Okay, thank you very much for the invitation to give this introductory lecture on resurgence. I'm very happy to be here. It's a big honor. And it's also very impressive to be here in front of some of the founders of this topic like Andre, who is here. So I will try to do my best. So let me start with a small introduction because my lecture will be quite introductory because resurgence others very basic questions in physics and the main question that you can start like formulating on it understand what resurgence is about is the meaning of perturbation theory in quantum physics Okay, so let me start with a very simple example, which actually was It was a kind of historically very important, which is the perturbed perturbation theory for the quartic the Anharmonic oscillator, so this is actually a basic problem in quantum mechanics that you can find in many textbooks and maybe many of you studied this in undergrad as an undergraduate student, so you consider the following Hamiltonian which is a harmonic oscillator standard harmonic oscillator perturbed by a quartic term and with a coupling constant g Which we're going to take positive And you asked yourself the question of what is what can you say about the spectral properties of this Hamiltonian? So so if you want there are some exact results that you can write And this is what we are going to call what I'm going to going to call in this talk non-perturbative definition because This type of results do not rely at all on any particular scheme So Yeah, yeah, yeah, well here, you know, I of course, you know You know that Q and P is yeah, it's a it's part of the it's part of the data. Okay Yeah, it's just just just going to I'm not going to do perturbation heating a spark here I'm going to do mostly the perturbation heating g here, but Later on we will do perturbation heating h bar so So one thing you know is that if he is positive, you know, this is a confining potential and If you want You know that this operator the inverse of this Hamiltonian is a compact operator And then you know that there is a discrete spectrum So a spectral theory guarantees you that there will be a series of Eigen values for this Hamiltonian Which depends on g so you will have a ground state energy At first it's at a state energy and so on so this is a quite Elementary problem and if you want to calculate for example the ground state energy you can always do it numerically now So this is you want the exact result then You cannot also do perturbation theory. So what perturbation theory? So this is one of the standard tools in quantum theory You know that this Hamiltonian when g is equal to zero is exactly solvable and you try to find So sort of formal power series in g to describe this spectrum So what you actually find when you do perturbation theory is actually a formal power series That's going but I'm going to call it like five. This is a purely formal power series which Has this form Well a series one half. So I'm going to put h bar equal to one Just to fix to avoid normalizations and you know, you can actually write this series You can amuse yourself to compute this series And you have something like this And what you know is that this series You know well, but you know many things how this is the first thing You know is that this is is divergent and actually the degree of divergence can be quantified very precisely This coefficients a n actually they were factor factorially So they go like n factorial times three fours to the n Times minus one to the n So this is a result that can be established in many different ways actually one of the applications of resurgence is to establish this result but Right now what I want to emphasize is that this is not only this is really a formal power series because it's asymptotically growth factorially, so it's divergent and what we know then is that The Granthus state energy which is a well-defined function by this spectral problem has an asymptotic expansion here I By zero G. So here we have two different words. We have the walls of extra results but we have the sort of Function of G and then we have the walls of asymptotic power series And we know that they are related by an asymptotic relationship in the sense of Poincare, okay? So this is all very standard, but now The G you said the G is positive. So G greater than 1 also Yeah, G is greater than zero and you say the final G is also So this series is also This is actually very for any positive value G Because it's factorial the virgins. So if G is less than one, can you say always that it diverges? It's the virgins for any positive value of G. Okay, but it has zero radius of convergence for any value of G That's why there was no this factorial There will be a final radius of convergence, but there is a factor So this is the virgins for any positive value of G any G different from zero gives you Positive G different from zero gives you so this the series formal process has zero radius of convergence So so the question one of the questions you can ask is the following, okay? We know that there is this asymptotic relationship, but is it possible to reconstruct This exact result from the knowledge of this perturative series That's one of the questions we would like to know the other question you could ask is how will you make sense of this? asymptotic series and These questions in quantum mechanics are relatively easy, but there are many other contests in quantum theory where these questions are very hard to To answer for example, let me give you two examples in quantum field theory You have some sort of non-perturban definition, but it's not at the same level of rigor in general that you have here So for example jamming theory you have some non-perturban definition by using the lattice regularization and then We know also that we have some sort of perturative scheme and it's a very important question We can't relate in some way the kind of non-perturban definition that we have in quantum field theory Say be a lattice regularization to the kind of perturbative series that we can find in perturbation theory Another even more interesting case is a stream theory because in a stream theory We just don't even have this type of Non-perturban definition in general so if you want to say something about perturbation about the stream theory beyond perturbation theory One good way is to start looking very carefully at the perturbative series and see what the perturbative series can tell you about Some sort of non-perturbative result. Okay, so so in a sense resurgence Answers gives you three different Three different General strategies the first one is to make sense of this perturbing series The second one is to relate the perturbing series to the non-perturbing result In case you don't have a clear non-perturbing result resurgence tells you As we will see in a moment that there is a further instructor here Which is going to be the trans serious Structure and This actually allows you to have some hints of the non-perturbing structure of a theory even if you don't have an unperturbing definition So so in a sense resurgence is the art of exploiting as much as possible the information Containing this to a perturbed in this perturbing theory and another very important conceptual question for me in resurgence is What I'm going to call the semi classical decoding. I mean Semi classical decoding as I will make a clearer later is The question of how much we can say about an exact result in quantum field theory by relying on son Approximation scheme based on semi classical intuition It's not obvious at all that you can always make sense of results in quantum field theory using semi classics in a sense Resurgence will also try it's also the an approach We try to make as much possible use of semi classical intuition in order to get exact results in quantum in quantum theory Okay, so these are some questions that I will address later on But the first thing that resurgence tells us is that in general if you want to obtain the exact result For example here in this kind of problem From a perturbative approach you have to enlarge your perturbative approach So the the way to do this is to introduce what I will call trans serious. So this is the first Thing that we will need in in this particular problem that I will show you here You don't really need to know serious as we will see but in general you will need we need trans serious So this is the kind of first technical aspect of Resonance theory that I want to introduce and this is yes. Yes Where Yeah, I mean in supersymmetric theories you have some types this kind of I know this is like a Special case in which the perturbation theory terminates So you cannot think about this Yeah, but then but then it's then perturbation theory is relatively easy in a sense because you know you have truncated it So you have another so you can define a trans serious in the quantum cyber within theory And then the question is how you resound the all instant on perturbation, okay, but each series itself is fine Yes, this is you have the same problem in the harmonic oscillator Terminates absolutely Semiclassical approximation that you have other other things in which there is a non-trivial Yeah, yeah, I mean also super symmetric quantum mechanics. You have similar behaviors You know, you know these are a special cases in which Terminates but this in a sense Well, actually, you know when you have when you have a sum over instant on sectors and the perturbation theory in each Instant on sector is fine and then the problem is how to resound the instant on series But this is typically easier as you know very well. This is actually it's a different series So so actually giving a day a general definition of trans series is Is not what I would like to do So let me actually start giving you some sort of what is a trans series in the case of ordinary differential equations So trans series will give you know, let's actually look at an ordinary differential equation With an irregular singular point So I'm going to give you Some examples of this so one example Which is also very useful is Oilers equation, so I'm going to give you two examples one which is very easy and another which is more complicated So this is just this equation And we are going to so the regular singular point is at infinity The other example it's much more complicated, but can be treated with the same tool, which is the Parliament to equation So these are differential equations Again, you know, we're going to study this equation here the regular singular point kappa goes to infinity So what can we say about this equation as well? If you try to solve these differential equations in perturbation theory You will find a solution again a formal solution for the for this one The order equation Which is of this form and actually these coefficients you can see That they're given by this expression So this is exactly the same type of behavior we have before we have a Protervative series we have a formal power series in one over z because we are starting to be here near setting the infinity and the Coefficients diverge factorial. Okay, and a is this constant appearing here so a is a constant and Again, if you also look at this palette equation, you will find something similar slightly more complicated. There is a formal solution Which actually grows like a square root of kappa and then you know you can Compute this up leading corrections to this This is for This is an asymptotic expansion around kappa equal to infinity So you have these two formal solutions And these are the kind of expansions that you find when you study a differential equation in a regular singular point Now it turns out that it's very easy to see that you can actually Write down a more general solution to these equations by writing In the case of all your equation you can write an equation more general solution basis on this asymptotic series Which I'm going to write like this Okay, so this is actually for any c is a constant. This is a formal solution of equation And this is actually my first example of a trans series so this is actually a sum of two formal power series one which is An infinity power series and this is again a truncated power series because you have no term but this is a formal solution and You can also write in the case of panleve 2 There is also a general truncated series solution which is u0 of kappa and then you add again with a arbitrary constant here you can write down the series We have this structure and then you know you have here this is So this is much more complicated as you can see Sorry Sorry, yeah, it's kappa. Sorry. Absolutely And here a is actually a constant as well four thirds so this In this case you can see that this is a formal solution by plugging this differential equation And here you can see that these ul functions satisfy linear Differential equations And they can be solved formally in power series And actually one very important aspect of this series is that it's also factorial a variant divergent And all the series appearing here all deception on series are also factorial a variant But this gives you an infinite family of solutions of formal solution to these differential equations And this type of solutions is what I call here trans series So what characterizes trans series? It may actually give you some characterization of trans series. What are the characteristics of trans series? Well, the first thing which is very important is that trans series involves typically more than one small parameter When you write the standard asymptotic expansion solution to an ordinary differential equation of this type You have typically one small parameter here. It will be one over z here will be one over kappa and so on But trans series have more than one small parameter Because they have say the one over z or one over kappa of standard asymptotic solutions But they also have typically in these examples small exponentials You have here small exponentials with an essential singularity Here you also have small exponentials and these small exponentials typically involve a Argo verb by some sort of constant which depends on the equation you have. So here we have an a is a constant Which is characteristic of the question you're using So that's the first important thing when you consider trans series You have to your large your set of solutions to your differential equation by including not only the small parameter Which is the standard in this asymptotic expansion, but also in a small parameter, which is typically in this case exponential exponentialism And this is actually what in physics is interpreted as an instanton effect So these are instanton corrections to your perturbative series So this is what you can regard as the perturbative solution to the problem in the sense that Here's your formal power series expansion in the original small parameter of the theory But then you can enlarge this by adding these additional formal terms Now the second thing is that all series appearing here are factorially divergent in general So here we have a special example in which the series gets truncated But in the example of panlevé is very interesting because first of all we see that you need an infinite series of Of this trans series involves an infinite number of terms And actually each of these terms involves a series An asymptotic a series which is itself asymptotic So all these numbers u l n for l fix all grow like n factorial Generically Are you respecting the z being real? Yeah, no, no here everything it's what do you mean by small exponential?  Well because In many problems you want to actually construct a solution say on the real z axis near z Going to plus infinity you want to construct a real an actual solution of the differential equations And then these guys give you small corrections to this When you try to construct a solution in those regions Okay My question was actually the other way around when five zero truncates and the other one is infinite series Yeah, yeah, yeah, absolutely. Yeah, yeah, yeah No, but this is no, but this is this is very similar to server with it This is they have an an instanton effect, you know, and then you have no perturative series around it So that's one instanton effect. Yes But the average is infinite sum of instantons and the perturbative series The perturbs is truncates and no, but you have many series, but each instanton series is even with it is also fine So when I hear call call call about call this small is of course assuming that z is real and large But as a formal construction you can you know, this doesn't depend on on on the values When we construct a real solution, we will see what is the effect of this This and now a very important property is that all the different series The different formal series appearing in a trans series are related and this is and this is Why the subject is called resurgence I'm not going to discuss in detail the The structure the structural aspects of this of these relationships actually Ricardo I think will discuss this a little bit more in in detail But one of the one of the ways you can see this is the case is that the large order behavior of this coefficient So to see here The behavior of this coefficient a n because I a to the minus n and a is the same quantity that appears here And you can also see that if you look at this coefficients here So you write this series Why zero as U and U zero n Kappa to the three n over two the square root here then These coefficients u zero n actually they are only norm zero when n is even actually Behave like two n factorial and then a to the minus two n when a is this quantity here So you can see that the And this is only valid when n is very large So the behavior of these coefficients when n is very large is controlled by the small Exponential that appears in the other sectors of the trans series And actually this type of relations can be Enlarged to a very precise description of the full asymptotic expansion of these coefficients when n is large In a one over n series and the asymptotes of these coefficients involves the coefficients appearing in all these trans series here So so more precisely So to make three more precise Late terms in the series The perturbative series Are controlled By information in the trans series in the other series of the series Marcus, we don't want to put any condition of Picking the great functions like l two functions or something. No no here Wedding No, no, this is No, I mean here here in the first example I was giving here of course here you need to you have to work in But yeah, yeah, you have to work in l two, but this is a far more power series here I don't have to worry about this the question is how these two things are going to be related there There you have to be careful about how you define this But here, you know, this is purely formal construction. You don't impose Any l to r condition when you there is but when you find that it will construct easy Absolutely. Yeah We will we will yeah, we will see how how this this comes out, of course, you know, you are absolutely right And actually essentially this is going to fix the value of this coefficient. Absolutely. Yeah. Yeah. Yeah, but we will come to this later What is very important in this subject is the formal construction in trans series, you know the The fact you can construct these trans series in a purely formal way It's very important because then you are going to use this as your building blocks in the solution But it's important already that you are able to construct this trans series formally You know from the point of view of quantum theory This is highly non trivial because this actually means that you are doing perturbation theory and then you are adding A known a new set of sectors to the theory which in many cases can be interpreted as perturbations around instantons or of different subtle points that we will see But this is actually what gives the name resurgence resurgence means that The coefficients here Are related to the coefficients in the to the information in the other terms of the trans series. So The information appearing here resurgent in the other terms and and there are A sequence of relationships that you can describe in some detail This is not the thing that I want to emphasize in my talk But it's very important when you construct a trans series that these resurgent properties are there And one way to make sure that you are constructing the right trans series is for example to study a Synthetic properties of the coefficients here and make sure that they are related to the information appearing in the other trans series This is very important. Otherwise, you are not really constructing the kind of trans series that appear in resurgence here Now the fact that this is the case in the case of all these is a theorem as I will make Clear later. So, you know, you can prove that this property in the case of all these But in other examples in physics, we don't have the same technology as you have in studying all these and then we have to work More experimentally and the fact that As synthetics in one series is controlled by the other series in trans series Is a very important tool to know that you are actually constructing the right object Okay, so so this is these are the the main example. Yes What are the electrons in the expansion? I mean that for example here you look at this series So this is the original a synthetic solution that you can have to this equation What I mean is that when n is large these coefficients have an asymptotic behavior, which is controlled By quantities that appear in the other turns in the trans series. You see If you construct this solution, for example, this constant a Doesn't appear but it appears as a fundamental constant controlling This small instant this small exponential effect in the other turns of the trans series in the same way that here, you know This a small term knows about The behavior of these coefficients when n is large So this is a general property of trans series That these series and these series and the rest of the series in these trans series are not completely independent but are related by by some sort of relationship that you can Make precise But one of the most important consequences of this relationship is precisely the fact that the large the late term behavior of the coefficients Appearing in the original series is controlled by the information contained in these trans series You see if just write this coefficient here, which is Just write this series here, which is what would you would do in a standard solution of this differential equation in a regular singular point You wouldn't know What is this a related to You would maybe find it experimentally, but you wouldn't know what it is. But if you construct the trans series You will see that this a is fixed by the construction of the trans series And this a is telling you exactly how your first series behaved when n is large So this is the kind of and this is what is This is where the name resurgence comes from resurgence comes from the Relationship between different threads in the trans series and this is not the kind of solution that you would construct by using elementary means In order to have complete analogy with the oscillator case The third oscillator case you have to have on e0 of g differential equation Because you have all these here you're talking about Properties of evolution once there was in the other question was properties of energy Yeah, I mean when you construct trans series The the best case scenario is when you have some sort of recursion that you can solve by enlarging your answers for the recursion That's technically the case. Okay, but I will now come back to the to the quantum mechanics example and you will see how Now let me let me give you other examples of trans series. So this is Other examples of trans series Which appear very often The first example is a saddle point approximations to integrals so remember that you have an integral Define integral that appears in saddle point approximations So where z is an appropriate contour in the complex plane So you know that if you want to calculate this integral in the saddle point approximation You have to look at the critical points of f So this will give you typically a finite set of saddle points S these are saddle points and now depending on the value of h bar You will find steepest descent contours associated to this associated to this To this other point. Let me call them z n Which depends on the value of h bar actually depends on the argument which bar there And when you do this kind of analysis what you do is actually to decompose your integral And then you will have here Formal power series if I'm going to call like this which are Which is the come from the for the expansion saddle point expansion of the integral on the saddle point And then you write the integral original integral as a linear combination. So these are coefficients a linear combination Of these saddle point expansions And this is actually also an example of trans series This is another example of trans series in which The formal power series which are typical adiabatic appear as the saddle point expansions of these integrals Around the saddle points and then you have a number of coefficients here And of course these coefficients are not arbitrary You can fix them by comparing with the original definition of the integral But the structure that you find Is very much the same and actually there are simple examples like the ad function for example Where you can analyze the problem In both ways. So for example the ad function you can think about it As solving a differential equation Or by doing Or by using the contour integral representation of this ad function, and then you will see that you get two power series One which goes like two thirds to the three to the three halves and another which Goes with e to the plus two thirds of the three halves So you get two formal power series expansions with different asymptotic behavior And this can be regarded as trans series for this differential equation or they can be regarded as Two different saddle point expansions in the integral representation of the ad functions So you see again that here in already in the example of the ad function The trans series have different Exponential behavior like here here you have A series which goes like Like a square root of k and this goes square root of k times an exponential So the different turns in the trans series have different exponential behavior And this is exactly what you have in the elementary example of the ad function You can think about the two formal power series solution to this ad equation As two different trans series having different exponential behavior again, okay So this is a kind of very an example where Trans series are actually relatively simple You have a typically finite number of turns in standard integrals and then We can now think about what happens In quantum mechanics again, so Because once you understand that trans series appear when you treat Standard saddle point expansions in standard integrals You can boldly go to Path integrals And then you can see that in quantum mechanics You will also have This type of trans series. So let me give you an example, which is slightly different from this one In this is that it's much easier to see how to construct the trans series And this is an example Which is well known Which is the The double well oscillator So you have Something similar Again g greater than zero And this is the standard potential With two degenerate minima So this is minus one over Two square root of g and this is plus Another two square root of g These are the two Critical, this is the two minima of this potential And in quantum mechanics, you can now again ask You know that this this potential has the same has Similar properties from the point of view of the Schrodinger equation that the previous one. So in particular, there is a A ground state energy here And then you could ask what are the kind of what is the kind of information that you get from perturative series So there is a perturative This is that you can calculate very easily by doing small oscillations around this minima This is something again that you can calculate Very easily in quantum mechanics So you get something like this But we know that if you want to actually Complete the Informate you can if you want to actually consider What is you want to reconstruct the actual ground state energy From the information that you think in perturbation theory. This is not going to be enough And you actually need precisely to go to this trans serious To this trans serious And what you find here is that There is More complicated funds and more complicated trans series which involves The perturative trans series as you the perturative series as you computing a standard quantum mechanics And then you will have Other corrections And here f1 of g again involves Non-perturative involves these small exponential effects. So This is of the form 2g to the one-half. Let me just write the first order 1 over 6g Over the square root of 2 pi And then this multiplies An asymptotic series in g and we see again here The presence of an exponential is more effect And this is one in one important case as In contrast to the previous case Of the oscillator with Positive quartic potential here In order to reconstruct this function as you as we will see in among as I will make you in a moment You only need this Perturative series But here with this perturative series you are not able to reconstruct the sub ground state energy You really definitely need this further Trans serious here And physically it's very nice. This has a very nice interpretation when you consider the path integral approach to this problem So one way of doing this is to consider the thermal partition version of the problem Which is the trace of e to the minus beta H this has information about about the spectrum of the theory because It It contains information about the ground state energy when beta is large And you consider the path integral calculation Of this Of this quantity you have to actually consider The Euclidean action and to consider periodic trajectories The path integral of the periodic trajectories so q zero Is equal to q beta so beta here becomes sort of Euclidean time And actually this Perturative series Comes from constant periodic solutions. So solutions which are localized here and here And when you do the Perturative expansion of the path integral in the same way or in a generalized way As you would do a subtle point expansion in standard In standard integrals you will actually generate the series here So where does this new series come from? Well in order to find it you can actually look at the At what is called the twisted partition function Which calculates The same type of quantities but including the parity And This actually It's a path integral of the same type but in which now you impose And periodic Boundary conditions in which the nth point of the trajectory is minus the nth point the original point of the trajectory And you can see that here constant trajectories do not contribute anymore But there is a trajectory that contributes which is the one that interpolates between this point and this other point So This path integral the leading order contribution comes from a non-trivial saddle point Yeah, yeah, this is the turning point and For instant turn. So this is the tunneling effect But precisely the tunneling effect if you want to relate it to this the tunneling effect is the actually physically the physical mechanics which is Responsible for the appearance of this trans series This trans series is precisely due to the tunneling effect and this exponentially small Quantity appears here is precisely due to is actually the value of the classical Euclidean action on this classical solution So that's how Yeah, absolutely. C is With some subtleties I'm going to come back in a moment. Okay So Yeah, yeah here the problem is that there will be also multi-instantan contributions because you see you can go here But you can also go and come back. You can go and come back and come back again So there are multi-instantan sectors that will give you precisely higher order Serious terms in this formal series Because there are two zeros and there will be tunneling effect between these two zeros. Yes. So how come this? Well, you know you can have for example Now these kind of solutions are solutions which go like this so But then you can have Solution which goes like this stays a little moment there and then contact, right? And and then you can repeat this structure many many times and this will give you actually Higher order terms here which give you more and more exponent exponentially more smaller smaller terms. Remember that here for example In this series here, you know This exponentials actually get larger and larger by an integer factor So the first transition is correction has exponential with l equal to one and there is l equal to l equal to three and so on So you see that this in this quantum mechanical examples This the higher order transducer skill will correspond precisely to these instantons that To these multi-instantan sectors. And the coefficients would be the difference. So this would be different Well, no, no actually c is going to be the same always like here. It will be half c square c cube and so on. You know, yeah, yeah Yeah here You will see that you can actually You can actually you don't need to have differences I mean once you have one ambiguity and then the rest of the transducer will inherit this ambiguity Okay, like here, okay, you have one ambiguity, but you see that at higher orders. You just have c square c cube and so on So this is one example Actually, so I wanted to give you some elementary example Some example in quantum mechanics and now I want to give you an example in quantum field theory. So this is much harder I put the eraser Oh here, sorry. Thank you So let me give you an example in quantum field theory So of course in quantum field theory Again, you can think about So this is an example in quantum mechanics So you consider say In quantum field theory you will have the path integral for in the path integral formulation You can for example consider The analog of this quantity The partition function of the theory at a finite volume And this is going to be a path integral of this type with say Euclidean With an Euclidean action Here I'm being very schematic For phi are the fields of the theory and this is just And then, you know, uh, there will be always typically a trivial configuration Say phi equals zero, so we will see an example of this and there will be non-trivial saddles The perturbative expansion of this path integral and this trivial configuration gives me the perturbative series, the standard perturbative series That's actually how you generate perturbative series in quantum field theory But if you do the perturbative expansion around the non-trivial saddle As it happens in this, as it happens in this example, you will get actually The trans-series structure of quantum field theory So it's nothing very special, it's just a trans-series in quantum field theory It's just a perturbative expansion as a non-trivial saddle point solution to the theory So for example in Jammel's theory at finite volume, you can consider instanton solutions of Jammel's theory And then the perturbative expansion in the background of such an instanton field will give you a generalized series Which will be the starting point for building a trans-series solution In the same way that in quantum mechanics looking around things to understand your trans-series Now this program in quantum field theory is very hard But there is a very nice example which is trans-simon theory Where this can be made very concrete So trans-simon theory as a quantum field theory was introduced by Witten So here the action here, so this is a theory for a say a g connection On a three manifold a on a three manifold m So the action of the theory can be taken as k over 4 pi And then you have the trace of a, what's the a plus two-thirds of a q So this is the trans-simon's action And then on a three manifold m you can consider a function of k Which is given by the formal partition function obtained by doing this So have something like this So here you need an i in order for this So here actually there is no mass decision between Euclidean and Minkowski theory So you have an i here in order for this exponential to be gating bit dependent So this is how you define the path interruptions in most theory And actually we know that this, well it's conjectured That this is equal to the Witten-Rosketti-Hildturaev invariant of m Which can be defined by purely combinatorial ways Okay, so in a sense this is an example in which This Witten-Rosketti-Hildturaev invariant of m plays the role Of E0 in the case of the quantum mechanical problem So if you want, this is the non-pertorative This is the exat or non-pertorative answer And actually the Witten-Rosketti-Hildturaev argument in 1989 That this invariant of three manifold m can be obtained By doing this as the partition function of this quantum field theory So this is an excellent laboratory In which you can try to compare this exact answer With the sort of pertorative expansions that we get in quantum field theory And actually there are many interesting results along these lines And you can actually construct very explicitly trans-serious Starting from this exact non-pertorative answer So let me give you an example which I think is very nice So let's suppose that m is a safer manifold Characterized by, say, some integers pr So I'm not going to get into the definition of this mathematical object Just suffices to say that this is a three-manifold m Which is defined by a series of coprime integers r from 1 to s You can actually construct this manifold by doing Sergeant in a certain node with certain sergeant coefficients Which are related to this pr And then you get some manifold m Which is defined by these coprime integers And actually Lawrence and Rosarski Computed this invariant wrtm for this safer manifold And they showed that it's a sum, it's a finite sum of terms And the first term that you find here Is actually can be actually written as an integral Of certain function f y Is just very simple It's a Gaussian integral y square over 4 g d y, where g is actually 2 pi i over k times p Where p is the product of these prs And then there are other terms here But this integral here Let me write what is this function Because actually this is a result which we have with Sergei Which is a very nice example of this construction So f y can be described very explicitly 2 minus s And product of k equal 1 to s 4 2 signs So this is this function And you see that in a sense This reduces the problem of calculating This contribution to the written residue to write the invariant To precisely the kind of saddle point calculation That you do here Actually here c is just e to the pi i over 4 times r So it's this diagonal contour in the complex plane And you can see that This tells you that this invariant can be written As a perturbative series Plus other terms here And not writing these other terms I could write them But the importance of this term is that Which is what I showed here So this is this I'm going to call z perturbative of k The importance of this term is that this was identified As the contribution by Lorenz and Rosanski As the contribution of the trivial connection A equals 0 to the path integral So you see just to recap a little bit In the case of terms and most theory We have a very special situation In which this can give us This actually invariant gives you the saddle perturbative answer And then we can compare to what we would obtain From a path integral tripping of this theory And you can see that this gives you Power series that can be obtained From the perturbative expansion of this saddle point integral Around the trivial saddle point by equals 0 So here we have a series Which corresponds to the analog of 5 0 in these constructions that I showed you before And then there will be other contributions Which will be the analog of Lorenz number k enters k enters here and here Is the level But it enters in the definition of the rest of the pink toilet Absolutely, absolutely You need a level here This also depends on k Absolutely So this is a case in which You can actually ask yourself If you can decompose this Exactant perturbative answer In terms of a trans serious expression And it's one of the few examples In quantum receiver you could do that Okay, so this is kind of a quick tour of trans serious And the other ingredient that we need In order to make sense of the Retriever resurgence Is actually how to make We have to make sense of this trans serious So so far I was writing this trans serious As formal expansions are actually here I have a question Yes So cypher pinfolder constructed using filling of the knots Yes So point is do you have any idea how to construct Over the three mini quadrilateral vibrations In general it's an actual question Yeah So this theory actually works there I mean you can define this in mind for any For any three manifold right Now the why I'm choosing here cypher manifolds Is because these cypher manifolds can be constructed You see very simple ingredients In the Whitton resetting future F construction If you think about the conformal field theory Associated to this terms and most theory Here you only need In order to calculate this in money You only need the S and T matrices Of the waste to mean a Whitton Yeah yeah yeah just sorry Here G S S U 2 Absolutely For this results G S S U 2 This is for S U 2 Okay So this is for G S U 2 Absolutely So so what makes these cypher manifolds Particular is that you can construct this in money Using the simplest ingredients In this in this theory Okay which are the S and T matrices For all the three manifolds You will need more information Coming from the Six quantum 6J symbols and so on But cypher manifolds only need the simplest ingredients Okay that's why they are particularly Used nice to study Because you can make everything very specific But this is just a technical issue Now of course you know You could the question is If you can do this the composition In other cases Now so here in this case we have We have this equality But the question is Because here the perturative series Appears as just dispensing of this As other point in there But the general question remains If I give you a theory And you construct the trans series How you are going to Recover the original answer Using the information in this trans series And the answer of the theory of reason Is that you should use Some sort of Borel resumption procedure Okay so this is the next technical ingredient That I'm going to introduce Yes This is the TQFT example Not QFT example Yes Yeah but I'm going to I'm going to give some historical remarks Why this is an example that That we can actually Because in a sense the program of resurgence Has been going on under other names In physics And the main question of resurgence For me Is the question of Can you actually write Exact quantities in quantum theory Using just semi-classical analysis And we know that for example For Jamil steering infinite volume The most likely answer is no You cannot And this was like a main Drawback for the program Of semi-classical decoding in the 70s You couldn't make sense of Instanton calculus in Jamil's theory Because of the infrared light values Of instanton calculus In a standard Jamil steering infinite volume And this is a case where Essentially the full program of resurgence Cannot be pushed So So in a sense what we are doing here Is looking at an example where it can be pushed But in a sense the whole program of resurgence was Came to a stop in the 70s Because this problem of Jamil's Of Jamil's So this is the reason why I'm looking at TQFT If you do supersymmetric QFT You are also in business Because you can't pay instanton I actually meant what I wanted to say At this I would call TQFT And the supersymmetric one I would call QFT Okay But still QFTs, supersymmetric QFTs From the point of view of resilience Are also very tame Right Because instantons can be made In N equal to super Jamil's You can do instanton calculus Instantons That's a quote for you Instantons and instanton It's become trouble When you're only instantons In N equal to 0 They're only In order in Jamil's theory Just the instanton expansion Doesn't make sense Just in general Right Because you don't have infrared cut-offs Essentially While in supersymmetric Jamil's You have So that's the main problem So the fact that Since this point emerges But I just asked you to know That previously When you were considering the quantum mechanical The point of view So there You calculated the card ending contribution Yes So you could have Also considered the supersymmetric quantum mechanical Yes Absolutely Yeah Face of that thing In the previous Yeah Yes So that would give the same reasons or something Well, actually Again, supersymmetric quantum mechanics There are Is non-genetic situations So for example You can have situations With the ground state energy Is 0 In perturbation theory And the first contribution Is purely non-pertorative So these are kind of non-genetic cases In which These transits are essentially Truncated things In kind of peculiar ways But this is essentially The non-genetic case Okay And people have actually looked at These cases from the point of view of resources But here I want to focus in the Generic case In which you have Infinite series appearing Everywhere Okay Okay So So the technical ingredient That makes sense of So far My transits are purely formal So Are objects that I can compute Order by order In my coupling constant In my small parameter And order by order In my exponential small parameter So how do you make sense Of these formal series Which are Divergent have series of covariance Now The trick is very old It's called Borel resumption So Let me just remind you the definition So You have given you Given a series A formal series A n that n Where a n diverges factorial So Has this sort of behavior So let me assume that The coefficients grow like this Then the Borel transform Of this series Is defined By a very Kind of a stupid way Of maybe series convergent Which is just Removing the factorial That makes the series divergent So you remove the factorial And then you get A series like this Which has a finite range of convergence Just by construction Right Because this is The range of convergence Will be actually A here Okay Now Now once you do this You go from a Divergent Formal power series To a power series Defining a A function In a neighborhood of the origin So typically this defines A function in the neighborhood of the origin And Typically you can extend Analytically This function To a larger domain So you can So is it weighted Zeta function regularization? No No this is just They No this is not Zeta function regularization This is just the Weighted Just flattening the weight With the Weighted zeta function The zeta function has nothing to do with Zeta Let it be Yeah yeah yeah Now this is just the inverse of the Of the Laplace transform As you will see in a moment So So you can Now if I give you a transidious Psi of z Which I'm going to write like m0 of z Plus sum of n equal 1 cn l cl pi lz You can of course Apply this weighted transform here But remember that these transidious Are up to an exponentially small term That multiplies them Are also power series So you can also do here The weighted transform Maybe sometimes with some small Modifications because sometimes there are Over all factors in front and so on But you know this can be This can be Actually modify appropriately So from a transidious You can also do a formal Borel transform And then you will get You will get Some sort of Borel transform Transidious And now all the ingredients appear Here are functions Which have a finite range of convergence And typically they can be Analytically extended to a larger domain So that's the kind of the first Now This is not the Borel Some the Borel resumption of the series The Borel resumption of the series Is a part of the ingredient Which is precisely the Laplace transform Of this Borel transform So but before Before Before going to this Borel resumption Let me give you A nice example of Borel resumption Of one of the examples that I Considered before So remember that here I define this Pertorative series here So appearing in terms of theory So this is maybe something that Sergei will discuss More carefully So this was Defined as the As the asymptotic expansion Of this integral Fy Into the minus y square Or for g Y And you can see that This is a series Which up to a trivial constant Is given by n Fn gamma m plus one half And then for g The m plus one half So you see fn here Is the power series expansion Of this function That was involved in this thing And you see this function Is holomorphic at the origin So this gives you a convergent power series But you see that the divergence Of the asymptotic divergence The factorial divergence Of the Pertorative expansion in this case Is due to this gamma factorial thing So what you can do now Is to do A Borel resumption Of this Pertorative thing In which I do a slightly Slightly So let me call This A n k over n minus one half Remember that g Is plus proportional to one over k So let me write it like this So this is a trick That we did in our paper With Sergi and Pavel So what is important Is to remove the factorial So let me call this z Psi So here this is a slightly Modified version of the Of the Borel transformation In which I remove the Factorial by dividing by gamma n plus one half Not by just n factorial And you can see that in this case This function is very easy You just recover The original function f that appeared here So this here z is Two pi i Psi over p So this is a This is kind of trivial exercise In which you have An asymptotic series defined By a Gaussian integration Of a Olimorphic function at the origin And then the Borel transform Essentially gives you back This Olimorphic function So this is a very simple example Now one of the consequence Of resurgence Remember that resurgence Told me That the large order Behavior of the coefficient that appeared here Was related To the information contained in this general trans series One of the nice Consequences of that Is that This Borel transform Will have a singularity structure Now the singularity structure Of this Borel transform Will be related To the information contained in the trans series So a statement About large order behavior Of this Divergent series Will become a statement About the singularity structure Of this Borel transform And you can see If you remember The definition of this Fy Which has these Hyperbolic signs The singularities of this function Fy Are precisely telling you information About the other Non-trivial connections Of these trans simons invariant Of this With the regentine For F invariant So When you do the Borel transform Of this Of this divergent series You will get This holomorphic function Which will have a set of singularities And the singularities of the Borel transform of this Are telling you information about the other connections Non-trivial connections of this Of this invariant And this is Again the power of resortions Resortions is telling you that Even if you do Your expansion on the particular saddle point This expansion will know About these other saddle points And in this case In the case of trans simons theory Is actually very beautiful Because Essentially by construction We know that this function knows About the other non-trivial connections So You could just do perturbation theory Around the trivial connection And you will discover the whole structure Of non-trivial trans connections On this And this manifold Just by looking At this perturbative series So this is Another Consequence of resortions Okay How I'm doing time maybe Okay maybe do you want to do a Small break Okay Let me then Continue So the Borel transform So this is This procedure is It's called the Borel transform So It starts with an Divergent series This gives you a series With a finite range of convergence Which You might be able to extend The analytic energy to a larger domain And now the question is How to recover Something related to the original series From the Borel transform And this Gives us This suggests Doing this procedure of Resumation So this is The Borel resumation This is what I'm going to define I'm going to denote the s of phi Of z And this is Defined by this Laplace Transform So you do This Laplace transform So if this integral exists The original series Is called Borel resumable And the integral is called the Borel resumation So you see in order for this integral to exist You have to be able Essentially to extend The Borel transform To a Neighborhood of the positive real axis And you need some Decay properties of this Transform on the positive real axis When this Internal exists is called the Borel resumation And why is this so? Because if this Why is this object interesting? Because if this quantity exists Then you can see that It has an asymptotic expansion Which is actually the original Series five set It's very easy to check this In the definition So this When it exists it's a function It's a function of z So this Internal will exist in a In a range of values of z And when it exists It gives you an exact A true function Whose asymptotic expansion is five set So this is The basic idea of Borel resumation Is a very simple procedure But it's important to Extend it To a more slightly general case Because it's very often the case That This function has actually singularities On the positive real axis Okay So very often You have Very obvious obstructions To extend this Because you might have Singularities on the positive real axis Say Poles of round scots And in this case You can This integral will not exist in itself But then you can Default the contour And define Instead of doing the integral On the positive real axis You do You consider two different integrals In contours slightly above or below The positive real axis And this gives me The post The lateral Borel resumation Of this series So So here there are two of them Okay Okay now Now the question now is the following Can we Can we once we do this We can take a trans series And consider this Borel transform And in principle You can also consider The ratio Each of these have same asymptotes, right? Yes, yes They also have the same asymptotes, yeah They differ on I mean Both of them have the asymptotes Two phases, absolutely And they They differ Absolutely, absolutely We will make this Explicit in a second Okay So now the question The question we can ask is the following So let's suppose We have a trans series But I'm going to denote Like psi of z Z z Which is of this form that I was Can we do before So this is not the Let me point out that This is not the most general trans series You can consider One can consider Most more general trans series This is for example The one that was Appropriate for part level two But in general You might have Many different Constants Many different exponential terms And so on But let me just focus on this Which is already Soficially complicated So I have This trans series So what I can do now Is to do Where is the exponential term? Here is here The oldest are exponential terms I'm not indicating them But all these All these objects I'm going to Have This sort of Include this sort of exponential Be here When you wrote Boel transform under that You were not Very clear But I Absolutely I was telling here That in order to develop transform I was seeing this verbally Okay But What happens that here You will have an exponential And typically You also will have to have some Some So maybe some fractional powers here Okay But then what you do Is essentially you remove this exponential And then you do some sort of Variant of the Boel transform That applies to all these things But you know this is This is not This is not crucial So let me Let me now Do define So sort of Lateral Boel resumption So assuming that We're in a situation Which we have to do a lateral Boel resumption Of the whole thing So this will be just S plus minus Of f0z And then I can consider This thing Again with the proviso That I have to remove this exponential And then when I do the Boel transform Okay So this object is slightly different Already because at least You have still an infinite power In an infinite This is still a formal object But now its ingredients Are actually true functions If this Lateral Boel resumption Converts Even assuming that there are some Singularities on the positive real axis This object After doing the control resumption Might define A true function So you have This sort of Situation And now the question And this actually defines The question you see This defines a true function Okay And now what I'm going to state Once So if you want this is the Boel This is the Boel resumption Of the trans-series Lateral Boel resumption Of the trans-series Sorry? You see a true function Now what I mean is that Each of the components Is a true function Now it's not obvious That the whole thing will converge Right? Absolutely This is not at all But at least we have improved The situation in the sense that Each of these trans-series Is now maybe a function If these Boel transforms Are well defined So this is the Boel transform And now let me state a principle Which is for me the actual True conceptual core of resurgence And I think that's what Guides the whole subject In a sense And this is the principle So what I call semi-classical I refer to this before Semi-classical decoding Okay So what is this? Now I can sort I see Comes like this Like this Oh lift it The middle block where I'll do it Okay Like this Okay No shadow Okay So the principle Like this So semi-classical decoding What I mean by this Okay So let me try to State this mathematical Let's So let's support that you have a function A true function Well defined function With the asymptotic expansion Given by a formal power series Okay So fz Has an asymptotic series f0z This is actually what happens in This is in a sense The common situation to To everything that I have seen We have seen here Or z equals zero Or z equals infinity Well let's say z equals zero Okay Around z equals zero Okay So this is the common situation To everything I have discussed here In quantum mechanics You have perturbation series In ordinary differential equations With regular singular points You have the formal power series Around the regular singular point In quantum field series z is the coupling constant and so on And this is typically what you have You have a perturbative series And then you might have A non-perturative answer That you try to reconstruct So we say That fz Admits a semi-classical decoding If Phi zero z can be promoted To a trans series So that's the first if You have to be To a trans series Xi of z Which is bottle summable Which is lateral bottle summable Means that This whole object exists So at least for some range of values of z This whole object exists This whole sum infinite sum converges And such that You can actually reconstruct fz By this procedure So this is equal to s plus minus Of xi of z For some value of z And actually there are typically Two values of z When this actually happens Typically both lateral Assumption exists But with different values of z And actually z plus minus z minus Is actually the Stokes constant Of the problem So this is a kind of Generic situation Now So what is what is the This is This is not something that you find In the articles of resurgence You know this idea of semi-classical decoding But I think it's the core of the problem So the idea is that you have to be The idea is that you have an A quantity in quantity that you want to reconstruct You typically only have the perpetrative series Semi-classical decoding means That you are able to reconstruct this whole thing By using the perpetrative series Plus a trans series And then doing border resumption And this trans series Means typically you have to enlarge The kind of sectors that you consider So for example in quantum physics You have to consider Other sectors like instanton sectors And then you must be able to Reconstruct exactly the subject From this border resumption And this at least for some values of z Now This is very important because It's not at all obvious That a quantity That you have defined by Whatever procedure Can be decoded in this way And in quantum field theory We have a very good example Where this doesn't seem to be the case Which is pure YAMMILS Pure YAMMILS theory In infinite volume You have a renormalised perpetrative series But nobody knows how to do this Nobody knows how to reconstruct Observables in pure YAMMILS theory Just doing some principle Some sort of semi-classical decoding In terms of trans series You can do it at finite volume There is a well defined instanton Procedure to do in this infinite volume But nobody knows how to do this in infinite volume So this is by far not obvious Now what has been happening in the last years And why there has been this resurgence of resurgence It's because people have decided That YAMMILS theory is too hard And we have looked at other examples Where life seems to be simpler And these examples involve Models of string theory We have revisited many cases in quantum mechanics And there has been a lot of work Of people trying to look at quantum field theory In finite volume but with different boundary conditions Which are closer to the infinite volume limit And this is work by Gerald Dune and Mita Tunzal And collaborators Which have revisited many standard quantum field theory programs Still at finite volume Because nobody knows how to do this in infinite volume But changing boundary conditions And finding interesting trans series structure And in particular having able to identify Semi-classical models for renormal and divergences Which is one of the main problems in YAMMILS theory So this principle semi-classical decoding It's very interesting And let me make a theory And this is a theory and I'm not going to state it But this is a theory Semi-classical decoding Is true for ODE's with irregular singular points So this is a theory which I guess goes back to a call With important contributions by ODE costing And probably others So this principle semi-classical decoding And this is highly non-trivial Because for example In mind that you look again upon level 2 I mean it's not obvious at all That from the trans series I wrote You can actually reconstruct the whole solution to pan-level 2 So one thing that you can do is for example To consider what is called the Hastin-McLeod solution To pan-level 2 Which is a very interesting solution Which has this asymptotic This chi When chi is bigger than 1 Much larger than 1 This actually goes like a square root of k And then it decays exponentially At minus infinity With a sort of 80 behavior This one is very important Because this is kind of the building block Of many problems involving pan-level 2 Like the tracing with a low and so on So this decays at minus infinity With some kind of 80 sort of law So this is when k goes to minus infinity So this is Hastin-McLeod solution And actually you can As an application of this semi-classical decoding You can write this function As a Borel transform So it's s plus minus of Let me write yours the plus solution Of chi Or actually say minus Of k c And c is a value here Which is actually the stone constant divided by 2 So you can actually Psi here is the transition solution That I was writing before for pan-level 2 And c has a fixed value Actually I think you can use s2 Something like that Okay So you have actually two different You can reconstruct this in two different ways If you use one lateral Borel resumption You will have to use c plus plus s over 2 And you use the lateral Borel resumption with minus You will have said minus minus s2 And they differ in the stone constant And this stone constant has been computed by people So you can actually completely reconstruct this solution From this semi-classical decoding And this is what semi-classical decoding is Semi-classical decoding is telling you That this Hastin-McLeod solution Can be thought as As an asymptotic series Plus a series of instanton corrections And the structure of instanton corrections Is completely encoded in this trans series And I, yes? The way you wrote the Stokes content It's really symbolic You were saying c plus minus c minus is the Stokes time So that's just the contrary But I guess what you mean Is you're looking at some fundamental system And there is Stokes matrices and so on Yeah, yeah, absolutely Yeah, yeah, yeah But my question is The way you describe it There just seems to be one Stokes ray The real line But Oh, you know, in general Absolutely, yeah, yeah, yeah, yeah, yeah Absolutely, you know Here I'm assuming that, you know This series is divergent around this ray And then there will be all the Stokes constant Along all the rays Yeah, absolutely, yeah, yeah In this case One irregular singularity Or there could be none I mean here I'm considering One singular regular singularity And then you have the Stokes Here I'm considering implicitly assuming That the Stokes line is around The positive real radius Okay And but you know Because I'm doing the lateral Resumption in that direction But you could do lateral or resumptions At different angles I mean this is just an extension of the theory I have presented you here And then you could consider This phenomenon along all the rays In the compressed plane I'll obey two from the zero I'm going to do six Stokes sectors Yeah, yeah, yeah, yeah, yeah This is the Now in order to In order to reconstruct These hasty molecular solutions around the real axis You know, you are actually You know, here you are You are considering the ray Which goes along the positive real axis This is what I need in this case And this is the Stokes constant Appropriate for that axis Okay For that direction So this is of course A very particular example But what I want to emphasize So here this is the Stokes constant Associated to that line Okay Yes But if you speak of decoding It means that you have a way to Tell us what the C plus and C minus Absolutely, absolutely I don't think you have it always Or Well You might not have An explicit way of finding it And this is one of the main difficulties of the But already the fact that this can be Decoded in this way With just a choice of C I think this is highly non trivial Okay But still if you can't determine C then it's C is There is a unique value of C Which is fixed by If you know of that Of course But at least this tells you something Okay, imagine that you don't know how to How to define C At least You know that this quantity Defines you a one parameter family In this case because again And it's a simple setting A one parameter family Of functions Built upon your perturbative C Okay And I think this is already Very important for example Sorry Is there is an independent principle Which determines C Independent from the fact that it has to Give the final answer correct No That's a very important thing And here I differ from other people Which think that already this is already You know resurgence Doesn't fix the value of C So resurgence If you want to I mean Sometimes people say that Resurgence defines nonpertorically the problem And this is not completely true Resurgence defines a nonpertorative object Up to Up A multi parameter ambiguity Which is in this case In this simple case Is just encoded in this constancy But you see any choice of C Will give you a well defined function With that asymptotic With that classical asymptotic So resurgence by itself Doesn't tell you what is the right object here On the contrary If you know the right object here Resurgence By how do you find in this constant here So the principle of semi-classical decoding Tells you that if you know this quantity Then It might You might be able to decode it in this way And it's not going to give you the precise value of C You would have to find it by fitting And this is one of the main technical problems You'll find in semi-classical decoding And even if you have In QfK we would know what C is Because it's instant on counting parameters In every QfK we know what it is It's exponential Actually You cannot know C If there are singularities in the variable plane Because the value of C Depends on the choice of lateral resumption This is something that Parisi for example I think understood very clearly Already for a standard general theory You know this quantity is ambiguous Unless you tell me what is your prescription For affordable resumption And this is actually The meaning of Stokes' phenomena In the context of resurgence Means that There is no single value of C But are you always assuming that there's some Differential, some ODE with No, not ODE No, this has not to be an ODE I mean The main problem in resurgence is first To construct the formal transidies So if you have a setting similar to an ODE Or a difference equation or a recursive equation You typically can construct Construct the transidies But even the formal construction of transidies In many cases is very hard In standard quantum field theory In perturbation theory An instanton is technically very, very hard Already competing in higher order terms In quantum theory is very hard Imagining an instanton triviality An instanton background is even harder So even the formal construction of transidies Very often is not very simple Now once you have solved the problem of Constructing the transidies formally You have to go and stay further And do the overall resumption And then you have the same kind of problems here And finally If you know this answer You might be able to try to find What are the values of z Which actually fit this non-pertorative value And this is what I call semi-classical decovalent Maybe this lateral-borel condition Is a little bit restrictive Because in this case When we know c for the quantum field theory Which is instanton boundary parameter It might be that it's not a lateral-borel transform Which gives you f but something else Maybe just some contrast Yeah, yeah, yeah, absolutely Yeah, there might be Yeah, yeah, yeah, yeah, yeah, yeah Again, you know here I restricted to a very kind of Simple example of resurgence I mean I could have much more complicated set of c's You could have, you know, depending on the sort of simulator You can have contours And turn these simulators in different ways and so on But I mean in the case of I mean this is if you want A simple example of the semi-classical decoding In this, which for example for panlevet 2 is sufficient But of course, you know, in general You might have to enlarge very much this But the idea is that you always have to be able to Relate an exact quantity to a borel resum trance series That's I think the main moral issue If you have to do something different Then, you know, you are leaving the realm of resurgence Yes? So this ambiguity of choice of c that we're talking about Can I say it differently that this is basically choice Of the contours, say in a path integral Or a regional path integral Or a reduced set of point approximations Yeah, yeah, yeah If that's a question of which contour we choose Yeah, yeah, yeah, I mean if you know If you have a path integral And well, for example Let's say even standard interiors If you choose the contouring If you have a well-defined contouring And a standard integral And you do the saddle point approximation These constants here will be fixed by your choice of contour Exactly, it's unfunnable There are usually we have real slides Or some first given God-given choice of contour And therefore that's the additional information Absolutely, yeah, yeah, I mean If you know how to define this non-partuitively You will find some way of going from here to here That depends on the problem Sometimes you have to be So maybe if in the time I have left What I would like to do Is give you some sort of recent examples of this So I want to revisit two things Two examples, one in quantum mechanics And another in extreme theory In this statement over here You really got in mind to go well beyond Borel summations And sort of case summation And multi summation as well Yes And also in this effort Would you expect the space of Cs to be infinite dimensional Once you go beyond the ODE picture or not Yeah, I mean In principle, you know, for example If this is not It's clear that in ODE picture If I impose this asymptotic condition This is a one parameter problem And this is where Cs stay in 4 here Now it turns out that in many problems In quantum, in even in extreme theory You know, this answer is sufficient To reconstruct your answer here But as I said, you know In general, you might have First of all, you might have a multi-instanton problem This is having different instanton sectors Here I'm assuming there is one single instanton sector Parameter by a single C But you have a different instanton sector Which is very often the case Then, you know, you will have to have here C1, C2 And you will have to have a much more complicated structure So there will be more constants in there You have a higher order ODE You will also have to consider a higher amount You know, there are examples of the high rank The or product of gauge ropes Each will have its queue Absolutely Absolutely You will have many queues Depending what is your gauge Yeah, absolutely So basically counted by Schoencklach Yes, absolutely So this is just to kind of not complicate my notation So let me give you Let me now discuss two recent examples In which these techniques have been applied So let me mention that it might happen That the value of C that you have to use In this story is zero Is there some sort of pole that I have to use? Oh, here So it might happen that the value of C that you have to choose here is zero So for example, in the original example I started with Which is the quartic oscillator in quantum mechanics It's known that just by choosing C equals zero Means that here just the perpetrative series is enough And this is what happens in the example of the quartic oscillator I started with It's known from the work of Simon and Grafian Grecki That in order to reconstruct the ground state energy of the quartic Of the quartic perturbation of the harmonic oscillator It's enough to do Borel resumption of the perpetrative series There are no singularities in the real axis So you can just do S plus and S minus are the same So for the quartic oscillator Is a very simple case of semi-classical decoding I mean the quartic You know you take C plus minus is equal to zero And you have a very simple case of semi-classical decoding The simplest case of semi-classical decoding It's when the original non-pertrative quantity Exact quantity can be recovered Just by Borel resumption of the perpetrative series But and this happens so often But not always So for example, for the double well oscillator Is no longer the case Double well oscillator You have to use a non-treatable series And of course every time you have You have to use lateral Borel resumption You will have to have a non-treatable value of Z Okay, so that's a But in some cases, you know Semi-classical decoding is very simple Okay, so let me give you Well, I don't have much time left But let me give you an example in quantum mechanics So which is the So this is something very well known Which is the WKB expansion So let's assume that you have Let me start with a standard thing So let's assume that you have a standard Quantum mechanical Hamiltonian So I put the mass equal to 1 half Or to save a little bit of time So what is the WKB method tells you The all orders WKB The all orders WKB method tells you that There is a There is a Simple differential So you take You consider the classical The classical curve defined by this equation So this defines Faction P of Q up to a sine So you have a non-treatable differential here And WKB thought actually gives you a serious A serious of differentials And then If you let's assume that this Hamiltonian has A discrete spectrum as before So let's assume that we have a confining potential As it has been the case In all these In all these examples So you can actually compute a perturbative series in h bar In actually in h bar square Out of these Differentials So essentially As follows You have your potential You have your constant energy So this defines a cycle On this curve So let me call this The B cycle for historical reasons And then what the WKB method tells you is that You can construct a perturbative series in the energy Which only involves the even Differentials here The others give you trigger contribution So and let me call this The perturbative quantum volume Which is a function of A and h bar So this is a formal power series in h bar Where each coefficient here is a function Let me call it A n A function of the energy So this is interesting because In contrast to simple series in quantum mechanics This is a parameter series So it's an series in h bar But each coefficient is actually a highly non trivial function of the energy And actually this is called the I call it the perturbative quantum volume Because the first term A0 of E Is actually The integral of P dq Over the B cycle Which is the available The classical available volume in phase space So WKB method defines a formal power series in h bar And the reason why this power series is important Is because This can be used To find a semi-classical approximation to the spectrum So Yeah, yeah, yeah, they are metamorphic And is it the contact I mean, is there a genus? Yeah, yeah, of course there is a genus Right But you only have one B cycle? I mean, in this case And of course there are many cycles in the complex space But you know You want to consider the perturbative Actually, André is a whole expert on this He can tell you everything about this But if you want to consider the perturbative Approximation to the volume In this case in which you have a confining potential This is the only cycle that intervenes At the perturbative level Now the question is When you consider precisely the trans-series Attitude to this object You will have to consider all other cycles In this Riemann surface Not only this one This is I will only hint at this When you say volume you mean Or in the phase space below the energy E Yeah, yeah, yeah This is the volume This is the volume of the region with energy less Or equal than in phase space Okay, that's This is just a name to To name this quantity Because Because this quantity I mean this quantity Actually It's very interesting But because this quantity Doesn't have as far as I know There is no Non-perturative definition of this quantity In spectral theory So WKB defines this quantity But as far as I know There is no Exact Generic Generic I mean Exact definition Of this volume Of volume Of this volume function Which is a function of H bar In quantum mechanics That's a very interesting thing And actually Andre knows a lot about this Because In his paper with Balian and Parisi in 1978 They actually pointed out this They actually said You know We use this function all the time Because Why we use this function? We use this function because The generalized Bore-Sommerfeld quantization condition Tells me that This perturbative function Is equal to 2PH bar M plus Say K plus 1 half Where K is the quantum number And this defines A formal power series Which is Again Of this type As a function of the quantum number But so we use this quantization We use this formal Volume function all the time As a form of power series But there is no Exact definition of a volume function In quantum mechanics For some potential theories Yes I know I know that But But there is no It's not like in spectral theory I mean spectral theory you know That if I give you the Hamiltonian Hamiltonian has a spectral properties In this case if the operator One over H bar is compact One over the Hamiltonian is compact and so on You have defined non-pertotaphyases of But this function There is no procedure to compute it Well there is something that Generalized bit on that square In fact that's what you want to construct Yes Yes That's one of the things that I'm going to Also I would propose to them For all algebraic integrables Absolutely Absolutely Yeah yeah yeah absolutely But there is no general definition Of this in general Okay so I give you an arbitrary potential An arbitrary Hamiltonian Even in one dimension It's not so easy to compute this function So the question now is then If one can actually define Is there a way of defining this In maybe in some cases And if so Is there a semi-classical decoding for this function Because this is precisely the kind of situation In which semi-classical decoding It's important in the sense that We have the perturbative part In some cases we might have an exact definition of this So the question is posed Can we actually find a completion of this Protortive series to a trans-series Such that this Borel Resumation Gives you the exact volume And actually as you were pointing out There are some cases in which this is the case But as far as I know There is no general statement About the semi-classical decoding of this exact function So for example To refer to the work of Of Samsung For this potential For this potential For example you can define an exact volume function But as far as and of course this volume function Exact volume function has this asymptotic expansion But it's not obvious that How you would reconstruct this exact volume function With resurgence using semi-classical decoding Okay This is something that has not been completely done Okay But so you see that already in quantum mechanics The potential semi-classical decoding Once you go beyond energy levels and so on Is interesting and precisely in that type of problems In which you have Serious Formal power series in which The coefficients are then cells functions Of another parameter say like the energy source So these are actually two parameter problems For these two parameter In this case the answer in a bar Is exact and expansion is in the C What you called C Right, right But Convergent expansion in Absolutely Yeah, yeah But this is the exact definition of volume But the question is for example There is one example that André studied In which for example Which is not so different Because you know this also has No, you know this also has no Just two turning points And nothing else In this problem This trans-serious structure Of this volume function Is actually quite non-trivial Right As André studied So So it's not obvious You know what is I mean Moving interesting I think to Kind of find some semi-classical decoding For this type of integrable models In which you have an exact volume function Okay The same way that André has its own approach to this Based on the volume function As the solution to a fixed point equation Right Which is slightly different from here Because you need parity to actually You have two different functions defined by parity But again, you know These are examples in which Already in quantum mechanics Which seems to be a very kind of Well understood system In which There are still many interesting questions In which new results New non-protective results Actually open again The question of semi-classical decoding And on trivial way And now in the five minutes that are left I just want to mention Of course street theory Also Ricardo will talk about The street theories in detail But But now in street theory We have In a sense a similar situation to what happens In quantum mechanics for this volume function Because We don't have non-protective definitions Of this Of quantities So let me mention just five minutes Topological street theory So again I'm going to consider a simplified version So I'm going to consider Topological street theory So x is going to be a Calabi-Yau 3-fold Also in most of my cases In most of the examples I'm going to take x to be toric So we can do a lot of progress here So in In topological street theory You define a formal power series For the closed stream sector Which is Again this is purely formal And you see it's not so different from the quantum volume In the sense that Oops I already raised that In the sense that you have a power series In gs gs is called the string coupling constant But each coefficient is itself a non-trivial function Of the moduli of the Calabi-Yau So again I'm just considering a simple case In which the Calabi-Yau has a single modulus And this is the string coupling constant But of course this extends straightforward to the most general case So topological street theory actually produces this for you And we now We know for example in the case of toric-calabi-yau manifolds There is a very systematic way of constructing this function As a function By using for example the topological recursion Or you can think about You can compute this by using a large radius In the large radius of the Calabi-Yau 3, 4 by using grommawit and theory and so on But the question is that this series is also factoring a divergence So it's known That these quantities diver like 2g factorial And this is one of the reasons why String theory and particle debris string theory Are not defined in an operative level Because the only thing you get from the Polyakov definition Or from genus expansion is this formal power series Now the question is There are like two questions here The first one is can we promote this to a well defined function And second if you can promote this to a well defined function Has this function a semi-classical decoding In which you get this as the perturative series And then maybe other trans-series terms And this is very interesting because We know that the Polyakov string theory gives you a lot of Geometric information of the Calabi-Yau manifolds So if there are non-trivial trans-series involved here We are opening the way to having new sectors Of string theory Which might contain very interesting information About this Calabi-Yau We know that this quantity is essentially Count bullshit instantons But maybe you find a trans-serious object You know, as in this object The trans-series will count Maybe we will have some enumerative content And we may be able to count Some objects in this Calabi-Yau manner Okay So there was a lot of work in the recent years In trying first of all already to define Trans-series for this object And for me the most successful attempt to do that Was the paper by Koso Santamaria Esquiapa and collaborators And what they did was very clever What they did was to use The BCOV equations A set of what are called the holomorphic anomalous equations So the holomorphic anomalous equations The nice thing that they have Is that they are recursive They have a recursive structure So every time you have a recursion You can try to enlarge the recursion To an ansatz of the form of a trans-serious And they were able to find Formal trans-serious extensions Of these quantities In general The only price they have to pay Is the same price that you pay With the holomorphic anomalous equation BCOV That these didn't really define the FGs Or the trans-serious extensions But they defined an up to a holomorphic pattern Big with it But modular that They were able to actually do this And so this paper gives a trans-serious Formal trans-serious construct What is the meaning of C? What? The instant uncounting question What is in this case? There has to be another parameter for it Yes, yes, yes, yes What is that? Yes Well that's actually for me It's essentially an open question What is the geometric meaning of these trans-series? There is no parameter in this theory What? Parameters are already there You have moduli, you have three coupling concepts What happened? No, but this is also in quantum field theory You know Instant was also involved for just Calculable If you have an action Yes, yes, but here we don't have a straight field theory action That's to do that Exactly Is straight field theory defined? Absolutely But then you know what you can do In the absence of straight field theory You can try to see how much the Pertorativistroto you already know That's what it is And what is nice about trans-series is that Even if you don't know What is the organizing principle behind straight field theory You can still compute Using these sort of formal methods You can compute a formal trans-series I'm not saying that we say We know everything about this formal trans-series But at least now We have some tool to compute it And this tool produces concrete things And one of the beautiful things that they did Was for example To check that these trans-series Actually control In extreme beautiful detail The large term behavior of these quantities So this is the first test That you have to do When you want to understand a trans-series I guess Riccardo is going to tell us more about this So here we have at least The ingredients for a semi-classical decoding And I guess Riccardo will also Tell us more about this In the sense that We now have at least Some sort of non-partuitary definition of this quantity Using essentially quantum mechanical models And spectral theory And again the question is posed of How can you relate A non-partuitary definition of this guy To these formal trans-series And Riccardo I guess will tell you more about this But this is just to show you That these tools of resurgence Which go back to the very elementary Aspects of quantum mechanics Have been extremely useful to understand problems At least to make some concrete progress In problems which are seen much harder Which is for example What is the non-partuitary definition of stream theory And also there have been many attempts To solve this problem The people who work in resurgence like me What we wanted to do was To follow the advice of Toft When Toft was looking at the normals He said We have to use as much as possible The information contained in perturbation theory And resurgence is in a way doing that You try to see Everything you can try to Use as much as possible the information In perturbation theory to try to see What is beyond perturbation theory And trans-series are a way to formalize this And in the case of stream theory There was an actual progressing understanding Concretely these sectors beyond perturbation theory Used in terms of trans-series But of course many problems remain And I hope that in the future There will be also some answers to these problems So let me stop here A few questions here From the point of view of identifying these two Geometric engineers that we have made so far Yes, you calculate that There is a meaning of Q Because in the case of series there is Q Yes Are you mean C? Yes So is that Q coincides with C of No, no because you see When you do geometric When you try to use Is true that this is related to Information in supersymmetric theory Yeah, yeah, yeah But these extra ingredients are not Part of cyber-witting history So as far as I know It is not No, no, no, no Because the instant of information Of cyber-witting theory Tells you world city instant of information Into political stream theory So it's not telling you a space time Information into political stream theory So it's different information Into political stream theory It's also new from the point of view Of the cyber-witting connection But I remember you personally Did you reconstruct non-perturbative answer From the From cyber-witting with one epsilon? Like you had a claim That you wrote a non-perturbative conclusion Yeah, yeah, yeah Using this quantum mechanical model Of this cyber-witting theory You can't reconstruct for example What is then this C there In your reconstruction You can tell us what we're doing Well, this has not been There is no It's still not clear How you semi-classically decode this And what is the geometric information In this I can't write for you this trans-series Now that I think for me One very important question The Q is that C is fixed Yeah, yeah, yeah This C is fixed Exactly This C is fixed by my definition But even in simpler cases You know Let's take 2D gravity Let's take panel level 1 Panel level 1 You know that the Coefficient of panel level 1 Of this Perthroat disease Half interpretation as some Sort of intersection numbers In the modular space of curves Now take the trans-series solution to panel level 1 Have these coefficients there some interpretation In terms of intersection theory With boundaries or something like that I mean Even in that case Which is much more elementary We don't I don't know at least what is the If there is a geometric interpretation For these trans-series And this is a contesting Which this series has a geometric meaning You see as a non-critical string theory So yeah In the quantum mechanical case On the left hand side Yes You will have a better equation Lambda's will be faster Quantities Yes You don't have to Sure You don't have to work it But in a way Absolutely Yeah Lambda's But in the topological string Yes There is no reason For lambda's to be quantum Yeah, absolutely Absolutely Yeah, yeah, yeah From the left hand side I would say that In things I understand There is a way to go of shell Yes Which is not unique And the right hand side It's always of shell Yeah, but let me point out Yeah, that's It was that this is actually What André remarked in 78 Because he was saying The volume function You see It's quantized as well Because k is quantized So this only defines e for a single You know for But if you have The exact volume function This defines the volume function For all values of e So how is this possible So when you have an exact definition In the You know You might have a way Of going Beyond this discretization Which the quantum mechanical Structure imposes So in some cases Actually even in this quantum Mechanical definition Is true that In the first approach In the first approach You have to quantize lambda But then you can go beyond that And write a general answer So physical is only quantized So physical Think is only quantized But this is But this is like in large Endualities where you know Every time you have like You try to construct a At stream theory With a gaze theory You know Gaze theory is discretized Because the rank of the gaze group Is an integer And And then you know That this should give you A geometric parameter in a space time So it's always the same problem here It's not different And it's also the same problem In the WTD formula Of the volume So That's always the case Okay