OK. So let me resume then my lectures. So to summarize what we learned in the first lecture, so we learned that most of perturbative series factorially divergent, so typically the coefficients grow like n factorial e to the minus n, and this leads to non-perturbative. So there are, let's say it's in a less committed way, there are typically non-perturbative effects of order e to the minus a over the coupling constant z associated to non-trivial saddle points. Sometimes one can make sense of the series by this procedure of resumption, by what resumption say. And when you do this, as I was emphasizing, you don't have to consider only, you not only have to consider the first perturbative series, but you also have to consider the series associated to these non-perturbative effects. And the goal at the end of the day is to reconstruct the true observable, which is behind this series. And this cannot always be done. In some cases, you are lucky, and you don't even have to include these non-perturbative effects. This is actually important to keep in mind. Sometimes you don't even have to include these. For example, you consider the quartica and harmonic oscillator in quantum mechanics. You just do a border resumption of the perturbative series. You get the right energies on the spot. You don't have to do anything else. In some other circumstances, you will have to add explicitly these non-perturbative effects and add, on top of the original perturbative series, these additional series, which I call trans-series. And you also have to do border resumption of them. OK, now, so this is a quite generic story. But what happens in string theory? So how all this story applies to string theory? Well, the discussion I was making this morning assumed that I only have one coupling constant. For example, in the case of the quartic oscillator, I have one coupling constant, which was just the quartic perturbation of the oscillator. Now, the most important aspect of string theory, from the point of view of this discussion, and also from other points of view, is that there are actually two coupling constants in string theory, which govern two different things. One coupling constant is the string length, S. And the other is the string coupling constant, which I'm going to call GST. And the string length actually measures how these measures quantum fluctuations are on the point particle limit. So strings are standard objects. And this string length, when it goes to 0, is the limiting which you recover point particle theory. And the string is just what describes the strength of the splitting and joining of strings. Every time you have a three vertex interaction of a string, this has the weight GST. So these are a priori quite independent quantum parameters, in the sense that you can formally at least have a string theory in which you take the string coupling to 0 and then you have G0 string theory. Or you can actually consider a theory in which you have GST, but you can take Ls to 0. And this is some sort of supergravity limit plus interactions, and so on. So you can, in principle, play with both of them. And then both of them will lead to different features. In particular, as I mentioned this morning, that is the most usual and tractable number of the effects out of the form e to the minus a over z. So here, since we have two different coupling constants, the nonpertorative effect in the string length will be of the form minus m. Let me call it like this, Ls squared, where Ls squared is also called alpha prime. And then here, we will have another.  It's typically proportional to an instanton action a wall sheet divided by Ls squared. And then there will be a nonpertorative effect in a spacetime, what is called a spacetime nonpertorative effect. And this is a nonpertorative effect in this coupling constant. And this is typically of this form. I'm not trying to be exhaustive here. Probably there are more complicated nonpertorative effects. For example, you can have nonpertorative effects in string theory, which go like e to the minus 1 over g string squared. I'm just giving you some of the most important ones and the ones that will appear in the concrete discussion that I will make later. So these two things will appear in a concrete model that I will discuss in these lectures. And this is what I'm discussing right here. But there are other nonpertorative effects in string theory. Now these effects are actually relatively, these effects which are involved in string length are relatively easy to understand. Because you can describe very often a string theory by a sigma model with a given target. And actually, these nonpertorative effects are typically due to what are called worst-hit instantons. So you have a map from a Riemann surface of genus g to a given target. And then you have a nonlinear sigma model. And then this nonpertorative effects correspond to a standard worst-hit instanton in a nonlinear sigma model. So you have here a target. And then if your, for example, if your map, if the image of your map wraps a non-trivial cycle, the area of this cycle will be given by a quantity called AWS. And this gives a physical interpretation to this quantity in string theory. So typically, this corresponds to the area of a worst-hit instanton in a nonlinear sigma model. So these are relatively well understood nonpertorative effects in two dimensions. And I have to say here that if the scale of the target is associated with the length scale of the target is given by a radius L, as happens, for example, in the ADS if the respondent.  Then this guy has to be measuring units of length squared. And then this exponentially small effect goes like minus L over LS to the square. So actually, weak coupling here, so in this case, weak coupling in this quantum theory, describing fluctuations of the string around the point particle limit, weak coupling means that weak coupling is actually the point particle limit. And this is then when the length of the string is very small as compared to the typical scale of the target. And of course, in this point particle limit, you see that this guy is going to be very suppressed. So nonpertorative effects in the string length are very suppressed when you have large targets, large target scales as compared to the length of the string. Now, these effects are slightly more complicated. It was actually pointed out by Polchinski that the source for these effects are d-brains. So actually, one of the motivations for the introduction of d-brains by Polchinski was actually to be able to account for these nonpertorative effects in string theory. And I have to say that historically, it's very interesting the way that these nonpertorative effects were discovered. Because you can already look at these two equations and see something very interesting. Imagine that I give you a pertorative series, just the series that comes from a pertorative calculation, and I tell you to estimate in some way what are the type of nonpertorative effects that you will find. Well, actually, the general arguments I was telling you this morning tells me that if the coefficients growth, like n factorial times a number to the minus n, you should expect the size of a nonpertorative effect to be of this type. Remember that you can see already this with a procedure of optimal truncation. Because in optimal truncation, this growth led to an error in my calculation with optimal truncation, which is exactly of this size. So you can always relate the large-order behavior of a pertorative series to the appearance of these exponentially small effects. So of course, if you have a nice path integral formulation of your theory, you can derive these things also from, say, non-trivial saddle points of your theory. But in string theory, we don't have a path integral of the full space of string fields. So it's not so easy to find this by looking at saddle points of a path integral. So actually, in string theory, the way these effects were discovered, where in some toy models of string theory in less than one dimension, what are called non-critical strings, people observe that the typical growth of the pertorative series was of this type and deduced the existence of these effects from the growth of perturbation theory. So this is actually what Polchinski noticed already in 1994 that he introduced the branch of a natural source for these type of effects. Now, let me see what we are going to do in this lecture. We are going to focus. We are not going to focus on the most general quantity. In string theory, we are going to focus on a peculiar quantity. And the reason why we focus on this is because we have a lot of control, analytic control on this quantity. There is no reason for that. And what we are going to focus is the total partition function or free energy of super string theory on some backgrounds, on backgrounds involving ADS space. But before getting into the details of that, what do you expect for this type of functions? Well, the free energy will depend on two variables. Let me actually call one of these variables lambda for research that we will see in a moment, and then g string. So lambda here should be understood as a function of L over LS. So it could be, for example, L over LS, but in the examples I will use, when there is some duality with ADS, between ADS backgrounds and gaze theories, lambda will be the top parameter and will be typically a power of this quantity. But it is just a way of encoding the dependence on the two coupling constants of string theory. Now, what per torative string theory tells me is that I can't, yes? Oh, sorry. Oh, sure. I will try, but it's hard. It's hard, yeah. Maybe I don't know if all these things are filled, so yeah. OK, I will try my best. So let me start with them with this equation. OK. So this is the total free energy, and if you do a calculation of this quantity, in per torative string theory, what do you find? Well, you will have to sum over the free energies of different genera, so weighted by the coupling constant to this power. So this is the typical dependence, and this is the contribution of genus G Riemann surface. So if you can think in terms of embedded strings, what do you have? It's just simply that you are calculating this is a vacuum to vacuum amplitude. It's the free energy. So this is the logarithm of the partition fashion. So you have to do the path integral of all possible string configurations, and we know that the space of string configuration is labeled by the genus. So you will have contributions of genus G Riemann. So you will have spherical strings. Then we will have genus 1 strings, which are donut-like strings, and so on. And for each of these quantities, string perturbation theory tells me a priori that there is a path integral over the space of strings with a fixed genus. And then I have to put all these things together with this power GST. So this is a formal power series, by the ones that we have been studying. And now you can ask questions about what are the convergence properties of this power series. And this is a slightly more delicate problem than what we had before, because before we had a formal power series, but the coefficients were just numbers. Now the coefficients are here, functions of lambda. So you have to be very careful about, for example, what kind of function you have here, and what kind of function you get when you sum over all these things, and so on. Again, I'm not going to give you a universal answer. I'm going to give you the answer, which is relevant for the purposes that I'm doing here. Let's first focus on this one. And this is a function of lambda, and it turns out that FG of lambda are essentially analytic functions at lambda equals 0, in many cases, in many examples. And this will be actually much clearer when we look at these functions from the point of view of the ADSF correspondence, because if this string theory has a gauge theory dual, these quantities can be a priori computed in gauge theory by summing over genus G diagrams, over tough diagrams of fixed genus G. And these functions are essentially analytic at lambda equals 0. What I mean, essentially, I mean that they can have some leading singularities coming from zero modes, but essentially they are given by a power series function around lambda equals 0, which is analytic. So they are analytic functions, means that they have a finite radius of convergence, a priori, around lambda equals 0, and they have a common domain of analyticity for all genus. That's also a quite non-trivial thing, because you could think that, depending on the genus, these functions would be analytic in a different regions. But no, they have a common domain of analyticity. So let's say lambda smaller than lambda star. And in this common region of analyticity, when you evaluate this thing, you get numbers. Now, you can ask, and this is an important question, what happens if you want to go from lambda equals 0 to lambda equals to infinity, for example? And this is the famous problem of weak strong interpolation that you appears in case theories. But notice here that the problem of going from, say, a small lambda to large lambda is completely different from what happened in the divergence series that I was telling you about, because here you already have to start with an analytic function. Now, the behavior of analytic functions is actually much more under control, because analytic functions typically have, for example, analytic continuations to larger regions in their domain, in their space domain. This is actually what we did this morning with the Borel transfer. The Borel transfer was analytic in a neighborhood of the origin, and they were using some principle of analytic continuation in order to be able to calculate this Laplace transform. So these functions, eventually, you can ask the problem of what is going to happen if you try to go to large lambda. And this is the problem of a strong weak interpolation. But you are in the domain of allomorphic functions. So at least of metamorphic functions are functions with branch cut, but at least funds which are allomorphic in a neighborhood of the origin. So this is a completely different problem. OK, so now once we fix lambda, so we fix lambda inside this domain, we will have now a series, a numerical series, in which you can now ask for this fixed lambda. So now we have here numbers. Now you give me a lambda inside this domain of analytic. Is it, and you ask me, well, how these numbers grow? Because the growing of these numbers, the growing of these coefficients, the crucial thing that will tell me what's the nature of this perturbative series. Now it turns out that these guys also diverge factorially. So they diverge factorially, and they actually diverge for a fixed lambda as 2G factorial times A, a space time, so AST of lambda to the minus 2G. So this is the typical behavior of these quantities. So you see that extreme perturbation theory is also giving the same problems that you found in quantum mechanics, in the sense that if you want to evaluate the extreme perturbation series, you are going to have to do something about it, because it has this factorial, which is going to kill any region of convenience for the series. And what is appearing here, this A instanton, depends on lambda, is precisely the function that appears here. So the appearance, this divergence here is related to this non-pertructive effect here. In the same way that this morning we were seeing that the divergence of the coefficients in standard quantum mechanics was related to the appearance of this instanton corrections. So, well, notice that here there is a 2G factorial, not a G factorial, and there has been a lot of fuss about it. Well, this is not as important. Essentially, it's due to the fact that this series is actually an even series in G string. But the non-perturban effect, the leading non-perturban effect, is not in G string squared. It's in G string. So this actually was a little bit puzzling. And one of the reasons, one of the things that people discuss in the early days of this non-perturban effect is that also the series was in G string squared. Essentially, in G string squared, because you have always even powers, the non-perturban effect still was involving one over G string and not one over G string squared. So in a sense, these non-perturban effects are larger than what you would expect. And this is one of the reasons that Polchinsky was mentioning that you need some sort of different kind of object in string theory, which was given bigger instanton effects than what you would expect, say, in quantum mechanics. Because in a sense, you have to, this has to go with the square root of the effective coupling that appears in this series. But this is what you find. And this has been verified in many examples, from examples in non-critical strings. And now we can, as I will explain, in theories with a large end double, we can actually check this with a lot of accuracy. And you see that, conceptually, it's not very different from what we saw before. Now the question, I mean, also this is not very different from what happens in quantum mechanics. Now this is raising a very deep problem. And the problem is that if you want to make sense of string theory, at some point you have to make sense of this series. In quantum mechanics, you have the Schrodinger equation which saves the day for you. But in string theory, we don't have all the time, we don't have in all cases something like the Schrodinger equation. And so you really have to make sense of this thing. And one of the virtues of ADSEFT is that a priori, it gives you an a priori answer that this, you can make sense of this by using a gauge theory. And this is what we are going to explore also in these lectures. Now, let me make a final comment on the relation to string theory. And also I have here given you two different kinds of number of the effects. It turns out that in certain special cases, these things can be unified in a single description. And this is when you discuss tight way super string theory. So in tight way super string, you can go to M-theory. And in M-theory, both wall city instantons and space time instantons are manifestations of membrane instantons. So here, there is a unifying notion of membrane instanton. And I think this was observed first by Becker, Becker and Stromminger, sort of the introduction of M-theory. So what is going on? Well, remember that this is a theory in 10 dimensions and this is a theory in 11 dimensions. Now, tight way theory has two couplings, as any string theory has the string length and the string coupling constant. In M-theory, you really have one single coupling, which is the plan length. But when you go to 10 dimensions, you have to introduce another quantity which is the radius of the living dimension, okay? So now you have one single coupling coming from the M-theory plus one radius that comes from the compativification to the dimensions and with these two quantities, you make up these two quantities by a relationship that you can find already by working out the supergravity and these are these famous equations that you can find in many books. For example, in Tomas or Tim book on D-brain strings, gravity and strings, there is a very detailed derivation. So this is the relationship between the single coupling constant of M-theory and the radius of compativification to the two parameters of tight way super string. Now, how do we see these two effects coming from a single source? Okay, now you're adding 11 dimensions. So you're adding M-theory in 11 dimensions and now you want to understand this in terms of some topological object. Now, there is something in M-theory which is called the M-to-brain which is a three-dimensional object. So here, let me imagine that I have a three cycles on three objects in three dimensions with three dimensions inside 11 dimensions that a membrane can wrap in some way. Let me call this M. Now, what is the kind of non-pertorative effect that you, what is the strength of a non-pertorative effect that you would expect for a three-dimensional object? Well, you see here that you have a two-dimensional object. The intrinsic scale of a three-dimensional object is going to be its area divided by the length of the string. Now, oops, I shouldn't have erased this. In three dimensions, what you would expect is something that you have a membrane like this. You would expect a non-pertorative effect which weight is going to be the volume of this guy which is a three-dimensional cycle. You measure in units of the plane length Lp to the q. So this is the typical weight of a membrane instantonium theory. But now when you go to two dimensions, you have two choices. When you go to two dimensions, you have to do a compactification of a cycle of radius R11, and then you have two choices. The three-cycle can actually wrap this extra dimension. And then what you get in ten dimensions is a two-dimensional object. So it can happen that this M splits into a Riemann surface of genus G plus a third dimension wrapping the circle. Or you can have that the three-cycle goes directly to the ten-dimensional space without wrapping the extra direction. So you get something here which is still three-dimension. So let me call it S. I think it's the opposite notation to my notes, but that's fine. OK, so what happens then? Now we can use these relationships that I got here. And then we see that when you wrap the three-dimensional sort of the compactification in this situation, the volume of M will be proportional to the volume of this Riemann surface in the two-dimensional setting times R11. While here, since you are not wrapping the 11 dimension, the volume factor doesn't involve R11, and you just have volume of S. So you see now that you are in the first situation. We'll recover this formula in a moment. What you get is exponential of minus volume of sigma G times R11 divided by Lp to the cube. But now, if we use this relationship here, you see that we get exponential of minus the volume of sigma G. R11 is G string times Ls, and Lp to the cube is G string Ls to the cube. So what we get is exponential of minus volume of sigma G times Ls squared. And this is the size of an operative effect coming from a wall-city instanton in tight way. Now if we don't wrap the 11 dimension, what you will get here is just simply volume of S divided by Lp to the cube, which is, notice, Ls to the cube divided by GST. So you see that we got the typical 1 over GST dependence that I was telling you is typical of an operative effect in the string coupling constant. And then the space-time instanton action comes from an intrinsically three-dimensional object. So what we expect from this picture is that when there is an M theory behind our super string theory, the effects that are due, the number of effects in the string coupling constant, are actually induced by a three-dimensional object in M theory. This is what we would expect. And notice that from the point of view of M theory, you cannot really distinguish these two objects. They are just different geometric types of membranes in 11 dimensions. The fact that this membrane is wrapping this direction or not is just a topological feature. But there is nothing to distinguish them. You have to take it on equal footing. So in a sense, M theory doesn't make a fundamental distinction between a space-time instanton or a wall-city instanton. It's only when you go to tie to a theory that you get such a picture. Of course, when an M2 brain descends to a three-dimensional object in string theory, we know that this has to be interpreted as a D2 brain. And this makes contact with the observation of Polchinski that in tie-to-A theory, you will see this three-dimensional object of M theory as a 2D brain. So this is a 2D brain effect. So this finishes a little bit my quick review of number 30 effects from quantum mechanics to M theory in one single strike. So are there any questions? Sorry? Yeah, yeah, five-branes. Five-branes would actually lead to this effect in which you have E string square. So these five-branes will give you another type of number 30 effect, which is actually not seen in the example I want to consider. But you would get also a different type of effect, but now you will have here a square. That's why I was mentioning at the very beginning that I'm not really being exhaustive with respect to all possible number 30 effects. And let me point out something which I think is important. Now, when people talk about number 30 effect, they always want to find a semi-classical realization of these effects. Now, when we talk about instanton quantum mechanics, we want to think about them as subtle point solutions to the path integral. Here, we are thinking about this, for example, as Quartis instantonus in a nonlinear sigma model. Here, we are looking at this in test of membrane. So we are thinking always as a sort of extra topological sector that leads to these objects. But it has to be clear. I mean, I hope it's clear. And if not, I'm saying it now, that in principle, you can have a number 30 effect which is not associated to any semi-classical effect. And for example, in quantum theory, there's a type of number 30 effects known as renormalons, which are not a priority coming from extra subtle points in the path integral. They are really coming from strong quantum fluctuations or they have some sort of diagrammatic origin, but it's not clear at all that it can be associated in general to a semi-classical object. No matter what we will see, and I think this is something that you can really see in the CFT, is that things which in extreme theory are coming from subtle points in field theory are not coming from subtle points. They are actually due to strong quantum fluctuations. So sometimes these effects are emergent. These numbers of effects are emergent. They are not really associated to a subtle point. You cannot say, oh yeah, this exponential comes from this particular subtle point configuration. So no, the path integral can do crazy things. And sometimes quantum fluctuations around the vacuum can lead to something which is exponentially small and is non-analytic. So here we are very semi-classical, well semi-classical, and then we always try to fit our intuition, to physical intuition to semi-classical objects. But quantum theories do crazy things and sometimes you will find numbers of effects which are not due to semi-classical descriptions. So in extreme theory we are lucky because supersymmetry claims the quantum fluctuations for us and then we can more or less associate semi-classical objects to all these configurations, to all these effects. Okay, so if there are no more questions, I want to get into the details, so I'm going to focus on one example. So as I said, I don't want to make a general theory of non-partisan extreme theory. This is probably very difficult and I guess it's much more useful to see how all these things can be made concreting at the context of a single important example. And this is the example of ABJM theory which is a gage theory with a known large endual where all these things can be made concrete and you can calculate them, okay? And then we will learn some new things about them. In particular, what we will learn is that M-theory is extremely powerful and essentially all the story that I told you about perturbative divergent series and so on is only true in tight-way super-stream theory. In tight-way super-stream theory you are stuck with these expansions. But actually what we will see is that M-theory essentially give us back converging objects and give us back all these which are much more powerful, much more nicer analytic structures. So in a sense, the genus expansions is really an artifact of tight-way theory. In M-theory there shouldn't be expansions. There should be only well-defined objects that then you expand asymptotically if you want and you can make contact with tight-way super-stream theory. But in M-theory you are able to overcome all these expansions of divergence and I hope I will be able to show you that this is the case with concrete examples. Okay, are there any questions? How about tight-way B? Well, tight-way B is a very different world, right? I mean you have face duality and the coupling constant and so on. So there I would expect, of course you also have instanton Fs in tight-way, right? And people have studied it, but they don't have anything concrete to say about tight-way B.  This has been studied by Yasok Sen, for example, recently how face duality puts constraints on this sort of asymptotic expansion and so on, but I'm not going to tell you anything concrete about tight-way B. This is a general story, however. Okay, so then let's come back to examples and then what I'm going to do is a very quick review of ABN theory, probably quicker than, but I have given lectures on ABN theory and I have put them on the archive, so if you want to learn about the details, I refer you to my lectures on ABN theory, but let me just give you some summary of what is ABN theory, where ABN is from Aharoni, Berman, Jaffer, and San Maldaceno, okay? Okay, so this is a gage theory, but it's not a gage theory based on Jambil's action, it's a gage theory based on the Trincymon's action in three-dimension. So just in case you are not familiar with it, the Trincymon's functional of a gage connection in three-dimensions can be written as trace of A which the A, so this is a three-dimensional connection plus two-thirds of A cubed, and this is a very peculiar theory, it has a kinetic turn which is first order in derivative, it's not of second order, so this gives a theory which has really different flavor from Jambil's theory and then it has a cubic interaction which is close but not identical to the one that you find in QCD, and this action is actually, this action leads famously as Whitton explained to us in 1889, a topological field theory, so the quantum field theory based on this action gives you a theory which observables like the partition function for example, do not depend on the metric that you put on your manifold and so on, and in physical terms you can think about this action as an action which has a huge group of symmetries, and then it's a very good starting point to construct conformal field theories, because conformal field theories are not topological, they should have a higher degree of symmetry than the standard theory, and actually in three dimensions this is the starting point for constructing conformal field theories. Now, in ABJM theory, what you have to do is to take this action and supersymmetrize it, and I will be very, I will give you kind of few details of this, mostly because I actually don't remember all the details about all the super fields and so on, I knew them at some point by heart, but now I have a little forgotten them, so you have to supersymmetrize the Chern-Simon's action, and then you can add n equal to matter in three dimensions, so all this is in three dimensions, and then you construct supersymmetric versions of Chern-Simon's theory, which involves a supersymmetric version of this multiple, so this just involves the n equal to vector multiple, and then you have the n equal to matter like chiral multiples and so on, okay? Now, this doesn't give you ABJM theory, in order to get ABJM theory, you have to do something slightly more complicated, so you have to take two different Chern-Simon's theories, and I will represent them with nodes, so each of these nodes is a Chern-Simon's multiplied with gauge group UN, so let me, you want to call it UN1, and then UN2, now in Chern-Simon's action, as it's cousin Jamil's has a coupling, which is usually written like K, and actually in order for the theory to be well defined, you need this guy to be an integer. Now the reason is that if this guy is not an integer, this action is not invariant, the path integral is not invariant under global gauge transformations, so this is a peculiar gauge theory because it has a coupling constant, but the coupling constant is quantized, but it's also very good that you want to have a conformal field theory, because in a conformal field theory, the coupling shouldn't run. Now you have a coupling that for non-parallel reasons has to be an integer, it's going to be very hard that it runs, because it cannot run, it has to jump, has to jump, and then, essentially this means that at the end it won't move, so this is a very good starting point then, and I will actually have here a Chern-Simons theory whose action with a gauge group UN whose coupling k, and here I'm going to take the opposite coupling minus k, I can't take negative couplings, this is not a big deal. Now apart from the fact that this is quantized, in perturbation theory, you will never see that this guy is quantized, and then you can do perturbation theory in one over k, so if you want g jamils square is essentially one over k, you can do perturbation theory in one over k. K is very large, you don't see this, you don't see the quantization, you just see a coupling, a weak coupling, and then you can do perturbation theory. Now so you take two copies of these guys and then you connect them by adding four bifundamental hyper-multiblets, which sometimes are called phi i i 1 fourth, and then this theory has the property that it has super conformal invariance, and actually it has n equals six super conformal invariance, so half the maximum degrees of the maximum number of supercharges, and it's a very nice super conformal field theory, there's a super conformal field theory, and there have been at least 1,000 papers on this theory, so I cannot tell you all the details about it, but for us it has two important properties. One property you already can see purely at the field theory level. Now in this theory, the fields are going to be vector multiplets, which are in the joint representation of the gauge group, and then we have matter, but this matter is in the bifundamental representation, so since we have gauge group u n, effectively these are also fixing the adjoint representation. So here all the fields are Hermitian matrices, Hermitian valued, and then you can do the standard top expansion of this theory, you can do essentially the same top expansion for pure jam mills, and we know just by one of our encounter that for example you can now calculate the free energy of this theory as a function of n and k, and usually in order to have a well defined quantity with no infrared divergences, you decide to put the theory, so you take this a b gen theory, and you put it in a compact manifold in order to compute this quantity more in a better defined way, and I will actually put the theory in all this talk, I will put it in the three sphere, which is the simplest choice. So I take this gauge theory, I put it in a compact manifold, this regulates for me infrared divergences, and I can compute its free energy, as you would do in say thermal field theory, you put the theory in R2 times S1, here I put the theory on S3, and then the free energy is just given by the sum of vacuum diagrams, all the vacuum loop diagrams with no external lines. Now, tough tolas taught us already in the 70s that if I have a theory with two parameters n and k, remember that k is essentially g square, is the inverse of g square, so here I can do what tough tolas taught us a long time ago, I can actually choose another coupling, which is n over k in this case, and remember that this is equivalent to take n to g square jambiles, and what I can do is to reorganize my vacuum diagrams, so my vacuum diagrams are going to be like this, right? Something like this. I can reorganize it according to tough in a different way, so I can consider diagrams that go like n square, and they are given by planar diagrams, this and this, and then I will have also diagrams of one loop diagrams, so diagrams of genus one, in which I will have diagrams in which I have some twisted lines like this. Okay, well, this is really complicated to draw, but you will have diagrams like this, and so on, and then you have a full one over an expansion, so to summarize, I will have a full one over an expansion, which actually looks like this. Now, I did the expansion here in n, but I can do the expansion in whatever parameter which is k is like n, and then I will actually do my expansion just too much conventions in the literature in this of a parameter, which I'm going to call two pi i over k. Now, here, as I said, since not remember that k in this tough expansion, n goes to infinity, and k goes to infinity such a way that land is fixed. So I can do the expansion in powers of n, or I can do the expansion in powers, in inverse powers of k. All these are equivalent automultiplication by a factor of land, okay? So this is just the way you do the one over an expansion. You can do it in any parameter which goes to infinity when n is large, okay? So this is an expansion actually in k, right? Because, sorry, yes. So this is an expansion in, yeah, this is an expansion in one over k, which also grows to infinity in the same way as n. It grows at the same pace, because the question has to be fixed, okay? So this is something that tough dollars that we can organize whatever quantity we want in this way, and in particular we can reorganize the free energy, which is the most, the simplest quantity that you can compete in a quantified theory by just doing this rearrangement. And notice, of course, that this has the same structure that the genus expansion of a super street theory, and this is at the heart of the ADSE correspondence. Already formally, you see that this gives me the same type of quantity, okay? So this is the first thing that I wanted to mention, that this theory, you can do these calculations, and people have done perturative calculations of these sort of quantities in the one over n expansion for ABGN theory, so this is something that can be done. A priori, it's a complicated quantity calculation, but it can be done. Now, this theory then has a one over n expansion, in particular it looks like this, and it was already in the original paper by Maldacena in 1988, it was already noticed that there should be a large and dual for this theory. Now, actually, this works the other way around. This is not like an equal for Super Jamil's. When Maldacena postulated the large and dual to this theory, the theory itself, the history itself has not been constructed. So this was constructed later on because this, also the idea I mentioned here, this is a theory of M2 brains proving a singularity in C4 mod ZK, these are not going to use, but it's the basis for this large and conjectural Maldacena. So let me, instead of deriving for you how this goes, I mean this was postulated by Maldacena based on the standard near horizon and so on, let me explain to you what is the dual and how this duality works. This is important for the calculations that follow. So what is this large and dual? Well, the first interesting thing to notice is that in contrast to N equal for Super Jamil's, where you have a tie to B dual, here you have, actually, originally you have an M theory dual, and you have an M theory dual and a tie to A dual, but this interplay between these two ways of looking at this duality will be also crucially in what follows. Now, let me first write the N equal for dual, sorry, M theory dual. So to do this, you have to consider M theory on a non-dimensional manifold, which is given by ADS4 times S7 mod ZK. So this is the four dimensional space, and this is S7, and you have here a quotient of these, this is a smooth quotient, this is not an orbit for you, because this is like the three dimensional version of a lens space. So if you realize the seven sphere, a set one S square plus ZA set four S square equal to one, then the action is just e to the two pi i over K Z i, where i goes from one to four. So this is a background of the dimension of supergravity, and it has a metric, which is a product metric actually, up to a factor of one fourth. And very importantly here, there is a single scale length in the M theory dual, which is the radius of the universe. The size of the universe, the radius of ADS4 and it's also the radius of S7 up to a factor of one half. So to be precise, have this sort of thing plus DS square of F7, ZK. And on top of that, you have also N units of G flux. So remember that in supergravity, in 11 dimensional supergravity, which is the large distance limit of M theory, there is like a three form field, and then it's four form field strength and half a flux through ADS4. So you have a N units of flux. And then you see that here, we have already the two parameters that correspond to the gaze theory dual. We have a K here, which is in the quotient of F7, and we have the N units of flux here. And this corresponds to the rank of the gaze group here and to the coupling constant here. So this K is here. So this is what you have in this side and you actually solve the questions of motion. You actually notice that N fixes this side. Remember that M theory, you have only one length, which is the plant length, and we have this relationship here, which tell me that when N is large, L over LP is large. So this is the, this is the, yes? I don't, sorry, but I don't hear you, I don't hear you, sorry. Yes, it's in the four dimensions of ADS4, right? You have a four form, so you have to integrate over a four dimensional space. So yeah, so this is the flux of DC, I don't know. Sometimes people call it G or whatever. So this is a four form, okay? Okay, very good. So now the question is the following. What does it mean? Now, if you compare, if you take this at face value, what does it mean? But this actually is very interesting from the point of view of the gaze theory and the theory because, you see, here there is a single parameter in this theory, in the same theory dual, as I told you, there is a single parameter really. I mean, it's L over L plank. And K is a geometric parameter. So here you can think about L over LP and then when L over LP is very large, this is the regime. This is very important. This is the regime of large scales. So this is where the size of your universe is much larger than the plank length. And this is precisely where semi-classical gravity works. So if we have a universe whose size is much bigger than the plank size, we expect that doing general relativity, in this case, 11 dimensional supergravity, which is the larger scale distance from theory, will work, okay? Now this, from the point of view of the gaze theory, is a very interesting limit because here, if L over LP is very large, what we have is that N is very large. But K is fixed. So this is a limit in which the coupling constant has to be fixed, maybe to some value, one, two, or three, whatever you want. And then N is very large. So this is not like the standard tough limit. This is very important. In the standard tough limit, you take N very large, K very large, and N over K is fixed. Here is not the case. K has to be fixed and then N is very large. So in particular, for example, if K is of order one, this means that in this regime, from the point of view of the tough coupling, they are always a strong coupling because lambda is going to be very large, say if K is equal to one or two. Here, in this point of view, K can be any number you want, but you don't have to scale it. You have to really face the fact that you have to deal with this theory when K is equal to one or two. In M theory, they are always in the strongly coupled limit of the field theory. But if you want to make contact with semi-classical gravity, then N has to be very large. So this means, in a sense, that you are thinking about the gaze theory as a theory with fixed coupling constants, and you are taking the number of gluons, if you want, and superpands of the gluons, to be very large. So this is more like a thermodynamic limit of the gaze theory. It's not really, it's more like a thermodynamic limit, and it's not really a tough limit, a priority. We will be able to make contact with a tough limit, a priority, not a tough limit, and I will call this the M theory regime, or the M theory expansion. Now notice that here the natural, actually here the natural expansion parameter here is Lp over L. This is the natural small parameter here. So in the gravity theory, what you should expect is that when this parameter is very large, the leading order is captured by 11 dimensional supergravity, and then you will have corrections coming from loop corrections in supergravity. From the point of view of the gaze theory, this means that you have to keep K fix, and then you have to take M very large, and do an expansion in one over M, when we're K is fixed, which is different from the tough expansion that tough tell us has to do. Actually, from the point of view of the diagrammatics, I don't know of any argument that shows that this type of limit is actually well defined. It's just a very peculiar property of this type of theories. The top limit is a universal property of any gaze theory in which the fields take values in their mission matrix. This n theory limit is a very peculiar limit of this type of theory. I don't think it's a universal feature of any quantum theory, a priori. It's just the existence of this n theory dual which guarantees you that there should be some sensing with this expansion is actually appropriate. But from the point of view of the diagrammatics, you don't understand how this is going to work, because actually tough invented this type of scaling precisely to make sure that diagrammatics was going to work, and we were going to be able to reorganize the perturbation theory in good powers of n. So this is actually a very interesting property of this theory. Now, when L is of, or say, when L over LPS of order one, what we have is a universe which is Planck's size, the size of your universe of Planck's size, and this, from the point of view of the gaze theory, is precisely the regime in which n is small, but k is still fixed. So it's still a strong coupling regime of the gaze theory, and it's a gaze theory with few degrees of freedom. Okay, so this is just to understand a little bit of the dictionary, and what are the regions we want to look at? And as useful in the ADS50 correspondence, you see that the region where the supergravity is easy is the region where we have L over LP is very large, but then we have to really look at this gaze theory in this sort of thermodynamic limit. So what I'm going to do tomorrow is I'm going to start by looking at the title version of this, because as it was already known to the people doing supergravity, this background has a tight way version, and in this tight way version, we'll be able to make contact with the genus expansion of Tuft, okay? But we'll see that there are two actual, there are actually, in this theory, there are two stories. There is the n theory story, and then there is the tight way story. And part of the richness of this large sensuality in contrast to an equal for superjammies is how you go back and forth between the n theory description and to tight way theory description. And this is what we're going to exploit in the next tomorrow, okay? Thank you.