 Okay, thank you very much. First of all, I want to thank the organizers for the opportunity to give these lectures. The practical information, there are lecture notes, latex lecture notes that should be posted on the web page of the school, I guess. So I send them to the secretaries, I should be there. And my lectures will be on number two FFX in string theory and ADSEFT. So what I'm going to try to do, by focusing on some particular examples, to show you how you can use ADSEFT to learn about number two at the fast speed of string theory. Because the ADSEFT correspondence should, if you take it in a strong version, should provide a number two at the definition of string theory and some backgrounds, and in particular should be able to compute number two FFX in string theory. But since the issue of number two FFX is quite delicate one, I want to start these lectures by spending at least one, or maybe one or a half lecture, by telling you a little bit about number two FFX in general. So in general, here number two FFX, you can apply to some other problems of string theory. So I will then start with a cross-course on number two FFX. And the first thing that you have to understand, to see why number two FFX are important, the first thing you have to understand is that most series, perturbative series in quantum theory, and this includes quantum mechanics, are not convergent. So this is something that the people who taught you quantum mechanics in second or third years who have told you probably didn't tell you. And actually the simplest example of this situation is actually a quantum mechanical example that I always like to discuss. And this is an example that you might have studied in quantum mechanics is the quartic oscillator. So you take a Hamiltonian given by a standard harmonic oscillator like this, and then you have a perturbation that you can write it like this. You know that this Hamiltonian is exactly solvable, this is the harmonic oscillator, and then once you add this perturbation, you cannot solve for the energies or the eigenstates of this Hamiltonian in closed form, and then you use a stationary perturbation theory for dealing with this. This is something that, as I said, you must have done in elementary quantum mechanics. And when you do that, for example, you can calculate the energy of the ground state that I will represent like this, and what the stationary perturbation theory gives you is some sort of synthetic series that you can compute. So this would be the first term that you would get, and then there are other things that you can compute. So of course this is the harmonic oscillator energy, ground state energy, and then you get a formal power series, a priori this is a formal power series, and I will be clear about what I mean by a formal power series in a moment in the coupling g. So also this is an elementary example, it's actually the basic example to understand the generation of perturative series in quantum theory. So let me write this in this form, let me write this as a series of coefficients a n times g to the n, and then the first thing you observe is you actually compute enough of these coefficients, and this is actually something that was observed experimentally by Bender and Bou in the 70s, Bender and Bou to the harmonic oscillator, they computed 100 terms of perturative series, and then they wanted to understand what is the structure of this series, what is the nature of this series, and what they found is that these guys behave like this. When n is large, they grow factorially, much bigger than one. So this means in particular that this series that let me call it phi of g, so this is this formal power series, this is in principle different from this quantity, as you will see in a moment, so this formal power series, as I was saying, is actually divergent for all values of g. So it has zero radius of convergence. And this is important because this is the generic behavior of series in quantum theory from quantum mechanics to stream theory. So as we will actually see in a moment, also the perturative stream expansion also diverges factorially. So you have to actually wonder what you have to do, what you can do with this series, right? So this is the first question you have to ask. Now notice that here I wrote this as a formal power series different from this quantity. Now this quantity, the ground state energy is actually well defined quantity because in quantum mechanics you can actually solve the Schrodinger equation for the stationary energies and this defines A0 of g as a solution to a spectral problem. So the ground state energy is a well defined quantity. There are a lot of mathematicians that have worked to actually show that this is a quantity which is well defined for a potential like this, you have a confining potential, and then the ground state energy exists as a function of g. But when we use perturbation theory to try to compute this quantity, what we find is actually not something that can be used, but something that a priori is a formal power series. You cannot use it at phase value to actually compute energies. Now this is important because in a sense what happens in string theory is we have the right-hand side of this equation. We have a perturbative series in the genus, but we don't have the left-hand side. We don't have an operative definition in terms of a well-defined object. String theory a priori was defined as a perturbative series, so this problem of how to make sense of a perturbative series is much more important in string theory a priori. We will see actually how, we will see in these lectures, hopefully, that in some circumstances string theory provides an unperturbed definition of this right-hand side, and then we can try to make sense of the series that appear in string theory thanks to this ADS safety correspondence. But a priori in string theory we have something which is similar to what happens in quantum mechanics because we don't really have a non-trivial Hamiltonian and a well-defined Hamiltonian that we can use. Now, in any case, of course, you could have complained to your professor of quantum mechanics why are you studying stationary perturbation theory? After all, if this gives me a divergent series, what can I do with it? Unfortunately, this series is not completely useless, so you can use it. But before getting into how you can use this, let's actually give some general definitions which I think are good to keep in mind because there is a lot of confusion in the literature about this. So, given a formal series f of z where z has to be understood as a formal parameter, we say that a well-defined function f of z provides a non-partuitary definition of this series of phi of z if f of z has an asymptotic expansion given by this formal power series. So this means that f of z, and this is the sign that you typically implied, you don't write actually an equal here, you write a like symbol is given by f of z which is this series. So what happens is actually that many well-defined quantities, for example, the Rathus detergent, this problem, they are well-defined, but they have on top of it an asymptotic expansion in terms of a formal power series. Now, if I give you the formal power series and you can find a function such as it has an asymptotic expansion in this formal power series, I have found that well a good non-partuitary definition of this series thanks to this function here. So many functions, for example, the gamma function have asymptotic expansions, so the functions are well-defined, the asymptotic expansions are formal power series, and we say that we have an operative definition of a formal power series if we can reconstruct the well-defined function which is behind it. Now, this procedure, the bad thing about it is that it's intrinsically ambiguous. So there is an ambiguity because this is an expansion on z equals zero. So I can add, so if I have this formal power series, it will be also the asymptotic expansion for a function fz plus any non-analytic quantity as z equals zero. So, for example, if I have the asymptotic expansion of a function fz and I add to this function fz something which is non-analytic as z equals zero, I will get also, this function will have also the same asymptotic expansion. So, and this is just because these kind of functions are invisible if you make an expansion on z equals zero because you have an essential singularity. So this means that if I have a formal power series, I always have an ambiguity in reconstructing the well-defined function which is behind it. And this is a very important fact and it's called the non-pertorative ambiguity which makes life very hard for reconstructing well-defined quantities if you are only given a formal power series. So, in a sense, the goal of my lectures will be to show you how in certain cases we can make sense of the genus expansion of string theory by finding a well-defined quantity which has an asymptotic expansion, the genus expansion of string theory. And this we will do thanks to the ADSEFT correspondence. Now, as I said, once you have this setting, once you realize that formal power series and actual functions are different and they are related through these asymptotic expansions, you have to ask the following question. How can I actually extract information from an asymptotic series? This is a very important question because after all, this series would be good for something. Now, in order to extract information, and when I extract, when I extract information, I extract actual information. So you might want to calculate, you know, with some precision the ground-state energy in this quartic oscillator, how would you use this series in order to extract information about the value of this ground-state energy? Well, there are many answers to these questions and the answers are, as I said, there are many answers because after all, this is not a straightforward problem. You know, the series, you cannot just take the series and plug in a value of g because, you know, if you take many terms in the series, it will give you a big divergence. So you have to make something about it and one way of doing this, which is the most primitive one but this is also useful, is to what is called optimal truncation of the series. This is also a way to understand why non-pertructive effects are important. So what do you do in optimal truncation? In optimal truncation, what you do is you take your original formal power series, so you take this original formal power series and you truncate it. So you keep only a given number of terms. So you keep only n terms. Now, when I meet, I hear I introduce something that I didn't define which is asymptotic expansion. An asymptotic expansion is such that these truncations give you some good approximation to the original function fz if z is a small enough. So what happens is the following. Imagine that you want to calculate fz from fnz and let me plot, let us plot the difference between this function fz and the truncation of this formal power series. And let's plot this as a function of n. So here I plot this difference and I see how it goes. Now, if z is sufficiently small, as you include the, so you can include one term in the series, you can include two, and so on. So what happens is that as you plot this difference, it turns out that if z is sufficiently small, the difference starts being small. This means that as you keep, for example, the three or four first terms in the series, you get a good approximation because this is decreasing until you reach a point, a given value of n, which I'm going to call n star. And once you reach this point, including further terms as the series, doesn't make it good. So after all, remember that this series, the original series is a divergent series. So this means that at some point, as I keep adding terms, I will get something that goes to infinity. And this happens after a given value of n. So this is the typical behavior you find when you compare the truncations to the exact value. So there is an optimal value, there is an optimal value of n, which gives the best approximation to the original function. And this is why this procedure is called optimal truncation. So in a sense, when you have an asymptotic series, its use is very limited. You cannot actually use, if you use this procedure of optimal truncation, you cannot really use all the terms in the series. Because if z is not very big, this n can be actually very small. So for example, I will actually propose you an exercise so you can do some numerical experiment, but it turns out that the truncation here, n here can be estimated, and it's proportional to 1 over z. So when z is very small, of course you can include a lot of terms in the series, and your optimal truncation will be quite good. But if z is sufficiently large, maybe n, n star is actually very small. So essentially, most of the series is actually useless. You can actually make a very precise estimate of how this goes, yes? Yeah, yeah, this is actually a good question. You always make an error here. So this is never zero, and this is something that I will mention now, and it's actually related to why you have to introduce number 30 effects. This is the first way to see how you should introduce number 30 effects. So actually it's a very good exercise that is very easy to see that if you have some behavior for n, n, n of this form, you have a factorial behavior with a subleading exponential factor as we have, for example, for this quartic oscillator, you can actually see that n, this n optimal, goes like a over z. And the error, so the error that you make at this optimal truncation point turns out to be exponential of minus a over z. So that's actually something that you can do as an exercise. It's very easy because what is optimal truncation? Optimal truncation is that you should minimize the error. And then if I tell you how these coefficients grow, you can see that the error is minimized for this value of n, and then the error you make is of this form there. So this is very interesting because it tells me two things. As I said, it tells me that the smaller z, the more turns I can use in my truncation, and vice versa, the bigger z, the less turns I can use in my truncation, and it also gives me that I'm making an error. So this is the hate that you were probably asking about, and this error is exponentially small. So this is the error I make. So this means that if I use optimal truncation, if I take a perturbative series and I use optimal truncation to get the value, the best value I can get on my physical observable, I will always make an exponentially small mistake, and this exponentially small mistake should be corrected by something. And this is the first indication of a non-perturbative effect. And non-perturbative is just meaning here that this error is something that is a non... This type of functional dependence, as I was mentioning before, has a sensor singularity that says it equals zero, and will be invisible in any power series description that I make of this. So this is a first indication that if I use this power series, I have to add something to correct this error. Now, this is actually what many people do when they use divergent series, and as a second exercise, so to make this very concrete, I propose you to do the following exercise with a very simple example, which is if you want a zero-dimensional toy model of the quartic oscillator, you take what is called the quartic integral, which is a perturbed Gaussian integral. So this is very much like a perturbed path integral in zero dimensions. So you take a Gaussian integral perturbed by a quartic term. Now, of course, you don't know how to evaluate... Well, actually, this can be evaluated with vessel functions, but imagine that you want to evaluate this integral by doing a power series expansion in terms of g. Now, this will have an asymptotic... an asymptotic expansion in powers of g, just as the quartic oscillator... Okay, so you can think about this as a toy model for the quartic oscillator, and the coefficients can be calculated in closed form. So let me write in... This is part of your exercise, and you want to go through it. So these are these coefficients. And now, what you can do in order to verify this statement is actually do the procedure of optimal truncation for this series, and since this integral you can compete numerically with a lot of precision using mathematics, you can see for different values of g what is the error that you are making, you can actually see that this estimate is actually quite good. This gives you the optimal truncation, and you will be able to see this plot very clearly. I mean, if you want some help, I can even tell you how to do this with few lines in mathematics. It's very simple. So this is a general feature which also applies to elementary problems, and this is actually what many people do with quantum field theory calculations. I mean, quantum field theory, after all, this is what you generate in perturbative series. These series are also this type. They are also divergent, and sometimes if your coupling is sufficiently strong, you have to do this optimal truncation to use the perturbative series that you generate in quantum field theory. Yes? Sorry? Oh, yes. Well, it's in the notes, but I will do it. I mean, in the notes, everything I say is written here in the Latin note, so you should be able to download them or print them. So this is part of this exercise. So you have a double factorial divided by a factorial, and this grows in this way. This grows in this way, so the double factorial grows like a factorial squared, so you get a factorial. Okay? So this is a good exercise to see how this optimal truncation goes. Okay. One bad thing about optimal truncation is in a sense that, as I was saying, you cannot really use all these terms in this expansion, right? You always make an error, you cannot really use all the terms in this expansion, so there should be a better way to actually understand number 30 effects. And this is the technique of Borel Resumation that I'm going to summarize a little bit for you. So how can you improve? So improving optimal truncation, and this is what is called Borel Resumation. So the technique of Borel Resumation is actually very powerful because the technique, as we will see, allows us to produce a well-defined function in some circumstances starting from a formal power series. So it actually produces a function. It's not like optimal truncation that gives you some approximation. It actually produces a function in some good circumstances. Now you can ask yourself if this function is going to actually agree with the function that you want to reconstruct or not. That's the big question. At a priori, it gives you a way to make sense of the formal power series which actually includes information from all the coefficients. So it actually justifies the effort that people put in computing many coefficients in Borel Resumation. You use them all. In optimal truncation, if the parameter is sufficiently large, you cannot really use all these coefficients in the power series. So that's why Borel Resumation is so powerful. So what is the trick of Borel Resumation? Well, it's actually a very funny trick. So notice here that in the behavior of these coefficients you have two sources. You have the leading growth which is factorial plus a leading growth which is only exponential. Now what is making the radius of convergence of the series to be zero is the factorial growth. So what Borel proposed is something very simple. Take my original function my original power series f of z and let's define a modified power series in which I just remove by hand this factorial divergence. My original coefficient I divided by m factorial and then I create another power series. So since this power series has a sub leading growth which is exponential this series will have actually a radius of convergence set by a. So you have here this series now in this z plane or Borel plane which has a radius of convergence z by a. Now this means typically that this series has a radius of convergence by a and this means that typically you will find a singularity somewhere in the boundary of this radius of convergence. So for example a very simple example if your original power series formal power series is just minus n over n a on n n factorial z n then the Borel transformation this is called I don't know if I mentioned this is called the Borel transform so this series is obtained in this case you just remove the factorial and then you get minus 1 n a n z of n and notice that this is just the series 1 plus sorry this is minus n here minus n and then you get z over a. So you see here you have a completely meaningless function now you have a function which is well defined and actually this series has radius of convergence a and the reason is that you have a singularity in this function in the negative real axis here so is this singularity which is causing the finite radius of convergence of this Borel transform so this is a procedure as I said that takes the original bath power series which has no any sense having any sense as a function and produces a function which at least is converging in a neighborhood of the origin and then maybe you will find poles you will find singularities but at least you are safe inside a region of convergence around zero now the interest of the Borel transform is actually that is the inverse of the Laplace transform suppose so this is also an interesting piece of elementary mathematics that if you want to invert the Laplace transform you have to do exactly this Borel transform so let us suppose that this function i of zeta can be extended to a neighborhood of the real axis positive real axis in such a way that the Laplace transform let me write it I am going to denote it like this exists now if this is the case this function is called the Borel resumation of the original series so for example in this case here you see this function has a singularity here when zeta is equal to minus a it can be definitely extended along the positive real axis and this Laplace transform is actually well defined so in this example here exists and then you see that this quantity we have started with this power formal power series we got this analytic function and it is Laplace transform exist and actually this function it can be seen that this resumation has the asymptotic expansion given by the original power series so to summarize if this procedure works I start with a formal power series which is necessarily you produce an intermediate object something which is defined in a neighborhood of the origin and then with this Laplace transform you actually reconstruct a function and which has asymptotic expansion the original formal power series that you obtain so this is the best you can have in order to make sense of a formal power series in quantum theory for example and it's also a very interesting exercise to do it for this example here so here I produce you a series I propose you to verify that the Borel transform of this series is given by an elliptic function k square is equal to one half minus one over two square root of one so this is also an exercise that you can do in this case for the power series that was obtained by doing the asymptotic expansion of this integral you can also do this transform and you get a function which again is analytic at the origin and is given by an elliptic function this is the elliptic integral of the first class so it is also a very nice exercise and now coming back to the general theory the question is the following now you manage by this technique to produce a real function which is well defined obtained from this Borel transform from the original power function so the question now is what is the relationship between the original physical observable that we want to compute say the Borel resumption so this is a key question so here for example what did we do we had this integral which is well defined we do this procedure eventually we get this Borel transform and now you can actually see here I also suggest you this as a continuation of this exercise you can now compute this quantity as a function of say let's call it G again because here I was calling G the function so here we started with this integral we got asymptotic series and after doing the Borel transform I got another well defined quantity which is this Borel resumption so I have two well defined quantities the original quantity I started with and this thing that I reconstructed not knowing this guy but just knowing this asymptotic expansion so the question is is this guy equal to this guy or not that's the question you have to ask because if this is the case then it's very good because this means that this goes from I don't know if there is a good object behind but still I can work this formal procedure and I get at the end something that actually reconstructs my original quantity now in some good cases some cases this works so this means that perturbation theory reproduces exactly the quantity you started with so the examples are quite elementary so this is again as an exercise it works for the example of the quartic integral so this is something that you can check again I gave you all the ingredients here I gave you this Borel transform so you can just again do the following experiment in Mathematica you take this Borel transform you do this Laplace integral and you compare this Laplace integral to the Laplace integral I gave you and you will see that they match on the spot so in this case I was able to reconstruct this by doing this procedure now this is of course an academic exercise but actually a very important thing is that it also works for the energy levels of the quartic oscillator that I started with so in some cases you can actually make it work in quantum mechanics and this was a very important result in the 70s by Barry Simon I guess also and Grafi and other people which actually show that this asymptotic series that you get in quantum mechanics that you study in second or third year actually makes sense but in order to make sense of this power series you have to go through all this work this is something that they don't teach in quantum mechanics they should but it makes sense in the sense of this power series now unfortunately this procedure does not work in many cases so in many cases you cannot really reconstruct your original answer and this is where things become a little bit more delicate now why can this fail why can this procedure fail well it's very easy to see where it can fail well it can fail in a very obvious way it might happen that when you do when you consider this Borel transform here we were lucky and the singularity was here but now imagine that I removed this sign and A is a positive number I removed this sign now I have here a minus sign so before my series was an alternating series now the series is not an alternating now what happens when I do that well my Borel transform now will have the singularity precisely on the other side and now I have a problem because now this integral over the real axis will hit at some point the singularity so this integral is not well defined so this is the most obvious problem that you can have when you do this procedure and this actually introduces a source of worry okay so let me then name why this can fail so what time is it so I have 50 minutes so in some cases Borel resumption doesn't exist as I define it due to singularities in the real axis the positive real axis and actually this actually happens in physical situations so let me give you an example where this happens which is a relatively important example which is another example from quantum mechanics and this is the inverted quartic oscillator here what you get what you have is a Hamiltonian where you have an oscillator that is inverted so what you have is something like this well sorry well inverted in the sense that it's not so you have something like this and you have two minima here and now what perturbation theory tells you well perturbation theory tells you first of all that there are two degenerate minima this one and this one and they should have the same energy so there should be two wave functions here with the same energy and actually you can compute the energy of this perturbative states and you get something like one half minus g minus 912 g square and so on and notice something very interesting in contrast to the power series that you find for the standard harmonic oscillator with aquatic perturbation now the series here is non-alternated and actually this series of power series is not resummable with Borel transforms precisely because when you do when you do this procedure of Borel resumption you find a similarity on the positive real axis and then you don't know what to do and this is actually not a technicality this is reflecting something very deep about this problem and it's just the fact that perturbation theory is completely wrong because in perturbation theory you have two perturbative ground states but this is completely false because we know that in this problem you have only one true ground which corresponds to the symmetric combination of wave functions centered on this minimum and the reason here is that by tunneling effect these two vacchia get mixed so here the impression coming from perturbation theory is completely wrong in the quartic oscillator where you have just a single well perturbation theory is approximately right and then if you go through all these business of Borel resumption then you get things right but here you get things completely wrong and the mathematical way in which this is translated is that the formal power series that you get in perturbation theory is not Borel sumo so you don't know what to do about it now you can you can do many you can try to solve this problem and people have looked at it in detail and the answer what is going on here is already answered by this thing by the physics of this problem so you have a tunnel effect which tells you that you have to be very careful with perturbation theory now in terms of the path integral this means that you should include other saddle points you should include non-trivial saddle points in the path integral so the way you solve this by the way you go beyond perturbation theory in the path integral framework the way you would see actually that this problem is like this is the following well it's clear that when I look at the Euclidean path integral right I will have to invert my potential because I'm going to do another continuation and there here I will have two trivial saddle points in the path integral and I will have a minimum here now if I do fluctuations quantum fluctuations around this minima I get this guy here that's very good but there is a whole new sector in the path integral which corresponds to a non-trivial saddle point where the classical solution is not just a particle sitting here or a particle sitting here but a particle that goes from here to here and comes back sorry one goes here one goes from here to here another one that goes here that starts from here and goes here and so on so you have actually non-trivial saddle points and what happens is the following when you do the path integral around these non-trivial saddle points you find that let me call this F0 so this is the formal power series now if you include the additional saddle points and you do the path integral around these saddle points you get just by doing this path integral you get a new formal power series of the form so in this case for example you will get a power series which also depends on g it's obtained by doing the path integral around this non-trivial configuration and it actually looks like let me just write it schematically like this and then you get here a higher power series of g so what is this structure of this new power series well here you have something like you have a power series in g in the same way that you have here a power series in g but the most important fact here is that this power series now is exponentially suppressed with respect to the original one because it has here this factor and this factor in the path integral has a very clear interpretation in the path integral in the clear path integral this is the exponential of minus the classical action of the non-trivial saddle so here as I said there is a non-trivial saddle point which is a trajectory and if you calculate the action of this non-trivial saddle you get exactly minus 1 over 6g this is also a nice exercise so this means that actually just considering this power series doesn't help here you have to add all these additional things now actually in the quartic oscillator life is very different because here you have this potential and then if you go to the Euclidean formulation and you invert it you don't get here you cannot have a non-trivial trajectory results from one point to another you just add here g times q to the 4 over 4 as we were doing in perturbing the standard harmonic oscillator you don't get the positive of having new saddle so the only power series you have is the power series that you started with in this double well precisely the non-trivial structure here it's the non-trivial topological structure of this potential tells you that there are new saddle points but this additional series is not enough for the problem you have to add additional series and this is good because with this additional series the Borel resumption procedure doesn't work so something else has to happen in order that we actually solve our problem so in general so in general in the presence of non-trivial saddle points you have to consider this sort of series of more general series so these series are obtained by doing the path integral not around the trivial solution of the question of motion but a non-trivial solution of the question of motion so here for example it's also a very nice exercise that you can do is to show that the original perturbing series of this problem that we started with can be actually obtained by doing the path integral around the trivial solution to the question of motion in the Euclidean case which is just this one this is a very nice exercise if you want to know more details I can explain to you how it goes if you have non-trivial saddle points in the Euclidean path integral you will have new series as I was telling you because you have these new sectors and these general series in general look like this well this is already a simplification but it captures the structure so remember that z is my general notation for a coupling constant so you have something like this and then here you will have a new power series in N so this looks very much like a perturative expansion but here you have to be careful because you have the exponential factor which as we were discussing before is not seen it's not seen when you are looking at asymptotic expanses because here anything I add here will be surprisingly suppressed won't be seen in this expansion and it's signaling the presence of non-trivial saddle points in the path integral and you also have some typical typical exponent for z that for example in this case here was g to the minus one half so its value depends on the model you are looking at and L is an integer values 1, 2, 3 and so on and actually in simple cases it can be seen that these saddle points typically come in families so for example here you can have a saddle point which you go from here to here or you can have a saddle point which you go from here to here and then you come back and then this has twice the action that you had before or you can come and go many many times and this number L is actually labeling all these family of saddle points that you can have in this situation of course in general you can have more indices but in simple situations you have something like this so we see that actually just starting with a power series or a power series and doing this with resumption is very good in some cases it works but in other cases you have to change your view of the physics you have to take into account additional quantum effects and this tells me that I will have to consider other series and these series as I said are obtained by doing perturbation theory but not around a trivial solution of the question of motion but a non-trivial solution of the question of motion which is in this case for example is this trajectory which goes from one point to the other or it goes back or comes and goes many times etc now so actually what you should do now is to put together your original perturbative series and all these other things so let me call a big phi of z so in general if I sum over all saddle points in the path integral I will have the series that comes from the perturbative vacuum and then I will have to add all these guys with maybe some coefficient so this is typically a free parameter and the important ingredient here is that all these series are exponentially suppressed with respect to this just because of this factor so you will never see these series when you do the perturbation series because mathematically they have this exponentially small factor around an essential singularity so you don't see them now this is purely formal this is purely formal you see I started my lectures and I told you that this series is very you have to be very careful when you write it because it's just a formal power series now I'm writing a bigger formal power series which includes the perturbative vacuum and this type of series are sometimes called trans series in the mathematical literature these are series which actually includes two small parameters you see when you do perturbation theory you have a small parameter which is your coupling constant here z but this series has another small parameter which is different from z because as I said it's invisible when you do a perturbation expansion around z equals 0 and this small parameter is actually this exponentially this exponential e to the minus a over z so notice that if a is positive this quantity is positive this quantity is also very small when z is near 0 it's exponentially small here you have this exponent here and this corresponds to the fact that these contributions are typically exponentially suppressed as happens typically in tunnel reading effects so these trans series are our way or mathematical including the fact that in a general quantum theory you will have more than one small parameter you will have the coupling constant but you will have additional parameters which are not reducible to the standard coupling constant they involve the coupling constant which is very small when you have a small coupling constant but you have to include them to take into account the different phases of the problem now this quantity here is very very very ill-defined because this already was very ill-defined and this was already very ill-defined now it turns out that in many examples in which this object is actually is giving you the perturbative series by itself you cannot reconstruct the original quantity but you can reconstruct the original quantity the original physical observable if you add this additional quantity here so in many examples what you have to do to reconstruct the original physical observable one has to consider Borel resumations of the full trans series Borel resumations of this object phi of z so what you do from phi of z you do this procedure of Borel resumations apply to phi 0 z so this is a well-defined this this I will explain in a moment how this cannot be a well-defined quantity but let's assume that this is a well-defined quantity and then here you will have z to the bl u to the minus l a over z and then you have to do Borel resumations also of the power series that accompany these non-verdative effects now this is often well-defined and reconstructs reconstructs the observable for example in the case of the double well which I was mentioning before I told you that this quantity is not necessarily well-defined and the reason was as I was explaining before that you have a singularity here and you cannot extend the Borel resumations so in these cases what you have to do is to introduce what are called lateral Borel resumations so the way to do this is simply by realizing that one way to make sense of this Borel resumations is to jump over the singularity and this is the zeta plane in which we are doing this Borel resumations and here you have the singularity instead of going through the this is the singularity here at a you have the singularity here you can actually define now your Borel resumations by doing the same integral that I was doing before sorry I was doing the z z z but now you take paths that go up or below the singularity in this way what happens is that this quantity now is well-defined because you are avoiding the singularity but it's complex to see and it's a trivial residual calculation that the difference between these two the imaginary part of this lateral Borel resumations is actually proportional to the e to the minus a over z so when you have a problem with Borel resumations in which there is a singularity here you can avoid the singularity by using this path now you have a place to pay yes I'm coming back to this in a second but you will see you can make this well-defined and the price to pay is that you will get an imaginary part for example in the problem of this double well oscillator this is exactly what happens you have a singularity here and then if you want to use this you will get an imaginary part but ground state energies no no no of course the imaginary part the sign will depend on how you go from here to here so the differing sign there is a discontinuity the imaginary part here minus the imaginary part here is going to be proportional to the residue around this point and this gives you this kind of calculation now so I'm coming back to your question now this gives me an imaginary path so what should I do about it because ground state energies are real you cannot have an imaginary piece in ground state energy now what happens is presented the following this guy here now let me do here with whatever prescription you want plus or minus and let me do here the plus or minus prescription so the result that one finds is that the imaginary part that you obtain here will be cancelled for an appropriate choice of this constant by imaginary part of this piece in such a way that everything is real at the end and the further the imaginary part here the ambiguity in the imaginary part here is exponentially small is precisely what is needed for this cancellation to take place so here I'm really skipping many many of the details but this is really an important question in physics because you studied that very well in quantum mechanics and there probably you have been taught that the problem with perturbation theory is removed by tunneling effects but if you want to really see how these tunneling effects lead you at the end of the day to a real energy to a real well defined energy from perturbation theory and from adding instanton effects then in these new subtle points you have to actually take all this into account so the final conclusion of this in which the Borel Resumations of this is not well defined typically the standard Borel Resumations you have to do two things you have first to do this procedure of lateral Borel Resumations so these are called lateral Borel Resumations and at the same time you have to include all these new sectors of the path integral you have to combine them together you have to put them together and then you have to do lateral Borel Resumations everywhere and the result that people find is that if you make an appropriation of this constant this will be completely completely consistent and will give you a real energy which is actually agreeing with the real energy that you obtain from the Schrodinger equation ok so the lessons that you want from all these is that first of all you can make sense in good situations from the Borel Resumations through this procedure of Borel Resumations but in short circumstances you will have to add something else to the theory in terms of these new saddle points and then you will have to combine these things doing maybe lateral Borel Resumations or something more complicated in order to make sense of the Borel Resumations as you see making sense of Borel Resumations in quantity is a very delicate business ok let me finish here