 OK, so this is my last lecture. And I just covered half of the lecture notes, but this is fine. So let me try to at least wrap up many things that I want to tell you in one hour. Unfortunately, what we are going to do now also is the most recent developments. It's quite elementary, and I think it's good because it actually covers some nice elementary statistical mechanics. So what I'm going to do is the M-theory expansion of ABJM theory, because what I was reviewing for you this morning was the top expansion of this ABJM matrix model. So you learn a lot of stuff, but you don't really learn what happens in this M-theory regime. Remember that in this regime, you want to take n go to infinity and k fixed. So what I'm going to do is to introduce a new point of view on this matrix center, which turns out to be quite useful and actually has been generalized for many models. And actually, it points out to a more abstract point of view on this problem, where we're going to reformulate the gaze theory in terms of the spectral theory of an operator, and this turns out to have applications also to some other models, like topological history in theory and so on. But the way to do this is quite simple. So what I'm going to do is to introduce what is sometimes called the Fermi-Gas approach to these matrix models. So this was done first by myself and my student Putrov. And essentially, it's a reformulation of the matrix model. So you take your matrix model, and you give two equivalent formulations. One is in terms of thermodynamics of a Fermi-Gas, of a free Fermi-Gas. So this is a quite simple system. And equivalent, you can think about this. You can reformulate this in terms of the spectral theory of an operator. So these two things are equivalent, and you can change your piano view according to what you need. So how does this go about? Well, let me not write again the matrix integral for ABJM theory that I wrote this morning. Hopefully, you can check it on your notes. And then the first thing you do in order to introduce this Fermi-Gas approach is to use some non-trivial identity, which has been quite used in the analysis of these Chern-Simons matter theories in the matrix integral of these Chern-Simons matter theories. So remember that in this ABJM matrix model, you have terms involving a shinch, actually a pair of shinch. And then you also have a term involving a cosh. Well, it's a fact of the life of polynomials. This is actually a polynomial identity, essentially, that this can be written as a determinant. So this is the determinant of a matrix whose entries are just the inverse cosh of mu i minus mu j over 2. Now, determinants can be expanded using the permutation group. And this turns out to be a sum over permutations in which you have the sign corresponding to the signature. So you see already here the fermionic nature of the problem. And then here you have a product over twice the cosh of mu i minus mu sigma i divided by 2. So this identity is non-trivial. You can find proofs in many places, for example, in the book on representation theory by Fulton and Harris. And this is just the development of the determinant. So remember that this was one piece of the integrand of our ABM matrix model. And then it's an exercise. I can propose this as an exercise. And if you have doubts about how to proceed, you can go to the paper by Capustin, Willet, and Jacob, where this was first done. So what these people show, they show it only for k equal 1, but k is not important, is that you use this identity. You can actually write the matrix integral in this form. So you write it. You have to do more stuff here in order to just apply this identity that is not sufficient. You have also to use some sort of clever use of Fourier transforms and so on. But the final result is that you get such an expression where rho of x prime is a function of the form 1 over 2 pi k, 1 over twice the cos of x over 2 to the 1 half, 1 over twice cos of x prime over 1 half, and then 1 over twice to the 1 half, the cos of x minus x prime over 2k. OK, so this is already quite complicated. So the integrand is just this kind of orgy of hyperbolic functions. But this expression, you have taught the statistical mechanics I have to do in order to gain my living, it's actually something that you can recognize. This, you have studied the book by Feynman on statistical mechanics. This is the canonical partition function. When I mean canonical, I mean the canonical ensemble of a free Fermi gas with n particles. And rho plays the role of the density matrix for this Fermi gas. So the density matrix, just let me remind you that the density matrix, usually you write it like this, where beta is the inverse temperature. Here I'm going to fix beta equal to 1. So there won't be temperature, but it's actually very good to think about this as an exponent in Hamiltonian. So this is, as I said, relatively elementary statistical mechanics. And then this opens the way to actually analyze this problem with the tools of statistical mechanics. Now, why is this useful? Well, first of all, if you think about the large hand expansion, it's always good to think about the large hand expansion as a limiting with the theory simplifications in way. Now having a thermodynamic analogy for the large hand expansion is always useful, because we know that in thermodynamics, life gets simpler when it is large. This is called the thermodynamic limit. So we should expect that the simplification that happens in the large hand limit that we want to analyze can be actually put in a more intuitive basis by having a thermodynamic analogy. OK, now let's go back to statistical mechanics one on one. So I'm going to refresh some formula for you for statistical mechanics. Of course, you learned in statistical mechanics that the way to deal with a Fermi gas, an ideal Fermi gas, is not to use the canonical ensemble, right? Because in the canonical ensemble, things get relatively complicated. So what you were taught to do is to actually do the calculations in the grand canonical ensemble, right? In the grand canonical ensemble, essentially you get rid of all these sums over permutations. That's why you have to do that. Now the way you do this is you introduce this grand canonical partition function, which is going to depend on the fugacity mu. And k is there with us from the very beginning. And k only enters here through this density matrix. So the density matrix depends explicitly on k. OK, now what is the definition of this guy? Well, this guy is actually a generating function for all the canonical partition functions. And here you use e to the n to the n mu. And sometimes I will denote e to the n mu by kappa. Kappa, I'm going to have trouble in distinguishing these two guys. Kappa is a Greek letter, is the fugacity. Well, mu is the chemical potential. And k, I will try to put it this smaller. This is the coupling of terms I must hear. So it is the parameter appearing in m theory. So this is the definition. And usually you introduce also the grand potential, which is just the log of this guy. And it's a very interesting exercise that you can find also in the book by Feynman. So I propose this as an exercise. These are really good exercises. These are things that one maybe has not studied in the undergrad years, but are good to know. So this can be written in this way. So this is the fugacity. And here we have dzl. And dzl is actually very simple. It's just the trace of rho to the l. So that's why you introduce the grand canonical ensemble. You introduce the grand canonical ensemble because in the canonical ensemble you have a crazy sum over permutations, which is difficult to deal with. While when you introduce the grand canonical ensemble, you get a sum in which you only actually look at cyclic permutations. Trace of z rho to the l can be regarded as a cyclic permutation, just coming from a cyclic permutation. Actually, the way you derive this in this exercise is by the composing the permutations here in cycles. And then we assemble things in such a way that you get this sample thing. You do also this in classical statistical mechanics when you do the virial expansion. This is actually also in the virial expansion you want to go to the grand canonical ensemble to simplify life. So this is in a way a quantum virial expansion for thermogases. Very good. Now, of course, ensembles are equivalent. Sometimes they tell you that ensembles are equivalent in only the thermodynamic limit. And of course, the large limit is only, they are only extremely equivalent in that way. But you can always get one information from the other, information from the grand about this function for this function of isa versa. But here, we have one relationship. You know all the canonical partition functions for all n, the grand canonical partition function. But this can be inverted by a well-known formula. Maybe I should start right in there. Which is this one. So you just take a contour integral over the fugacity divided by 2 pi i. And you have this guy. And then you take kappa minus m minus 1. So this is a triviality while writing. The only thing I'm writing is that you can always extract the coefficient of an infinite power series by considering it's the residue with an appropriate power of k divided, kappa divided. This is really completely true. Now, why this is related to spectral theory? Well, it's related to spectral theory because the Hamiltonian, if you think about this matrix as the exponential of an operator Hamiltonian, you can actually look for this eigenvalue problem n goes from 0, 1, and so on. And actually this eigenvalue problem is very well defined because this operator, you can think about this as an integral operator, an integral kernel. And this actually has a discrete spectrum because this operator is what is called, in mathematics, a trace class operator. So this means, essentially the trace class means that all these ZLs are well defined. Actually, notice that this can be computed from the spectrum by summing overall energy levels. So you can think about this as e to the minus l en. And this quantity is well defined for any l, 1, 2, and so on. And it's a theorem that is an operator of trace class. And in this case, it's also positive. It has a discrete spectrum. So you can think about this as the spectrum of fermions. You have three fermions with a Hamiltonian, which is still didn't write for you. But you can think about this as defining a spectral problem. So you have fermions with energy levels. And this is a free Fermi gas. So you know very well how to do thermodynamics. You just have to feel. You use polyprinciples and feel the levels until you put the last fermion with label capital N. Now, it's also a relatively elementary result that the grand canonical partition fashion of this Fermi system can be written as a determinant of this operator kappa. So you have this equation e to the mu minus en. So this is the famous Fermi factor that appears in the statistics of Fermi gases. And this is sometimes called a first home determinant. So it's a beautiful fact that you can rephrase all your statements. You can rephrase everything in terms of elementary statistical mechanics. But if you are fancy and you want to do some things like more generally, you can think in terms of spectral theory. You have a very well-behaved operator, rho, which has a discrete spectrum. And then this guy is called the freth on the spectrum determinant of this guy. OK, now, so this is very good. But what is it good for? This is very fancy. But maybe it just tells me nothing new. Well, in order to make progress, let's try to see what is the Hamiltonian that appears here. Let's try to write this Hamiltonian. Now, the Hamiltonian can be in principle red because here, I know the exact kernel of this guy. The exact kernel of this operator was given to me by the matrix integral. So I probably can write this operator. And it turns out that rho can be written as follows. And here, you see, this is a one-dimensional gas. Because here, I'm integrating over the one-dimensional variables. So each xi is the coordinate of a fermion in one dimension. So it's a fermi gas in one dimension. Now, in one dimension, I can write this kernel in this way where x and p are canonical commutation, canonical Heisenberg operators with commutes to h bar. And h bar is not any quantity. What is h bar? h bar, I have to be able to read it from here. And it turns out to be 2 pi k. So it's a very excellent exercise that I propose to you that you can write this in this way. And the function ux is log of 2 cos x over 2. And t of p is the same function variable for p. And notice here that you haven't written this is not. So you can think about this as a potential. And you can think about this as a kinetic term. Now, this is very easy to see what it is. This looks like complicated. But if x is large, this is just x over 2. And if p is large, this is just p over 2. So you think about this as a kinetic and a potential term. You have a very simple physical picture for this system. You have here a potential which goes like this, goes asymptotically to x over 2. So you have something like this. And this is a linearly confining potential. So there will be energy levels in this potential. And then you put your thermions here. Now, the thermions are ultra relativistic in the sense that their kinetic term is a linear term in p. Of course, these, they will get corrections. So this is actually not very difficult system. It's a system of linearly confined fermions, which are ultra relativistic in one dimension, free fermions. So the thermodynamics of this system, at least in this approximation, in which you approximate this by this, you could do this in elementary thermodynamics or statistical mechanics. And this is what we're going to do. Now, you have to be careful because thinking about this potential and kinetic term is slightly subtle because this is not the exponential of minus ux minus tp. You see, there are some commutation issues. Actually, sorry, there is no half here. There is half here and there's a minus here. There are some commutation issues. This is not exactly like this. But this will differ of these in H bar corrections. So if you want to do, for example, a semi-classical analysis of this Fermigas or the thermodynamic limits of this Fermigas, this commutativity issue will be not crucial. It's only crucial when you look at sub-leading corrections. But there is something very interesting going on here. What is H bar? H bar is 2 pi k. Remember that k, k is inversely proportional to gs3. So this means that we have a quantum Fermigas and H bar is small when string theory is actually strongly coupled. So this is, in a sense, a dual version of the T2A string theory because then the easy regime for this Fermigas is the semi-classical regime. As always in quantum mechanics, when H bar is small. But H bar is small. It's a strongly coupled T2A string. So this picture of the matrix model opens the way to study strongly coupled T2A theory, at least for this observable, which is the partition function. By using just the WKV approximation to a one-dimensional Fermi problem. Now we'll see that this is very useful. So this is a little bit of a miracle that this is working this way. But now let's do this exercise. Let's do this exercise of studying this Fermigas in the thermodynamic limit. So as I said, this is an exercise that you can do in elementary thermodynamics. So let's do it. Let's assume, in my data, I give you this problem. I give you fermions in one dimension with this potential, this dispersion relation. And I ask you to calculate the free energy of this system at large n in the thermodynamic limit. How would you go about this? Well, there are many ways to do it. But there is a very easy way to do it, which is to consider some sort of classical Hamiltonian, which I'm going to put like this. So this is the approximation to this Hamiltonian. Also, of course, let me actually define the classical Hamiltonian. The classical Hamiltonian will be just ux plus tp. As I said, this is almost what you get, except for h-bar corrections. And on top of that, I can actually approximate these at large energies by this. Very good. So how do you actually go about analyzing this system? Well, there is a very easy way, which is very intuitive, which is looking at the Fermi surface of this gas. So the Fermi surface is the surface of Fermi which have the same energy. And this you can plot very nicely. So this is the available volume of phase space for a fermion with such an energy. So if you look at the Fermi surface, you get a square. Because you have this thing here. So this is 2e, 2e, minus 2e, 2e, minus 2e. So the Fermi surface of the system is this. So this is the Fermi surface. And you know semi-classically, this something I was knowing in almost 1900, that semi-classically, you must have one quantum state for every cell in phase space. So the number of fermions should be approximately given by the volume of phase space, the volume of e, divided by the volume of elementary cell, which is 2 pi h-bar. Now, this is very easy to calculate. So this is 8e squared divided by 2 pi h-bar. On the other hand, I know from elementary thermodynamics that this thing is n where you just change e by mu. So this is one of the relations that you get here. So this means that j mu is approximately given by 2 mu cube. Just have to integrate this, right? Divided by 3 pi squared k, when I'm already using the fact that the h-bar is given by 2 pi k. Well, so now I have computed the grand potential of this Fermi gas at large n. This is a large n approximation because I use semi-classics. And semi-classics is good. What happens here is that when you have many fermions, n is large, since you have a confining potential, the energy of these fermions will get bigger and bigger. So you are in a pruning which, effectively, the energy is large. And the semi-classical approximation, no matter what the value of h-bar, it's always both if the energy is very large. So this is a semi-classical calculation. OK, now you know that this relationship here can be approximated. And this is actually what you have done in statistical mechanics. You haven't used this equation. You have used an approximation to this equation which says something that you have used this approximation. And then you have said that a large n in the thermodynamic limit, so you have some sort of Laplace transform. And they told you that in the thermodynamic limit, what you have to do is a Legendre transform. The thermodynamic limit f of n is given by j of mu minus n mu, where mu is the derivative of j. Sorry, n is given by the derivative of j with respect to mu. So this is what you learn in the thermodynamics. This is how you relate in the thermodynamic limit the canonical ensemble to the grand canonical ensemble. So it's very easy now. You know j of mu. So you do this Legendre transform. And then you find, first of all, you find that mu, mu is determined by this equation, actually. And n of mu is equal to this. So you derive immediately that mu is given by a square root of 2 pi k to the 1 half n to the 1 half. Another thing that you learn from this relationship is that n large, the thermodynamic corresponds to mu large. The fugacity has to be large. And now you get f of n. And what do you get? Well, you plug it in here, and you get precisely, exactly, pi square root of 2 divided by 3 with a minus sign, n to the 3 half k to the 1 half. So this is the n to the 3 halfs of Clevanoff and Chaitlin on the back of an envelope, just using elementary physics. Let me just check that all the factors are OK. Yes. So you see, this n to the 3 halfs here is completely demystified. This is the scale that you would find in elementary statistical mechanics if I give you a thermi gas with a linearly confining potential and a linear dispersion relation. Actually, depending on the power here, if you change the powers here, alpha and beta, you get any power of n that you want, just by adjusting alpha and beta. So the n to the 3 halfs is actually completely expected from this behavior. And not that you even get the coefficient right. So the coefficient is exactly on the spot that you get from any SCFT. So OK, this is successful, but this is not really what we want to do. This is nice to do this, but we would like to actually go beyond that. So what's the next step? Well, remember that this result is really what I call, sometimes, the strict large n limit. So the strict large n limit means that you really take n to infinity. And this is the leading term in expansion at large n. And by the way, the study of this still larger limit that Kfix was first done by Herzog Klevanov, Pufu and Sabdi, essentially using this matrix model directly. Now, this Fermi-Gasaprost that I'm trying to sell to you is good in driving this in a very limited way. But its power is that it makes possible to actually get the full 1 over n expansion in the m theory limit. So how does this go? How do you do this? I'm going to summarize the calculation. The details are actually fun, because this is, as I said, elementary statistical mechanics. So you can do whatever you want very explicitly. So how would you go about improving this result? Well, if you want to improve this result, you have to go slightly away from the thermodynamic limit. And this means going slightly away from the leading semi-classical limit. Now, where did we make approximations? Well, we make approximations in various ways. We neglected here commutators. We just here took the leading behavior of these functions when x and p is large, et cetera, et cetera. And this can be actually, so we were not very systematic in one word, right? So you want to be systematic about this. You can be. And you can write down some much more precise formula. And the way to actually improve from this is to actually do a systematic WKB approximation. So here we took very large challenges as compared with h-bar. If we want to go beyond that, we have to incorporate corrections in the h-bar approximation. So first of all, let me actually explain how you would go about it. Well, what you find is that this potential of this Fermigas has a WKB approximation. This is by general reasons in thermodynamics. So you would have something like this. So let me use k as my expansion parameter. So k is h-bar. The leading approximation to j was what I computed here. It goes like 1 over k. But then you will have higher order corrections. So what we computed, what I just compute is j0 mu at large mu. So this is just what I did. I just computed this function at large mu. It turns out that you can't do much better. You can compute all these functions, jn at all n, if you want, because you don't get a closed formula. But you can push it as far as you want. And you can get the exact dependence on mu. So what do you get when you do this? I don't have much time to tell you the details of how you have to do this. But it's always playing in the notes. And if you have questions, you can ask me later on. So how this gets corrected? Well, I have to say that this is a very easy calculation as well. And the reason, the way you do this calculation, the easiest way to do this calculation, is to combine this formula, is to use this formula, and combine it with this formula. So how does this go a little bit? So let me just write capital O. Now the Zl's are just the traces of e to the minus Lh. Now if you want just to compute j0 exactly, what you have to do is to compute the exact semi-classical approximation. But in the semi-classical approximation traces, you can just do classical statistical mechanics. So how do you do classical statistical mechanics? Well, you integrate over phase space with the right one-to-measure. And then you use this integral here, the classical of xp. And now you don't approximate this by its polygon, but you use the exact log cost that I was using. When you do that, so by using this, this is really the exact, this will give you the exact leading approximation. So the details, as I said, are in the notes. You calculate this at L in closed form. You plug in there. You have an infinite series. You resummit, you do tricks from the trade. And then you get the following function. j0 of mu is equal to mu cubed divided by 2 mu cubed divided by 3 pi squared. So the k, I don't include it because it's here. Then you get the linear piece, which was sub-leading. So we didn't see this piece before because we were just leading, getting the large mu approximation to this guy. We get a constant involving, again, z of 3 for those of you who like number theory. And then we get an infinite series of sub-leading corrections. Let me write them like this. a0L b0L mu plus c0L e to the minus 2L mu. And these coefficients can be computed explicitly at any L. Now, what is this telling me? Well, this is telling me that this function is more complicated, but it's very interesting to actually look at these exponentially small corrections. Why? Well, because if you remember, I can get from j0, I can get f of nk by doing a Legendre transform. So at leading order, what I have to do in order to see what kind of correction this gives me in f and k is just to substitute the value of mu given by the saddle point approximation. But what is this? Look at this. These are exponentials of minus 2 mu. And then, if I put here the value that I just obtained from doing this trivial calculation of the Fermi gas, what I get is minus square root of 2 times pi k 1 half n to the n 1 half. So you have seen this before this morning. This is exactly the way of a membrane instanton. We saw this. I wrote this this morning. This actually is e to the minus L over Lp to the cube. So in the same way that the tough expansion allowed me to calculate all these nonpretative corrections in terms of what it is, this procedure in which I go to this other picture, in this procedure, remember, the natural regime is the strongly coupled string, super string, because h bar is small, is g string large. And then what I see is that the natural nonpretative corrections that appear here, the natural exponentials that appear here are not the Walsh instantons, but the membrane instantons. In other words, this is a formalism in which I can compute the membrane instantons among theory systematically. Systematically. You see? Here they are. All these guys are membrane instantons. And of course, this is an expansion around k equals 0. So this is still not yet in theory. In theory, I want membrane instantons, but I want them for finite k. But this is a good starting point. We get them for free. They appear here by doing this calculation. So you use the thermodynamics of this gas. And here you have a systematic series expansion in membrane instantons. OK. That's already very good. So what else can you do? Well, the next thing you can do is to try to go. I mean, this is very systematic, right? So every step gives me a gif turn, right? Very, very leading order. I get into the three hulls. When I do the first WKB expansion with corrections of finite mu, I get the membrane instantons. So now what kind of gif I'm going to get next? Well, we are going to see what you get by doing the next WKB approximation. This is the next natural thing to do. I did the leading order and a very serious worker. And then I go to the next order. So I go to the g1 of mu. Let me write it here. I have a question, right? Yes. So let's see what happens for j mu. This is a more complicated calculation. Here you have to use more technology. But essentially, all the technology was introduced by somebody called Bickner in 1930. Bickner was interested in how to calculate quantum corrections to thermodynamics. In order to do that, he invented the Bickner formalism, the Bickner formulation of quantum mechanics was invented to actually do exactly this kind of calculation. So it's not strange that the best way to do these calculations is Bickner's formulation of quantum mechanics. So this is really elementary quantum mechanical calculation. And then you get mu over 24 minus 1 over 12. And then you get essentially a series like this, but with new coefficients. So you get something that goes like this. And then you can now compute the n of mu. It's given by a coefficient. And then something which is already exponentially suppressed. So what happens is the following. You see, there is a diminishing of the intensity of these corrections as you go farther and farther to WKB expansion. If you are interested in polynomial contributions to these guys, already with the leading order and the next to leading order, you are done. Because everything else will be a number plus exponentially small terms. So this is a sort of non-renormalization theorem that tells me that, essentially, the polynomial piece of this function is exactly one loop. Because this is a one loop calculation in quantum mechanics the next to leading WKB approximate calculation. So this has been very useful. So what you get from here is that this JWKB has the following structure. It's given by a perturbative piece of mu. Perturbative, I mean, is if you want polynomially mu plus something involving membrane instantons. And this guy here is given my mu cube to mu cube over 3 pi squared k plus mu of 1 over 24 k plus 1 over 3k. And then there will be a function just of k. Why? Because there is this guy which has a 1 over k. There is this guy which has a k squared. There is this guy who has another power of k. And I put them all together in just a function which doesn't depend on me, only depends on k. So this is very nice. And now I remembered that I have a formula that I erased, which gives me, now I can repeat what I did before, I have a formula that gives me the free energy or the partition function when I know something about the grand potentiometer. Remember that when we had Joe's this piece here, I was able to compute these reading piece here. And now let's try to see what we get by including this new factor here. You can tell me, well, you are not going to get much, right? But actually, the answer is not true. This is actually not true. You get a lot of stuff, and let me explain you what. So I use this formula here. And here, approximation means that this formula is true up to exponentially small corrections. But it's going to be exact at all orders in 1 over n. So I get this thing here. Actually, there is a 1 over 2 pi i that I forgot. And here I have a new. Now, what's the function that when you do a Laplace, what's the function that you think when you do the Laplace transform of a cubic polynomial? Well, you have a quadratic polynomial. I get one function, but you get essentially a Gaussian again. But if I have a cubic polynomial, and I choose the integration contour in such a way that this integral converges. So before the integration contour correctly, what I get here is precisely an 80 function. So I get e to the ak. I get a function here, set k to the minus 1 third. And then here I get an 80 function. c to the minus 3 of k, n minus b of k. And these functions here, so this function here, so this coefficient here is c of k divided by 3. So you want c of k is 2 over pi square of k. And this guy here is b of k. Now, notice something very interesting. The 80 function is known to go. This goes to e to the sum coefficient minus n to the 3 halves. So the n to the 3 halves, that's clever enough, and Chaitlin found, is exactly the leading order, exponential behavior of the 80 function. But the 80 function has an infinite series of corrections, which actually scale like 1 over n to the 3 halves. So you have something like this of this form. And this is the full 1 over n expansion of the n theory partition function. So if you neglect exponentially small terms, so terms which go to 0 exponentially with n large, this is the full 1 over n expansion. And what is this doing for me? Remember that f g's, in the tough expansion, the f g's have a constant plus a polynomial piece. Now, this function a of k is taking into account all these constant map contributions c of g. And this 80 function, the sparsing of this 80 function, reproduces all the polynomial pieces of the tough expansion. So in one single strike, and resumming the genus sparsing of type 2 a super string theory, n theory is doing, for me, something remarkable. It's taking pieces of f g. It's taking pieces of this divergent expansion. And it's putting them together in an entire function, in a function which has no divergences whatsoever. The 80 function is a perfectly entire function on the complex plane. So you see, n theory is getting rid of these divergence of perturbation theory. And it's readjusting things for me in such a way that I get things which have analytically much better properties. Actually, it's very interesting that this function has an overall power here, which I remember correctly. So I guess this is 1 over n to the 1 fourth. This exponent here is a typical exponent of the 80 function. And you can see that when you see what it gets in the partition function, it gives you a sub-leading logarithmic correction in n to the partition function. Now, this 1 over n to the 1 fourth, this logarithmic correction. So this gives me a logarithmic correction to the free energy. These logarithmic corrections have been much studied by Asok Sen and collaborators in the context of logarithmic corrections to the entropy of black holes. But mathematically, the problem is very similar here. I mean, the way Asok and his collaborators are trying to reproduce these log corrections is by doing one-loop calculations in supergravity. So together with Asok and my student Alba Grasi and Sajantani Bhattacharya, we actually try to see if one-loop 11-dimensional supergravity reproduces this guy, and it actually does. So here, not only then to the three halves, which is the leading term, it's just the pure classical gravity result is reproduced by supergravity, but also the one-loop correction is actually accounted by 11-dimensional supergravity. Moreover, if you see what is an expansion into the three halves, remember my basic dictionary. n to the 3 is l over lp to the 6. So n to the minus 3 halves is proportional to lp over l to the 9th. And a power series expansion in lp over l to the 9th is what you should expect for the quantum corrections to a gravity theory in 11 dimensions. So all these series of corrections that the 80 function is producing for me are coming from loops in supergravity, at all orders in the Newton constant. So this is really a surprising result, because it's giving me an exact result in all order quantum supergravity. The only thing I'm missing here are non-pertorative corrections. But all the pertorative corrections coming from 11-dimensional supergravity are accounted in this calculation. And you notice that we didn't have to kill ourselves to this calculation. We just took this Fermi gas, and we did the next to leading order calculation. So this is the magic of the Fermi gas. By doing one-loop calculation in elementary quantum mechanics, well, not that elementary, but quite simple, we get an all-order result in quantum supergravity. Actually, it has been argued recently by Gomez, Davolkar, and Drucker that this 80 function, the full 80 function, they argue that you can actually calculate it, represent from a supergravity calculation using localization. So they really can argue that this result is also compatible. The all-order result is compatible with supergravity. Well, you see the power of these approach is that we are really using ADS-CFT to go really beyond the classical supergravity approximation. So we can get results at all orders in quantum supergravity. OK, now, what's next? What should you do next? Well, we have already noticed that n theory is reorganizing the perturbative expansion in a different way. So we don't have to longer worry about the divergence of the FGs because these pieces here are being taken into account in an entire function. So we should ask ourselves, OK, now it's clear that n theory is taking care of this perturbative part, this quantum gravity loops in a nice way. But what happens with the membrane instantons? What happens with all these contributions, which are exponentialism, or what happens to them? Let me give you at least a hint of how this works. It turns out that the exact n to function, this exact contribution of membranes, can be resummed in closed form. So remember that the leading order was a constant, AOL. But actually, if you include all orders W, K, B, you can resum them order by order. And then you get here a function of K mu squared plus BL of K mu plus CL of K. And this, it's multiplying E to the minus 2L mu. Now it turns out that these coefficients can be determined in closed form. This is highly non-trivial. I'm telling you that you can resum the full perturbative WKB expansion. There are just three or four quantum mechanical problems where you can do this. And this is maybe the fifth one, I don't know. But this is not common. It's not common that you can resum the full WKB expansion. And why is this so? Well, remember that in order to get this tough expansion, we use the fact that this matrix model could be related to topological string theory on local p1 and test p1. It turns out that these coefficients are also determined by topological string theory on the same kalabiyahu. But now it's not the standard topological string theory. It's something called the refined topological string theory in what is called the Necker-Sofs-Satashvili limit. You don't know what is this. It doesn't matter. The only thing that you have to know is that these coefficients can be completely in closed form as far as you want, as explicit forms of K. And the reason for those of you who know a little bit about this, let me just spend a couple of sentences, is the following. In the Necker-Sofs-Satashvili limit, it's not a refined topological string theory. It's related to the solution of spectral problems. It solves spectral problems for you. Now, it turns out that the spectral problem that appears here, the spectral problem suited to this operator can be matched in such a way that it's identical to the spectral problem typical of local p1 and test p1. And then you just can use all the technology developed for this model to actually read out the coefficients. So some of them are listed there. And then you are very happy. You are very happy. You are computing this full quantity. And notice that by inverse Laplace transfer, I can read from these main-bring instanton contributions. And then you are very happy. You are on the point of packing everything and go home. And suddenly, you notice something bad. You notice that this guy diverges. And this guy also diverges for physical values of K. When K is equal to 1, this guy diverges. When K is equal to 1, this diverges and so on. So you did something wrong. Because my matrix integral doesn't diverge. So for example, b1 of K is proportional to 1 over sine of pK over 2. So you see when K is equal to 2, this diverges badly. So actually, whatever rational value of K, you can always find one coefficient of diverges. So this has an infinite number of poles in the K plane. And then you are really depressed and you don't know what to do. Because this means that your calculation is missing something. But then you kick your head and say, of course I forgot something. Because this guy is only taking into account main-bring instantons. But remember that there were all these worst instantons that we computed already, actually. We already computed them. So we should add them. So it turns out that this JWKB is not the exact answer. Because this only takes into account perturbative corrections in WKB. But there are non-pertorative corrections in WKB. So the full JMU, and simplifying things a little bit here, the story is slightly more complicated. But let me just provide the idea, because I have just five minutes to go. The idea is that the full grand potential is the polynomial piece, which is responsible for the n to the 3-halve behavior and for the quantum gravity corrections, plus the m2-brain thing. And then you have to add a piece, which I want to call Walsh's new K. And it turns out that when you do this correctly, and again I'm glossing over the details here, the poles here cancel against equal poles but with opposite residues here. So poles cancel. And you find a function which is an analytic function at mu at infinity. So this function has a finite radius of convergence at infinity. So what is going on here? Well, this concentration of poles was found by a true Moriyama and Okuyama, based on the pre-issue that we had done. And it's a very remarkable phenomenon, because it's telling you that if you try to construct a quantity in m theory just by adding membrane instantons or just by adding Walsh's instantons, you are going to fail. This is another way, this is the way in which m theory sees that a theory based only on Walsh's strings and membranes is incomplete. The way tight-to-air super-stream theory sees that Walsh's a theory based on finite stream is incomplete is through another mechanism. It's through the factorial divergence of the series and the necessity of including non-pertoactive terms. When you go to m theory, you leave the real of the virgin series and you go to the realm of convergence series, because this series is convergent as a series into the mu. But its coefficients, the price you pay if you don't include the Walsh's instantons, is that all these coefficients have singularities. Now once you have the singularity, once you have both contributions, the theory is completely consistent and no longer you have cancelled these poles, but you also have obtained an analytic function at least in some domain of the mu space. But in terms of analyticity, this is much better than the FGs. Remember that the FGs, the sum of FGs was a factorial divergence series. This is really a convergent function. For any fixed value of k, this function has a finite range of convergence. So this encodes, some of the results here are conjectured, but this encodes all the information about the matrix model BGM at all orders in 1 over N and including, as far as we know, all exponential and small corrections. For example, you can check, remember that at the end of the day, you have to check that you recover the finite gauge theory results. After all these should be a way of computing ZN of k. Now you can actually see that this conjecture gives exactly this, this you have to do it numerically. We have verified this with 500 decimal digits in some cases, okay? So it's clear that there are, well I don't think we are missing any non-perturative contribution here. Everything is encoded here. Okay, now let me make a final comment. Now notice that in the large expansion of TOFT, what is natural, what comes to you naturally are the world city instantons. And then the membrane instantons, you have to add them as non-perturative effects. Here is the other way around. You look at this Fermigas problem and then this standard WKB expansion, so the perturative expansion, what it gives you are the membrane instantons. And the world city instantons, actually, if you look at how it depends on mu, it depends on mu in this way. And this is what it should be because if you look at mu here, it has a K over one half. So it's only when you divide by K you see that you can't trade this by a square root of lambda. So this is mu over K. But in this formalism, something that has K here is non-perturative in H bar, remember that K is in H bar. So the world city instantons, which were appearing for free in the TOFT expansion, here are due to quantum mechanical instanton effects in this Fermigas. So this Fermigas has the perturative WKB story and then you have to add quantum mechanical instantons and these are the world city instantons of the original theory. So if you want the Fermigas, it's a sort of S-dual version of the original string or the original matrix model or the original type of super string. So now just my last sentence because I still have maybe 30 seconds. This formalism has been extended recently in the following way. Remember that here we started with a problem with what's based on this ABJM matrix model. So we, you know, people do localization, got this matrix model for us, we started analyzing it and then we found in the process of analyzing this problem that we could use all these results of topological string theory first to calculate these world city instantons and then also to calculate the membrane instantons. So this was very nice. But then now you can ask the following question. Can we actually use insights from this thing to actually understand better topological string theory? The topological string theory is a simpler model of string theory but it also has the same problems of a standard string theory in the sense that it's perturative expansion is a purely divergent series and there is no known object that actually encodes from which you can obtain the standard topological string amplitudes by using, by doing an asymptotic expansion. So in some cases there are proposals for non-perturative definitions of topological strings but there has no completely coherent point of view and it turns out that this approach to ABJM suggests a way to deal with this problem topological string theory. So remember that here one key thing in our problem was to use this operator to define this Fermi gas picture. And this operator you can think about this as the quantization of the Fermi curve. Now topological strings come with curves in the non-compad case and these are called the spectral curves or the mirror curves. It turns out that these mirror curves can be quantized and this was guessed by people a long time ago and there was a few years ago there was a paper, there were a couple of papers where people did the WKB expansion of this quantized curve and they found this refined theory. Now it turns out that you have to do more than this. You have to take the mirror curve and produce an actual operator, not just a formal WKB expansion. When you do that you find that this operator is of trace class and it has a well-defined spectrum, a discrete spectrum and then it turns out that the the retorative WKB piece of this spectrum gives the refined topological string but then there are also wall-city quantum mechanical corrections to this spectrum and this quantum mechanical corrections give you the standard topological string. So by quantizing the mirror curve you get in the single strike both the refined topological string in the necrosis of the limit and also the standard topological string. In particular you can write down matrix models, explicit matrix models whose one of an expansion gives you the topological string partition functions FGs as an asymptotic expansion. And this you can do for essentially all examples in with low genus. So not only this formalism is useful for understanding the non-pretative completion of super extreme theory but you can also use it to give a non-pretative view on all the interesting models of extreme theory and the picture is always the same. You have asymptotic expansion and the asymptotic expansion you won't eventually realize it as the expansion of something well-defined. And this something well-defined in the case of ABGM theories, this matrix model and in the case of topological strings is an operator associated to the mirror curve. Okay, let me finish here. Thank you.