 In addition to participate and the possibility of giving this talk, I'm going to be telling you how to account for the certain part of contribution to the correlators of Wilson-Lupes in Ennicole IV, Super Young Emils. This is based on some recent work, mainly done with Pablo Pisani and Alan Rios Foukeman, both of them here in the audience, but moreover with some ongoing project, which is in collaboration also with them and Costia Saremba. So I guess at the starting point should be like try to motivate a little bit why to consider this problem. So while Wilson-Lupes are interesting gauge invariant observables in any gauge theory, because they come with very valuable physical information. For instance, you could use them to define the quark and the quark potential that describes the confinement properties of a gauge theory. You can also use it to define this Brestralum function that calibrates the power radiated by accelerated charges, or even in Ennicole IV Super Young Emils by considering certain life-like contours. You can relate expectation value of Wilson-Lupes with the gluen scattering amplitudes. So why to consider Wilson-Lupes in Ennicole IV Super Young Emils? Well, maybe the main reason to consider Ennicole IV Super Young Emils is that Ennicole IV Super Young Emils is a gauge theory in the prototypical example of ADSEFT. So if you want to learn about the correspondence, this is probably the case which is under good control to understand the structures underlying the correspondence. So and moreover, as I was telling us in the case of Ennicole IV Super Young Emils, you also have some Wilson-Lupes whose expectation values and even correlators between them can be computed exactly, exactly as a function of the coupling constant. And this is given that the ADSEFT correspondence is a weak and a strong coupling duality. Having exact results is crucial if you are interested in implementing some kind of precision test of it by comparing explicit computations in one side and the other of the duality. So maybe this is my main motivation, let me stress it. And then when you start to look for exact results concerning Wilson-Lupes in Ennicole IV Super Young Emils, as I was saying, there are many methods you could use. Maybe the most results were obtained using either integrability or localization methods. So what today I'm going to take a rather simpler and maybe bolder route, but very interesting anyway, which it has to do with just an explicit resumption of some kind of particular diagrams which we call ladder diagrams. Well, let me just sketch an outline of the talk, but I will start with just the definition of Wilson-Lupes and how are they defined in Ennicole IV Super Young Emils and then I will go into the details of deriving some integral Dyson equations for compute the contribution of ladder diagrams to the first, to the expectation value of a single Wilson-Lupes. Then I will extend just some ideas to write down Dyson equations for the contribution of ladder diagrams to the correlator of two Wilson-Lupes. Of course, you can go on and consider more correlators with more Wilson-Lupes and this is part of the ongoing project I was mentioning, but I will not discuss them today. One ingredient that is going to be crucial in what I'm going to be presenting today is that, as Nadab was saying, we will see when you define Wilson-Lupes in Ennicole IV Super Young Emils, you also specify, I mean, you also couple this external particle with some of the scalar fields of the theory, so you have to specify some orientation in the internal space. So when we consider the correlator of two Wilson-Lupes, we will take the two Wilson-Lupes with different internal space orientations, right? So we are going to introduce this internal space angle gamma and this is this gamma, it is this gamma that will allow us to go to specific critical cases or some parametric regime in which we can extract exact results. So to be more precise, you can find a critical angle in which the correlator becomes supersymmetric. So in that case, you can get exact results and moreover, you can show that the solution of your Dyson equation is going to coincide with the result obtained by supersymmetric localization. But then, oh, sorry, maybe another, maybe the most interesting part of our work is the existence of this other parametric limit, we will see it in detail, some limit in which of course ladder diagrams is not all you have to take into account, but in this parametric limit ladder diagrams are the leading order in some parametric limit, so they dominate. So if you, with some ladders, go to a strong coupling limit and then you compare with the string theory computation and then you can find an agreement between the two computations and this is our precision test. I was mentioning. Yes. No, we don't provide any physical interpretation, it's true, you need to, it amounts to some analytical continuation of this parameter, you need an imaginary gamma, I mean in the string theory it's weird because it's like an imaginary angle in the five-sphere, but from the perspective of the field theory it's just coupling the scalar field with an imaginary quantity or coefficient, but you might think of it if you want to, in the string theory side, you could think of changing the signature of one of the angular directions, but I mean it's like a trick to access a regime in which you can trust your ladder approximation, I'm not claiming more than that. So, well then I will just summarize and make some comments. So, okay, well what is the Wilson loop? The Wilson loop is just this path order exponential of the contour integral of the gauge potential. It measures some non-Avillian phase that acquires an external particle that is forced to move along some close contour and essentially it depends on two things, right? It depends on whether, on how do you take this trajectory and also it depends on what type of particle you make, you move along this close contour and this information is just encoded in the representation in which you take this trace. You can take this trace in different representations and this is what particle is moving around. So if everything I'm going to be telling you today has to do with the fundamental representation but you can consider other cases for those who were in the school last week, Diego was talking plenty of these other cases. So if you take the fundamental representation we were talking about quarks circulating in the loop. Okay, well, but then the physical interpretation of the expectation value of Wilson loop is typically depends on what contour do you take. So a few typical examples are if you take a very elongated rectangle you can, the expectation value you can use it to define the quark and the quark potential or you can take, sorry, this casp in the trajectory, this can allow you to define the casp anomalous dimension but if you consider like a wavy line I mean a trajectory which is slightly deviating for the straight line you can define the refraction function that was telling us about. There are many interesting contours but today I'm going to focus mainly on circular Wilson loops and the reason to focus in on this kind of circular Wilson loops is that for them you can get some exact results in order to implement some precision tests. So well, in an equal four super young means the kind is a little bit more than that because you have other fields in the action representation apart from the gauge potential. In particular you have six real scalar fields. So in principle you can define a Wilson loop in which the external particle also couples to the scalar fields and then if you do that you have to specify how do they couple so you have this additional if you want contouring internal space which is the coupling of the scalar fields is controlled by this R6 unit vector. Okay but then what would you like to do is to try to compute this expectation value of such Wilson loops in any equal four super young means well the reason to include in this I mean of course you can even in an equal four super young means you can consider the standard Wilson loops but this is just a matter of convenience if you include these scalars this coupling with the scalar fields you have a locally supersymmetric Wilson loops and this is the key I mean this is the reason why you can produce some exact results at the end of the day. So anyway you would like to compute expectation value of these objects in an equal four super young means and this is going to be in general a non-trivial function of the of the tuft coupling say and also the function of the of the rank of the gauge group but we will mainly work in the planar limit today. Okay what can you do? Well as usual if this coupling constant were small let's say much smaller than one you can just make a perturbative series expansion of it and you can compute it this is sketchily depicted by the contour with these Feynman diagrams accounting for different contribution of different orders you can do that but certainly this is only going to be valid in this in this machine but sometimes you could be in an opposite regime in which the tuft coupling or the coupling constant is very large and then you can this this perturbative approximation typically breaks down so but in those cases what you can do is to appeal to go and apply I use your ADSFT dictionary. So what is the dual picture of Wilson loop? Well we also heard about this last week typically in ADSFT it's a reformulation of your field theory in terms of some gravity theory in ADS in which the field theory input enters as a boundary condition for your gravity theory right. In the case of Wilson loops the precise prescription was given by Maldacena and Ray and G already in the early days in 1998 and they claim that expectation value of a Wilson loop in a fundamental representation is accounted by the partition function of a string whose open string whose end points and at the boundary of ADS on the contour that specifies the Wilson loop right. So if you want to compute now in the dual language expectation value of the Wilson loop you have to path integrate your string theory over all possible worship configuration subject to this boundary condition yes. Typically this is of course going to be very difficult but what you can do is as usual try to study this in some semi-classical approximation yes and then what is the semi-classic approximation is when this path integral is dominated by just the configuration that minimizes this action and this is like minimizing the area of the worksheet that ends on this contour but okay well this is in general not justified but it is so this semi-classical approximation is going to be a good approximation as long as you have a coefficient in front of the string action that becomes very large. So what coefficient do you have in front of the string action? Well you also you always have a one over alpha prime but also since you are now formulating your string or polyacrobrombo auto action in ADS you're gonna pull out the factor of r squared where r is the radius of your ADS space time. So the effective tension of the string is r squared over alpha prime and this is if you go to the ADS EFT dictionary the square root of your tuft coupling is defined as the Yang-Mills square times the rank of the gauge group. So then for this semi-classical approximation to be valid you need this effective tension to be very large and then this is the strongly capital regime from the field theory point of view. So of course you can go beyond this semi-classical approximation and this good amount to consider quant of fluctuations of the worksheet. I'm not gonna be discussing this but if you're interested in this kind of corrections there is a talk in the afternoon by Guillermo Silva talking about this. So but as I said my motivation was to implement some precision test but this is in general challenging because what I just said results typically are valid in opposite regimes. Then this one you can go to consider ladder diagrams. Ladder diagrams are their definition is they are five-man diagrams with no internal with no interaction that this is right. It's only propagator you consider. Of course this is a lot simpler than the most general case and in many cases this is a tractable problem and you can even resume it. The caveat is that in general this accounts only partially for expectation value of a Wilson loop. But as I said there are gonna be some specific cases in which the correlator for instance or the Wilson loop becomes supersymmetric and in those cases the ladder diagrams are the only contribution you get. All interaction diagrams cancel so the ladder diagrams are the exact answer for the expectation value of the Wilson loops. And also there is when we go to correlators and we play this parametric limit there are gonna be some regime in which although ladder diagrams are not the only things you get they are gonna be the leading order in some parametric expansion. So in these two cases you can resume it or compute the exact reservations of ladders go to a strong coupling limit and make a comparison with some string theory computation and you could observe that the two results actually agree. So yeah. So well this is I hope this is clear the idea of what I'm gonna be telling you now I'm gonna go to it's gonna be much more detailed so if there are questions about what I said so far or what is the motivation or the idea of the computation please. This is a good moment to ask. Okay so let us jump into the details. Let me let me define this non-Aurelian phase factor I called you is closely related to the to the Wilson loop but now it's an it's an in an open trajectory or an arc of some trajectory between two points is but it's still the path order exponential of this combination of the gauge field I'm gonna take I always take in this now the coupling with the scalar fields to be a constant along the along the trajectory and just take it to be in some particular direction. So generically this combination I'm gonna call it this operator O and we would like to compute the expectation value of the trace of this non-Aurelian phase. As I said this is gonna be directly giving you the expectation value of a Wilson loop if we now if you take T1 and T2 such that the two endpoints of the curve are the same so you are in a close curve this is gauging value and this is gonna be the the expectation value of the Wilson loop but in general is this other non-Aurelian phase U whose the expectation value of its trace is what we call W. So okay and then I would like to compute and I'm not gonna be computing exactly this but only the ladders diagonal contribution to this expectation value. So what you have to do is just to now you have this exponential you have to expand it the series of the Taylor expansion of the exponential and start to contract with propagators and never including any interaction diagrams. So well when you do that you of course now you have to take into account the propagators between scalar fields and gluons or gauge fields and as Nadab said you can combine them in an effective propagator so I think Nadab was calling it I am calling G but it's essentially the same the same object is so the propagator between two of these operators O between T and T prime and on the curve it's just an effective propagator G which typically depends on what what is the function X of T parameterizing this distrojectory. So I'm using this double line of tension to emphasize this the two indices of this of these operators O. Well let me just try to do a rederivation of the answer the result that Nadab presented that was the expectation value of a series of gluons was given by this special function. So this can be re-obtained with using these internal equations for the expectation value of this non-Avillian phase U. So the central thing is that this non-Avillian phase factor satisfies some recursion relation so the U between T1 and T2 is the identity plus this integral of U between T1 and some T that you're integrating between T1 and T2 of this insertion O. So what are you gonna do? Well you're gonna just take the trace compute expectation value and and start to do in start doing weak contractions so for instance if you take the trace of this recursion identity here you get just n here is the expectation value of the trace but since you would like to weak contract this O in T prime with some other O within U you have to split this U as a U between 0 and T second and an O in T second and then another U between T second and T prime and then you contract these two O's then you get this effective propagator and you get this and if you carefully follow the the contraction of color indices you get the expectation value of the trace of one U and the trace of the other. Now since you are interested in the large n limit you can argue that in the large n limit there is a factorization of this expectation value of two traces it's just the product of the expectation values of each of the traces independently. So but then this is again not this is what we already call W up to this normalization but this is a W of T second and this is a W of T prime minus T second so what we have obtained is that this W the expectation value of this non-Avillian phase has to satisfy this integral equation so still this is in general difficult to solve because of mainly because of this effective propagator here. So let me just because in the following examples I'm now going to go through the details but you can get an sketchy the diagrammatic interpretation of this and you could have a guess that this integral equation comes out. If you let me just schematically represent this expectation value of the non-Avillian phase W by this blue blob so which accounts all possible ways of contracting propagators planarly in this segment between zero and T. So this is what we would like to compute. So these diagrams are maybe a little bit confused this is an open arc and this dash line is just representing that the closure of the color index but it's not a it's an open arc if you want. But okay so you would like to compute this it could be that there is no propagator all along the line or if there is at least one you can call T prime the right most point in the segment that is contracted with the propagator. So if this is T prime so to the right of it there could be no propagators at all so you cannot insert nothing here and then you can call T second the other point that connects T prime with the propagator with another point it's gonna be to the left if T prime was the right most. And then you have this double line and then you can have propagators that start and end either in this segment or in this other segment but certainly you cannot connect them because if you have a propagator that starts here and ends here you will be crossing this blue line and that would be no planar. So this is a press this kind of diagrams would be a press in the planar limit so the only thing you can have is propagators within this interval and this is this W of t minus the second and you can also have propagators that start and then in the same in the segment between zero and t second and this is W of t second. And of course because of the of this the blue double line you have this this effective propagator. So you can if you see you can derive this kind of internal equation just yet by by looking pictures in this way. So but as I said this is in general very difficult but if when you go to the case of the circular Wilson loop there is some some magic and some simplifications due to that in the case of when you take this the paragraph to be a circle this effective propagator between O and T and T prime becomes just constant right and this is this constant. So when you go to this internal equation there's now the integral equations looks a lot simpler and then if you make just by simply making here in this case is the key or the the method to solve this this integral is the Laplace transform. If you do the Laplace transform of the integral equation what you get is just an algebraic equation which turns out to be a quadratic algebraic equation for the transformation of W of t. So it is what we are called W of z but it is very simple to solve this quadratic equation and then you you have an explicit expression for the Laplace transform of the function you would like to compute. Then you only have to do it to anti transform back and then you get if you anti transform this W of t is precisely this this Bessel function this this first this Bessel function I want. As I said if you would like to connect this with a circular Wilson loop now you have to take t such that the initial and the end point of this non-Aberian phase is the same and then you for the case of the circle this is when you evaluate the W and 2 pi now if you evaluate this W function in 2 pi you get precisely this Bessel function that Nadab showed to us earlier. So let me try to extend these ideas for the case what I said I was going to compute is the connected correlator of two circular Wilson loops. So the connected correlator is the correlator minus the product of the individual expectation values and I'm going to be considering two concentric circular Wilson loops like they here separated by a distance h in some transverse direction and you can even consider them to be of different radius they don't have to be the same but another thing that I'm going to introduce which is going to be crucial for our computation is that as I said the Wilson loops is is coupling with the scalar fields through this unit vector this R6 unit vector I'm going to take the coupling in the two loops to be in a different direction so let's say that we we we call that the first one is coupling in this direction the second one is coupling in a different point in the internal phase sphere forming an angle gamma with the the first one in the north pole. So and this gamma parameterized this difference in internal space orientation and it is that what is going to allow us to play all this I mean this this parametric parametric limit. So okay this this Dyson equations for the connected correlators in the case of a gamma zero has already been considered in the early days of ADSEFT by Sarambo but as I said with now we are going to turn on this internal space separation this is going to give you two things to to benefit from this it is possible to find a critical value of gamma such that this this correlator becomes supersymmetric and can be exactly computed for that particular critical case and we can also by taking this cos gamma very very large you can implement this this ladder limit in which you capture the full expectation value we just consider other diagrams. Okay but now still we are only considering ladder diagrams so we but now we have two different sort of effective propagators because say now we have these two loops we can have a propagator that connect two points on the same circle either the first one or the second but since they are circles these effective propagators which I called rainbows here and depicted with blue are gonna be just constant as before right those are gonna be the easy ones but then you also have the possibility of having a propagator that connect one point in the lower circle with one another point in the upper circle and these are the ones I'm gonna be calling ladders and of course for them in each now you have this non-trivial function of the difference between t and t prime and now we would like to compute this this the connected correlator of these two Wilson loops the quantity was closely related to that and this is the analog of what I defined w in the previous slide is this K of t with loosely you could call it a green function which is expectation value of two traces of two of any and two of any and phases one is the the complete contour so one is complete and that one is just an arc but then you take the trace and you close it then the connected correlator you want is going to be when you take this t parameter equal to 2 pi such that the two of are closed and this is going to be the connected correlator but then as you go you need that you will see that you also need to define this other green function which is looking similar but but the difference the main difference is that in this case the the two non-havillian phase factors are within the same single trace right and moreover then in this case no one of them is is is complete so it has an extension t this has an extension s and 5 is like a relative separation between the starting point of this one and the starting point of the other one okay let's see if we can derive some disson equation for these guys and eventually computed the the ladder contribution to the connected correlator to do that I'm not going to do in detail it just appeared to the schematic way of thinking so now I was graphically representing this green function k by a yellow blob and this other gamma green function by a purple blob so so this is what we would like to compute so at least since this is connected that there's going to be at least one point that is connected with some propagator going upstairs so there is going to be certainly at least one point in this segment zero and t with the propagator so let's call t prime the right most point in this segment that is contracted with one of those effective propagators so this is the prime so now that you have two possibilities this is contracted with a rainbow propagator that connects with another point in the same segment or a ladder propagator with that connects t prime with a with a point in the in the upper circle so when it is contracted with the t prime within the same segment you have also two different in principle ways of doing it because you can contract it like this or in this other way again since t prime is the right most you cannot have you have no propagators ending on this between t prime and t but you can have propagators in this segment and this other segment you cannot cross this other line the blue the other line otherwise it could be non-planner so for instance here the propagator that ends in points between t second and t prime they could only be this blue bubble the blue blue blob that keeps you a w u of t minus t prime and then what you have here it's just the same structure as here but now instead of being between zero and t it's just between zero and t prime and then you have like the traces for in each of them so this is like this the integral with of w of t minus t prime with with k of t second and of course you have to integrate for all possible values of t prime and t second such that t second is to the left of t prime this one looks a little bit different but if you change coordinate you can show that these two guys are exactly the same and that's why you have a factor of two here and this is if you want the first term is a the contribution of these two possibilities the third the second term is just the possibility of this right most point in the in the interval zero t connected with the ladder propagator we with a point bar five in the in the interval zero to pi but now if you see now you follow the lines you realize then you you can have all possible ways of throwing propagators here and there but now if you follow the line you see that they are in a single trace and that's why this is now a green function gamma this auxiliary gamma function and then whenever in the case is when we connected with the blue line we just get a factor of g here I don't have the definition of g g is this lambda missed it so I'm calling g this this combination is just this combination so whenever you have a blue a blue rainbow you have g and whenever you have a green ladder you have this effective propagator g large g so but then you see this is like a not it's not a closed integral equation because it relates k with an integral of k but then you have this integral that involves gamma so you need like to derive like an integral equation for gamma and you can do it in the same way so you start with the object you would like to compute and then think of all possible ways of contracting propagators within this single trace and it could be well maybe there is no propagator connecting the lower with the upper interval or arc and in that case if there is no propagators going from downstairs to upstairs the only thing you could have is a blob a blue blob downstairs and a blue blob upstairs so this is just this term w of t w of s but then if if if there are going to be ladders let's call t prime the right most so you're going to get a t prime that connects a ladder diagram that connects t prime with some five plus s prime point in the after in the upper segment but since this is the right most to the right you can have propagators but no one could be ladder so here you have a blue blob and you are gonna get another blue blob there and then what you get here is essentially the same structure but just we restricted restricted intervals and then you have to of course move t prime over all possible values between zero and t and s prime between all possible values between zero and s so this is the first time and here you get this these are the blue blobs w of t minus t prime w of s minus s prime you got the effective a propagator from this green line and then you got essentially the same guy but evaluated in t prime and s prime and this is a close integral equation because now these are known those are those special functions the only unknown is just gamma so if you manage to solve this that's the first step of your procedure you have to solve for gamma once you solve for gamma you could go to the integral equation of of k and then try to solve for k and finally evaluate it in 2 pi and you're gonna get the ladder contribution to the connected corrector of two loops as you can imagine doing this in the general case could be still very tricky and very difficult so let me discuss so not the general case but some specific cases the first specific case is this critical value I was mentioned so there is if you define this cost beta cost beta in terms of this geometrical parameters of the of the loops you see that effective propagators is written in this way and it is obvious that if you take cos gamma equal to minus cos of beta this become just constant and it's the same constant as before up to a sign so this is again a major simplification for your dyson equation for gamma for instance now you do the same you Laplace transform the integral equation for gamma you do Laplace transform for both variables s and t and then you get an algebraic equation for the Laplace transform the double Laplace transform of this auxiliary green function gamma that is those are the Laplace transforms of the vessel functions so those are known so you just have to anti transform this looks complicated but at the end of the day when you do this Laplace transformation anti transformation you get that this auxiliary gamma in this critical case is just the same vessel function but evaluated in t minus s so then as I said you have to go to the integral equation for k solvage and finally evaluated in 2 pi you do it and then you get some this quadratic expression in vessel functions right so this is the exact resumption or ladder of ladder diagrams in this critical case still you may be worried that this is not taking into account interaction diagrams but you can claim that interaction diagrams also for the connected corrector cancel and just to justify this you can show that this precise critical value of this internal space separation is the one that makes the corrector supersymmetric what I mean so when you have two Wilson loops silver Wilson loops with different orientations they are half VPS each one but they preserve different set of supersymmetries in general but if you take this critical angle that relates the internal space separation with the spatial separation space time separation it becomes supersymmetric so you can use the common supersymmetry to localize and this is what I'm already pointed out by pestum and it was working in detail in this other work by John B pestum and Richie for the connected correlator and essentially you localize to the same matrix model that we Diego Tranganelli was telling us last week in the school which is simply a Gaussian matrix model and then if you would like to compute the connected correlator you now you have to compute the expectation value of this matrix model of two insertions of two traces of this exponential of this this matrix of the matrix model and if you do that you get of course the exact expression that is quadratic on the vessel functions okay and of course this is indicating that interaction diagrams cancer so but as I said maybe the most interesting thing was to go into this ladder limit what is the ladder limit well it is it is the generalization of an idea that we implemented for the casp Wilson line but it's essentially the same so you could define a limit in which cost is cost of gamma goes to infinity and the coupling goes to zero such that lambda hard which is the product of the two is fixed so for instance consider the two loop contributions you have three possible kind of contribution you can you could have two ladders a ladder and a rainbow or some interaction so this one comes with two cost gammas so it's lambda hard square but this one cast comes with lambda square but only one cost of gamma so it's lambda hard over cost gamma so the same with the interaction there so if cost gamma is a very large number these two diagrams are suppressed in comparison with this one right so the conclusion is that if you're in this ladder limit you can dismiss not only diagrams with vertices but also any diagram and contain it some rainbow so it's not only that you consider propagators but you consider only propagators that connect the two loops so it means in our internal equation that whenever we we route W with that should be replaced by one because now rainbows are also sub leading so then the internal equation for instance for the auxiliary function gamma is very simple it's simpler so you don't have gamma here not in the interior not in the inside the integrals then you can just get translate the internal equation into a differential equation but taking partial derivative with respect of s and t and because now you have that this g function depends only on the difference you introduce this change of coordinates you make this ansatz and for this function gamma of x you get like a children equation where the potential is minus this g function that was given by the effective potential the effective propagator so of course solving this in the general case could be difficult this is a periodic potential but since we are interested in making a precision test we would like to study this not exactly as a function of lambda but it could be nice but we are at least interested in the strong coupling limit of this and when you take the strong coupling limit of this since you have a gamma in this in this g effective propagator is like a having a very deep well so this is just dominated by the semi classic by the classical approximation in which the particle is going to be localized at the minimum of the potential and and and the again but it's going to be just the the height of the potential so you have this gamma is just this exponential of the the height of the potential and then you have like a a periodic delta function and then when you have this approximate solution for this auxiliary gamma you go to the question for k solve it evaluate it in 2 pi and then you get that the connected correlator or at least the ladder the ladder contribution the strong coupling limit is this exponential of a square root of gamma and something you would have expected the square root of cos gamma and this is a square root of this combination of the geometrical parameters so I don't have time to go through the details but you of course you can try to analyze the same thing by doing the semi classical string theory computation now you have to compute it like a worksheet that connects to circles that are separated along this distance x but seems you do also have this this separation internal space you have to include this motion in this internal angle okay you make an answer you impose proper boundary conditions and then you realize that there are two constant of motion that essentially relate to the two geometrical parameters h and gamma and everything is related through some elliptic functions and I don't have time to discuss this but let me just go directly to the to the ladder limit you realize that now if you take one of these constant of motion is going to be very large then you this the relations simplified and then you got this cos gamma is 81 minus s s has to be taken this interval if you now look at the regularize action that was also an elliptic function becomes simply this so now if you use this relation to write the square root of t you get this and then if you look at what is the relation between s and the geometrical parameters in this large t limit you get this relation and then when you plug this inside this expression you get exactly exactly the same result that we we got for the strong coupling limit of the ladder resumption so let me just then go to the summary of what I said so the main the main thing is that the best moment for me there was a thing that I learned that the resumption of ladder diagrams can be always studied through some Dyson equations some internal Dyson equations which are in general difficult to solve but in our case there was this critical case in which the resumption of ladders could be computed can be computed exactly this case was you can show the it is super symmetric super symmetric and then moreover you see that the ladders resummations of course coincides with the with the it's the full answer and it coincides with the very study obtained with the matrix model and then maybe the most interesting thing is that it's the existence of this parametric limit in which although ladder the answer in this regime is not the the full answer is the leading order answer and then you you can go to a strong coupling limit and make an explicit successful explicit comparison with with the minimal area of the dual string so this is a precision test and let me emphasize that it is a precision testing in a supersymmetric case I made it for the circular Wilson loop but it's quite likely that these ideas can be extended for any arbitrary Wilson loop so that's in the sense is a very general kind of of precision test but okay that was all what I wanted to say thank you for your attention