 So yesterday where we introduced ABJN theory as our main example for this type of non-pertorative calculations and I was explaining that there are two duals, two gravity duals, one is an n theory dual that I was describing yesterday is this ADS-4 x SES-7 Mozart's K background but there is also a tie to a dual and this is well known from the early days of super gravity so this is a really a super gravity story and the best way of actually understanding this tie to a dual is to start with the ADS with the 11 dimensional background and remind yourself that the tie to a theory is obtained from n theory by a reduction in a circle so you have to find a circle in this geometry in order to do that and there is a geometric fact that comes that helps you which is this existence of what is called this half vibration and this says that S7 can be regarded as a vibration over Cp3 so this is the complex projective space this is a six dimensional space and you have a vibration by an S1 this is a non-trivial vibration and this means that there will be a non-trivial background for the Ramón-Ramón connection so there is a non-trivial connection associated to this bundle and this will become a non-trivial Ramón-Ramón one form but you can do the reduction this was done by Nielsen and Pope already in the early days of super gravity and you can use the dictionary that I was presented the other day to realize that you will get then a 10 dimensional background where ADS4 remains and then you get here the basis of this vibration which is Cp3 so this is the background you get and notice that this has negative curvature it has positive curvature so this is not, if you think in terms of Calabillo compactifications I like a lot to think about Calabillo this is not a Calabillo but it has positive curvature but this is because you have negative curvature here so this is a solution of the question of motion of 10 dimensional super gravity and then the dictionary tells you the G string should be proportional by the dictionary that I was telling before to 1 over K squared remember that K appears here and then you have the dimensionless parameter made out of those scales you still have the length the radius of ADS4 that now is going to become a radius scale for Cp3 and the dimensionless combination now involves the string length not the plank length so you have this relationship and you also have that lambda which is the top parameter in this geometry in this theory is actually equal to 1 over 32 pi square L over LS over 4 so now you have to, if you remember the M-theory modulus space the parameter space of M-theory is one dimensional but now here we have a two dimensional parameter space so we have a parameter space in which you can set here lambda and here G string so let's understand what is this now when L over LS is large so when lambda is large so this is this region here the string is small as compared to the length scale of the background and this is a point particle limit while here you have a non-linear sigma model which is strongly coupled because the string length is large of the size of L here we are when lambda is equal to 1 you have a strongly coupled non-linear sigma model but this is if you want, this is for the parameter that describes the fluctuations of the string in size but then you can have now joining or splitting of the strings and when this joining and splitting is very suppressed this is you want the genus zero theory so there is no joining or splitting of the strings and as you turn on G string you start having interactive strings so actually this corner here when you have the point particle limit of the strings this is the supergravity corner so supergravity is just a corner of this model space this corner here when lambda is small and there are no joining or splitting of strings corresponds to the perturbation theory perturbation theory goes from here to here if you want so this is the region in which you also have the planar limit and then as you turn on G string you can get as many complicated situations as you want if you are in an arbitrary point here you have a theory in which everything is strongly coupled the string here is strongly coupled from both points of view for the point of view of the string length and from the point of view of the string coupling constant so this is the parameter space and now you can try to extract consequences predictions from ADS CFT or what are the relationships that are expected between the gravity side or the string side and the gaze theory side now the simplest prediction of ADS CFT was already pointed out by Witten but it was not used much before this story with APN theory is an equality of partition functions so the APN theory partition function which we defined yesterday on S3 should be equal to the string M theory partition function on the corresponding background right so this should be X10 or X11 depending on if you are using type A theory or M theory now if we do it so let's actually do it for M theory so let's look at this for M theory and this is actually quite interesting relationship because we actually don't know very much about this side we know this side is well defined this is the gaze theory partition function but we don't know very much about this side so an interesting thing to do is to actually use this side to learn something about this side now you have to be very careful because of this equality because this equality is not completely trivial it could be a trivial equality and you could define this object by using this object sometimes people say this people say that you can define the M theory partition function with the gaze theory partition function but then this doesn't have any predictive power this is a definition this has predictive power why? because we don't know exactly this but we know something about this and what we know about this is that at large distances so in the weakly coupled limit in gravity this should be the supergravity partition function on accident so this is something that we know about this side whatever this side is it has to satisfy this it has to go in a weakly coupled limit to the supergravity limit that's why this is a predictive this type of equality has two sides has a predictive size in the sense that we can compute this and this is going to tell us information of this and it also has the ability of letting us prove all the regimes of M theory that we don't understand because we can understand this size very well and we will see we can actually get many ideas of this we can prove aspects of M theory through this equality so this is the status of this equality some things you can check this is really a prediction some things you can check and some things can be proven in theory by using the gaze theory equality now in extreme theory in principle the partition function is well defined on perturbatively in genus by genus expansion so in the case of extreme theory life is slightly better because a priori you could do the genus expansion but still this so in the case of extreme theory so in the case of M theory we can predict things and we can check things so in the case of extreme theory we know a little bit more because we can now write this ABJM of S3 equal to Z string on X10 and this we know has a genus expansion so this will be this expansion that I was telling you yesterday so we know how to calculate these things a priori you are able to do complex integration or modelling space we know how to do it but this as I was telling before is an asymptotic expansion so we should really rather write here design for an asymptotic expansion now this guy, remember depends on two quantities N and K and here lambda is related to N and K by this and G string is related to also the parameters in the gaze theory by this equality so what is this telling me is that the gaze theory is providing a non perturbative definition of this asymptotic expansion in the same way that the Schrodinger equation is providing me a non perturbative definition of the perturbative expansion that you do in quantum mechanics you have to think about this as an intrinsic perturbative expansion and this is if you want the well defined object whose asymptotic expansion in the top regime gives me this asymptotic series so this is one of the nice things of ADSEFT it really provides a non perturbative definition of Schrodinger theory this quantity, this gaze theory quantity is such that when you span it you get exactly the asymptotic expansion of Schrodinger theory and nothing else so there are many subtleties here because this side here it's a priori only well defined this is a partition function it's a function of N and K and note that these guys were integers to begin with N is an integer, is the rank of the gaze group and K is an integer here we have a function which has a formal power series of two variables which are continuous lambda here in the string theory is a continuous variable it gives me the radius of at least it's continuous in the super gravity limit this is a radius and K is also discretized here but the string coupling constant a priori is not discretized so this is really a non trivial equality in which a function which is defined for pairs of integers has an asymptotic expansion in terms of a formal power series involving functions but this is a well defined this is a really a good non plurality definition of Schrodinger theory and the question is what can we learn about this and we can actually, for example we can ask many questions about the plurality expansion of Schrodinger theory by looking at this subject and this is what we are going to do now as I said before there is a simple prediction of this ADSFT duality in this case because the super gravity action on x11 can be computed the only thing you have to do is to evaluate the super gravity action on shell this requires regularization this has been studied already in the early times of ADSFT it's called this thing called holographic renormalization and then the consequence that you get here is that the free energy of the gaze theory in this regime remember that this regime is enlarged on k fix sorry enlarge on k fix this behaves this is a calculation that you can do so minus pi square root of 2 divided by 3 k to the one half and n to the three halves and this is here I just define this as the log of z of A, B and C so this is just the log so this is a famous prediction of ADSFT correspondence and displays this scaling of the number of degrees of freedom as three halves this was done by Clevanof and Chaiklin long time ago and this was very puzzling for a long time because you see on the right hand side you have an ABN theory this is the free energy of a gaze theory and it's a gaze theory which as I told you all fields are by fundamental representation in a joint representation of the UN and then you should expect the number of degrees of freedom to grow with n square while this gives you a growth which goes with this point in the empty house so this was a slightly puzzling but it's not that puzzling because if you do this from the point of view of tie to A theory so from the point of view of tie to A theory you can see that this gives me a prediction for the behavior of the planar free energy so you you write F0 of lambda maybe I'm not let me not write F0 because in F0 I'm using actually I'm not using the same this is the string and yesterday I was using and noticing was spanning in powers of Gs which is not exactly that so so let's let me call this Fg just just to make sure that the span here is in a different parameter from what I was doing just today so but this is just related by an overall scaling an overall factor to the Fg that I was defining just today let me then define this in this way that's a better way to do it let me re-spress this quantity not in terms of N and K but in terms of N and lambda which I introduced there and I take the limit when I go to infinity and this I can, it's very simple to obtain because I just have to re-spress I have just to introduce a lambda here and you see that if I introduce lambda here if I write N as K times lambda I will get here a power of N squared of N to the three-halves and then when I divide with N squared what I get at the end of the day is one over the square root of lambda so I have a power of N to the one-halves down and I have a power of K to the one-halve up so what is this telling me actually is that the planar free energy of A, B and C because this is the strength of coupling here so this is for lambda much bigger than one has a non-trivial scaling with lambda and actually this is very interesting because if you actually look at this function when lambda is near zero this actually goes like minus lambda squared log of lambda so what you have it's a function so this is what you find from perturbation theory so what you have here is a function this planar limit is a function that if I write it here minus one over N squared F and lambda in the limit when it goes to infinity and I plot it as a function of lambda I have a function here which goes to zero as lambda squared log of lambda and then eventually it will go to a square function which is proportional to the square root of lambda here so we have a very different behavior with coupling from a strong coupling and in the intermediate region we have a non-trivial interpolating function from weak to strong coupling so again we have two pictures of the dual here we have a picture in M theory and we get very very precise predictions from ADSEFT the first prediction can be put in the language of M theory is telling me how the free energy of this theory behaves when N is large and K is fixed so this is in this M theory region that I was describing yesterday but this result can be translated in a prediction for the behavior of the planar free energy that you take in the top analysis at the strength of coupling and the interesting thing is that the function you get is a highly non-trivial interpolating function this has been seen of course before in N equal to 4 super y a mean we have many examples of this type of functions but this is a non-trivial interpolating function whose strong coupling behavior is predicted by ADSEFT so this is really a strong prediction of ADSEFT either this is true or is not so either you find these numbers or you don't so if you want to disprove ADSEFT it's very easy you just have to do a calculation in which you have to compute this and see if it agrees or not if it doesn't agree then ADSEFT doesn't work even in this large chain limit absolutely but then you get into epicycles it's good to be able to make a strong statement this is really a strong prediction of course you can always fix things but you can also fix the total make system so this theory was solved as the right description so we'll see that it's actually this prediction is actually right yes oh sorry yes yeah sorry it just depends on your on your normalization I mean yeah you are absolutely true and you see here different normalizations because sometimes I use n to the large chain expansion sometimes I do this so you actually write f0 of lambda as I was defining yesterday then you know so let me actually write here sorry by using different normalizations but this is just part of the so f0 of lambda goes like this when lambda goes to zero and if you actually see how it goes f0 of lambda is actually you have to multiply by lambda here and then you see that this is going to go at the square root of lambda if I remember correctly actually as this guy goes like lambda to the three halves so you have this sort of behavior sorry I absolutely right this is true so f0 of lambda is obtained by multiplying by lambda squared here because f0 of lambda was obtained by doing this expansion at the leading order gs minus the square f0 of lambda so this was my definition of f0 and then it's obtained from this definition by multiplying by lambda squared so this goes like lambda to the three halves and this was the thing that I was having in mind sorry so this goes like minus lambda squared log lambda here and this goes like lambda to the three halves up ok so these are as I said these are just different normalizations of fg because depending on how you do the expansion you have different normalizations different different choices when we do the expansion in string theory the natural parameter is use the string coupling constant when you do the gauge expansion you can use either choose n or use this parameter gs which I remind you from just there was 2 pi i over k so so there are different normalizations and they give you different functions but they all differ by powers of lambda so this is the interpolating function thank you for the remark very good so how do we test this is it possible to test these predictions well this is from the point of view of the parameter space you see we have the typical situation in which what is easy from the point of view of gauge theory is hard from the point of view of the string theory already if you just restrict yourself to the planar limit in which in gauge theory you only look at the planar limit and in string theory you only look at the g0 free energy you see that string theory is easier in that size because this is where the non-linear sigma model is weekly coupled while this corresponds to a strong coupling in the gauge theory side this is here here in this point here everything is coming from planar diagrams now if you want to get to this region you have to resum the full series of planar diagrams the full series and then you have to you will find a function of lambda which is typically convergent in a neighborhood of lambda equals 0 and then you have to analytically continue this function to a strong coupling if you can do it string theory tells us that we should be able to do this identity continuation to a strong coupling and this is the behavior that you will find so you have to resum over all the planar diagrams and this is the kind of thing that has that has stopped the possibility of developing large change gauge theories for a long time I mean people were not able to resum the planar diagrams of any gauge theory or any realistic gauge theory now what happened in the last years is that at least for the supersymmetric gauge theories using localization we can reduce this path integral calculating this partition function to an ordinary matrix integral and then we are lucky that people in the 80s were playing with resumming planar diagrams but for theories gauge theories in zero dimensions and they found many concrete results and now these concrete results can be used to in quotation marks realistic supersymmetric gauge theories so this is a long story and don't have the time to tell you everything about localization and matrix models so I want to I want to introduce another type of analysis which is based on a thermodynamic formalism which is useful for them theory for them theory story but let me quote at least what are the main results of having obtained using this matrix model approach so I guess couple of years ago there were lectures on localization of gauge theories here in Trieste so you can probably look at them and I also wrote some lecture notes that you can't find on the web on this theme so let me summarize although this is a long story, let me summarize it now what was found after the war by peston people apply his mythos of localization to a b gen theory what they found is that this this is this a b gen theory and the three sphere let me write it explicitly as a function of n and k can be written as a matrix integral so this is probably the formula the bigger formula we write on the blog where it takes some time to write it so let's just write it down just for once what they found is that this partition function can be reduced to a double matrix integral you want so it's an integral about over two sets of real numbers of families of vectors so these are n so these are n variables n mu i from one to n and this integral is from the value is nu i from one to n so this is an integral over two n variables when n is large this is a complicated integral and then you have a complicated integrand as well so in the numerator you have a factor of this type and then in the numerator you have a factor coupling the nu i and the nu i and the nu js so let's write it like this to the square so this is to the square and then this is also to the square and then this has some sort of imaginary Gaussian weight ok so this is the result of localization so let me tell you a couple of things about this thing also deriving this takes probably as many lectures as we have already in the course in the school so mu and nu are actually related to the scalar remember that there were these super sinons theories basis of vector multiplets these vector multiplets have a scalar fields associated to them and mu and nu are actually constant depths for the scalar part of the n equal to trunx sinons multiplets so these are related to the scalars in the n equal to vector multiplets of the trunx sinons of the trunx sinons size so if you want here we have these two nodes with un with coupling constant k another un with coupling constant minus k and then we have this for bifundamental multiplets link in the two nodes this piece here comes from integrating out the vector multiplets is a one look determinant coming from the vector multiplets here so it's giving you interactions between these scalars and then the four hyper multiplets that are in the bifundamental representation connect the two nodes so this leads to an interaction between the variables here and the variables here and this is the interaction you find finally when you evaluate the trunx sinons action on this on this localization locus you get exactly this term the super symmetric extension of the trunx sinons action and notice that this guy and this guy have a different sign with k and this is reflecting the fact that these nodes have a constant coupling constant k and these nodes have coupling constant minus k so this formula, this matrix integral is actually an amazing simplification of the real problem of calculating this path integral right so this is we have gone from an infinite dimensional path integral to a finite dimensional path integral in particular this is really very good because you know when we say that the gauge theory gives a number two definition of a string theory a purist can come and tell you yes but you still have to define the gauge theory the number two definition of a string theory is easier than the number two definition this integral gives you a perfectly well defined definition of this actually you can see that this integral is conditionally convergent and as we will see later on you can write it in a form which is even more convenient for calculations so this is the result and now the question is what can we extract from this matrix integral can we actually for example test these predictions from this matrix integral can we actually generate for example the full asymptotic expansion can we study the properties of this asymptotic expansion using this matrix integral and the answer is yes and as I said this would take a detailed analysis that I am not going to do here but I will eventually give you a first principle derivation of this from the matrix integral maybe today of course today maybe this afternoon but let me summarize what has been obtained from this matrix integral well as I said there are two ways of looking at this theory remember that there was this n theory regime in which n is large and k is fixed and then there is this tough regime let me call it the tough regime in which n is large and n over k equal to lambda is fixed this is the regime in the gaze theory where you may contact with m theory and this is the regime of the gaze theory where you may contact with type 2a theory they are related but in principle they are different now what was the study first was the tough regime and the reason is that this matrix integral has been very much studied in the tough regime starting with the work of Bresa Egyptian Parisian Souver in 1980 who studied a similar much simpler but similar matrix integral and what they show is that for this type of matrix integral which you can think about as gaze theories in zero dimensions you can actually resound the planar diagrams so the techniques coming from this paper and much further other papers were actually developed and we went only what was found were like close formula close explicit formula for all the fg's of lambda so for all the fg's obtained by the one over and expansion of this matrix integral now this formula are complicated but they should be because this means that they are they are really taking into account a lot of information let me give you some properties so let me list some properties of these functions and what was learned from them so here I'm using the I'm using the notation in which f and lambda is expanded in powers as I was spending yesterday in powers of these gs variables to g-2 so what were the properties that were found so let me list the highlights of this analysis so what was found for the top expansion of these guys well the first thing that was found was a close formula for fg's of g you can find it in the lecture notes in all its glorious details it involves various special functions hypergeometric and major functions but what is the physical meaning of this guy well the physical meaning is obtained if you want to make contact with the stream theory as I was showing you in the drawing that I just erased you have if you want to make contact with the geometric picture of the stream theory stream theory is geometric in the limit in which the lambda is large it's only when the size of the stream is as small as compared with the ambient space you can think about stream propagating a given geometry so you have to look at the you want to make contact with geometric stream theory you should look at the expansion of this function when n is much larger than 1 and it turns out that you obtain so let me introduce this variable lambda hat which is lambda minus 1 over 24 you get a constant so this is the leading term you get a constant and then you have a power series involving exponentials of minus 2 pi L square root of 2 lambda times a function of lambda a polynomial actually a polynomial in 1 over p square root of 2 lambda ok and this goes from lambda 1 to infinity so what's the meaning of this guy well, first of all you see that when lambda is large and after the shift you get exactly the power of three-halves so this is the first principle derivation of the three-halves behavior that Clevenoff and Chetlin were obtaining now you find a constant and then you find an exponential an infinitive series of exponentially small corrections that lambda was in my ADS dictionary lambda which is n over k is proportional to lambda over Ls to the fourth ok so what is this guy well, this guy apart from factors has the dependence of minus L over Ls to the square this is precisely what I was finding yesterday as the general dependence of a wall city instant so this corresponds to a wall city instant and that's actually very remarkable because this doesn't happen in equal for super jam you don't really find this type of exponential these small corrections in L over L string so how this is possible well, this is made possible by a remarkable property of the geometry remember that the background for tie-to-a theory is ADS fourth times Cp3 so Cp3 is a six-dimensional manifold which has a one-dimensional non-trivial cycle a two-dimensional non-trivial cycle which is a Cp1 and this Cp1 has an area and actually you can see that the area is exactly given by 2 pi a square root of 2 lambda in these units so this is exactly the area of a string wrapping L times these two-dimensional cycles here so what you have what is going on is that you have a spherical string because we are actually looking at the genu zero sector which maps to the Cp3 and wraps L times this cycle inside so this is an S2 which wraps L times this non-trivial cycle inside Cp3 and this is actually something that doesn't happen for example in ADS five times S5 because S5 doesn't have non-trivial two cycles where strings can wrap this is really a new phenomenon that appears in this duality now another important thing here is that for reasons that are still not completely clear this function is actually closely related to the genu zero free energy of a topological string theory topological string theory and this is a special topological string theory topological strings have a star yet Scalabiau manifolds and this is a topological string theory with a Scalabiau manifold which is sometimes called local P1 times P1 and for those of you who know a little bit about Scalabiaus this is the space, this Scalabiau six dimensional complex manifold which is obtained by taking P1 times P1 which is a four dimensional complex manifold and taking its anticanonical bundle which gives you two extra dimensions and gets you a six dimensional space so this is a very peculiar Scalabiau it's non-compact and for reasons which are not completely clear this theory which is a kind of simple string theory is governing the physical quantities appearing in super string theory so this expansion for those of you who know Scalabiaus well this is the typical cubic term this is the typical set of free term that appears in the zero free energy and this is the worst expansion and is closely related to the worst expansion that appears in the topological string theory so this is very beautiful because this is an application which is a simpler string theory which is this topological string theory gives you the answer for a physical super string amplitude and in particular the worst instantons which are very well known to appear in topological string theory now get translated into physical when I mean physical is worst instantons in a physical super string theory and they are describing these maps of a sphere into Cp3 so this is already a remarkable result because not only you prove the cleverness of scheduling scaling but you get an infinite number of corrections to it which are predictions from the gaze theory to string theory so this has never been tested all these coefficients so this is a prediction yet to to be established so what else do you get so this is just the first fact now the second fact you get is that you also can also compute all the FGs this is remarkable because I don't think there are many gaze theories in which you can compute the full one of an expansion you cannot compute closed formulae for all the FGs so you can compute them recursively with a gadget called the holomorphic anomaly equations which were introduced into political string theory physical string theory and can be used to actually decode these FGs so the FGs can be computed recursively recursively and they have the structure they have a following structure I want to be very schematic here they are given by you know son power of g let me write this make sure I guess it's 3g-2 or something like that yeah well it's g-3 halves and then you have a polynomial in landa one and then you have a constant and then you have a sum over instantas as something this is very schematic by L square root of 2 landa and then another another set of functions like the helpful ones so this is so this is how it behaves when landa is much bigger than one so it's very similar structure and these coefficients are actually for those of you who are familiar with the political theory are related closely related to these famous constant math contributions which were actually first found by Sergio and his collaborators in this holomorphic anomaly paper and they reappear here as the leading contributions to these ABJM free energy at genus g so ok so these are again what we have so this is a quantity that you can find in the supergravity limit all these things go away so this is the leading supergravity limit for this quantity Fg of landa and then you again have a worst instanton corrections that now come from sending Riemann surface of genus g to this cp1 are genus g Riemann surface can wrap a cp1 symmetrically and this is somewhat counting again this was it instanton ok so this is this gives you the full asymptotic expansion of this free energy at all orders in the genus expansion now what else can you can you get well remember something very important that we found as a general story occurring in street theory is that you can learn about instanton effects if you look at the synthotics of this quantity so we have to ask what is the synthotic of this quantity and we can do it because we know this quantity so we can study these synthotics and try to see how they behave remember that we were respecting basis on general arguments that this diverges 2g factorially and then you have a space time instanton action in this way now since you know these guys in detail you can actually look to verify these asymptotics and in fact not only you verify these asymptotics but you can actually guess the exact form for this function so AST of lambda can be seen to be given again up to depending on how you how you normalize it if you normalize this effect by using gs so remember that every time you have something like this this means that there is a leading non-perturative effect of this form and AST of lambda can be fine to be the I here is just trivial because it's just to cancel the I that introduce this definition so this guy has a real part which goes like 2 pi squared square root of 2 lambda and then he has an imaginary shift pi i r pi squared I and then he has exponentially small corrections in a square root of lambda so this is what you find and then well this is just the outcome of a detail analysis and now the question is well what can I extract from here what can I know from here what can I do with this well you see we have a very explicit formula for this guy so you can actually try to see what is in the same way that here we are able to identify this dimensionally with an object of dimension 2 and this was corresponding to a dimension 2 is here we are able to identify this with something of dimension 2 it turns out that here we can also use this thing to try to identify which is the underlying object responsible for this non-partuitive effect so you would think that this is also dimension 2 but it's actually not really the right dimension because you have to be careful with what you mean by gs an important thing that you can and a way to see what is the right dimensionality is to take into account that if you go to the to the variables n and k this guy has so this non-partuitive effect has these dependence pi k to the one half n to the one half and now so this is if you want n theory variables so this is n theory variables and now you remember that n was proportional to l over lp to the 6 so this object is actually e to the minus l over lp to the cube so this is an object which we measure in units of the plan land is an object of dimension 3 so what it is well it has to be a membrane instanton or a d to break instanton so a membrane instanton an intrinsic 3 dimensional object of n theory so we just see that everything works as you would expect for a gs theory with a non trivial string or m theory dual so let me emphasize that these are really in a sense qualitative predictions of ADS CFT now ADS CFT gives a very quantitative precise prediction it tells me what is the power here but ADS CFT also tells me that what I find in the gs theory should be of time should be something that should be should fit into the general scheme of string theory and it's very satisfying to see that precisely the non-pretative corrections that appear at the level of words at the level of the words instanton at the level of the membrane instanton actually fit what you expect and this is non trivial because this depends on the very precise scaling of these quantities with lambda I mean this is highly non trivial I mean if this object is scaling this way you wouldn't find the right behavior and you do so that's very satisfying now a third aspect a fourth aspect that I want to mention here and it's the final aspect for this so let me write it here a fourth aspect is the following ok now now you have all the gs lambda and they have a common radius of convergence as I was recognizing yesterday so can you do this Borel resumption story I mean you might be able to do it after all you know we have a full series of coefficients for this genus expansion and it turns out that the genus expansion in this example so I mean at least as long as far as numerical analysis tell you so you look at this and you say wow this is fantastic and in the same situation that in quantum mechanics with the anharmonic quartic oscillator should be able to do a Borel resumption of this series and I would get my exact non-perturative answer so you try to do that and remember that the exact non-perturative answer you know it are just the free energies of this gase theory and these should be obtained so let's put it as a quotation mark by doing this Borel resumption this sequence fg of lambda right so this is a very concrete thing you can do because we know this for example for any integer value of n and k can be computed and now you put just the value of lambda that corresponds to your choice of n and k you put the value of the string coupling constant that corresponds to your choice of n and k and you can see if these are equal or not with some numerical this is something I have to do numerical but you know you can do it and they are not equal so this is very important is that even if a series is Borel resumable it doesn't mean that the Borel resumption of this series will give you back the non-perturative object you had before and this is a not a common phenomenon but it was observed already in the 70s by Baleán, Parísia and Boros they observed that this happens and actually this happens when you have instances which are complex when you have complex other points this can be the case and this guy is complex this instance that we found here is complex there is this complex factor here so what is going on is that these membranes are lacking in your analysis you have to incorporate them you have to construct a trans-series incorporating these membrane instantons if you want the genus expansion of string theory to give you back exactly the case theory partition function so sometimes people say string theory is perturbatively not complete because it's not Borel resumable but this is not completely correcting this case because this series is Borel resumable still is lacking non-perturative information and this is due to this complex membrane instantons so you want to really so this shows you that you cannot just do string theory just with worship instantons and with worship fundamental studies you have to include membrane instantons and this is a way of seeing it that these things are not really equal there is a more dramatic manifestation of how these things work of how membrane instantons have to be incorporated but in order to do that I have to do the analysis in the M-theory framework and this is what I'm going to do this afternoon so if time permits I will present you a framework in which you can do the N-theory analysis because everything I have done so far is analysing the synthotics of the matrix integral in the regime, in the top regime in the regime in which the matrix integral is fixed and N is large and then I get these FGs in M-theory you don't get FGs in M-theory you have to do an expansion when N is large but K is fixed so you should get something different and it's not obvious a priori how you can analyse the matrix integral in this regime because when you look at what people did when you study matrix integrals since the work of Bresam, Paris, Estixón and Soubert these are analysis in which you always keep the top coupling fixed and it's only in this regime that they were able to analyse this matrix model so the question is how do you analyse this matrix integral in the N-theory regime and what can you learn from that so what I'm going to do this afternoon is to introduce a framework in which you can do that and after this framework is powerful enough so that you can actually compute these membrane instanton contributions systematically in the same way that in the top expansion you can calculate the wall city instanton systematically I will introduce a framework in which you can compute the membrane instanton systematically and as a first simple application of this method we will be able to derive this N-theory three-halves scaling on the back of an envelope which is probably the shortest derivation of this Clevon of Saitling in scaling so I will do this this afternoon thank you