot do this. You cannot do the same trick as in three dimensions for resumming these diaries. You try to do it, you find a divergence, you have to regulate it in some way. So it's just that you are lucky that in three dimensions, at least this one whole pair of ladder diagrams, you can resound them in a naive way. And then people use this in perturbation theory, just as true. But this is not universally possible. Already in one dimension, it fails. It looks like resurgence is very useful in those kind of physical systems in which there are phase transitions and it looks like this can be applied to many statistical systems. So is this correct, or is there something particular in superconductivity that allowed you to start applying resurgence to low-energy physics? And if there is nothing particular, why did you choose superconductivity to start with? I didn't choose, I mean, I just, no, I mean, of course, you know, this is the context of discovery and the context of justification. I didn't discover this in a sense by saying, I want to understand superconductors. I mean, I can't tell you why I happened to stumble on superconductors, but it was not because I said, oh, let me look at superconductors. I was interested in renormal. I wanted to understand renormal physics in these models in some precise way, and eventually ended up there. Now, in terms of phase transitions, all everything I did here is at zero temperature. Now, I think a very interesting question is the following. Now, starting the system at final temperature, it's clear that at some point, the energy gap will vanish, because this is what the superconductor is. As you start hitting the superconductor, this energy gap vanishes and the system is no longer a superconductor. So, if you do perturbation here at final temperature, would you find a similar situation? I mean, would you find that, you know, as you hit the system, the Borel singularity starts moving away in the real axis and then disappears to infinity precisely at the transition temperature? That is a very interesting question. And this is how you would understand the phase transition, the superconductor phase transition at final temperature as in terms of resurgence. Now, that's very interesting because we know that at final temperature, the perturbative series is going to change. But precisely because there is such a phase transition, it will be very interesting to monitor the location of the singularity. But here everything is zero temperature, okay? Yes? You had two non-perturbative factors, one of them, can you write them, can you show me again? Yes, what do you mean? One of them was coming from the gap and the other one is 8 pi divided. Yes, absolutely. So, this, okay, this is the effect for Godang Yang, okay? That's very interesting because, you know, in some circumstance, you see one of them is controlled by one over gamma. Yes. The other one is controlled by one over gamma to one half. Absolutely. So, there are some circumstances in which this gamma to one half comes about as an instanton in the effected field theory. Yes, yes. So, it just reminds me that I just want to comment on that. So, for example, even if, in for taking meals, you know, there are usual instantons. Yes. You know, if you look at the, you know, tunneling is related to center symmetry, okay? Minimum. This is the work of Lusher and Nobel and others. You can show that the tunneling is exponential minus one over G, not G square. Oh, I see. So, and that comes about because of, this is really an instanton of the quantum action. I see. It is speaking. I see, I see. That's interesting comment. As I said, my first guess would be that this one half would be related to a finite action configuration of this bosonic theory. But I don't really know. I don't have anything concrete to say on this.