 Thanks a lot to all of you for participating in this workshop. It has been amazing the whole week. I learned a lot of new things. So as the last speaker of the conference, I have the pleasure to summarize all the talks. No, joke me. So I'm going to tell you about a work which is in this papers with amazing collaborators. So the people in green are somewhere in the room or used to be in the room at some point in the workshop. And then there will be a little bit mentioning of some kind of work in progress. So the motivation slide is not really needed in this workshop. The basic idea is that as you've heard in many talks, we have many, many calculations of kind of exact correlation functions observables in QFTs with different amount of supersymmetries in various dimensions. And one thing that you can hope is that in some appropriate large end limit, these calculations can make contact with type 2a or type 2b supergravity and eventually, hopefully, with corrections to the supergravity limit. So eventually, contact with some kind of quantum gravitational theory. So the goal of the talk is much more modest. I will kind of explain on a couple of examples how all of this works to leading order in n and lambda, so in the supergravity approximations. And in the talk, I will not have conformal symmetry. So the kind of examples I'm going to study will be the dimensional QFTs on a round sphere, so no squashings, no water, topology, none of that. And the kind of examples will be the formations of the maximally supersymmetric 4D theory with different amounts of supersymmetry or the maximally supersymmetric Young-Mills theory in dimensions between 2 and 7. So those will be the concrete examples I'm going to study. Good. And I want to emphasize that in kind of example 1 and 3, we do have a lot of calculations with supersymmetric localization. But example 1 concerns an n equals 1 QFT, so massive theory on a force sphere for which as far as I know, there's no published at least results for QFT observables, exact observables. Okay. Okay. So what's this kind of examples I'm going to talk about? So this is an overview of kind of n equals 4 Young-Mills. So the content is a gauge field, some scalars in the adjoint and fermions in the adjoint. And I can decompose all of this into multiplets of n equals 1 supersymmetry, which will be a vector and a bunch of chirals, all of these in the adjoint. And so this thing has an SO6R symmetry in general at the conformal point. But of course in this formulation, I only have an SO3 times U1 manifested. And so the deformation I would like is to turn on superpotential masses for the chiral superfuel. It's kind of an obvious deformation that you can study and people did a lot in the past. So the new kind of ingredient is that I will do this on S4. So this is how we connect to the topic of the workshop. And so if you want to do this, you are kind of immediately faced with an obstacle that I have a massive theory on S4. So I don't have conformal invariance anymore. And it's not obvious how I'm going to couple it on S4 preserving supersymmetry. Well when there is a will, there is a way. And that's the way basically. So what I've written over here is the Lagrangian schematically, a thin equals 1 star on S4 with all of these masses turned on. So the couplings in red are superpotential masses. And the couplings in blue are due to the fact that I'm on the sphere. So if I take Mi to 0 and R very large, I'm back to the flat space conformal theory. So localization seems to work in the case when M3 is 0 and M1 is equal to M2. This is known as N equals 2 star theory. And this is actually kind of an example in the Pasteur's original paper on localization for gauge theories on S4. Okay, so I'm going to focus for the next five slides or so on this kind of example. Good, so this is a slide which most of you would know about. This is essentially borrowed from Pasteur's paper. So the localization answer is that the partition function on S4, for this example N equals 2 star with an SUN gauge group, the path integral localizes to a matrix model of a kind where I have to integrate over the kind of eigenvalues of this adjoint scalar AI. And this is the measure in black. Then I have a one loop piece and an instanton piece and a classical piece. So I've spelled out the classical contribution and the one loop one. The instantons are harder. So this is known as a necrosis partition function with some specific values of the epsilon parameters. And so for me, this will not be an important player because what I'm interested in is a large n limit and the common lore is that a large n instantons don't contribute, but this is not entirely obvious. So these people studied exactly this matrix model in the large n limit as a function of the coupling lambda. So I still have a parameter and interesting things can happen as a function of this parameter. And so one of the arguments in their paper is that at large n instantons are not going to be contributing and if you wish them, calculations I'm going to show later on give you a further kind of argument for this, holographically. Yes? Yes. But I don't know how the instanton modulize space volume a priori scales with n, right? So it could be that I have an exponentially smaller factor and then the volume grows with n in an exponential way, at least in principle. So I'm cautious here with this, but of course that's the usual argument is what you said. Okay. So in addition, I'll be interested in this dependence on lambda only when I'm in the large lambda limit because eventually I'd like to simplify calculating supergravity. So as a function of lambda, things are complicated and interesting and I encourage you to open up these papers which are beautiful I think. But I'm interested in this limit where I do have an analytic answer for the path integral or the log of the path integral which I'll call the free energy on the force here. That's the answer, but since all of you have had a QFT course, you should be bothered. I have a gamma here which is kind of an artifact from the fact that I've calculated all of this with some scheme, regularization scheme. I think this one is dimwreck, but it doesn't matter. I have dependence on the scheme in this final answer. And so I should be aiming to understand what is the scheme independent part of the free energy and the answer can be formulated over here. So if I differentiate this F three times with respect to the dimensionless mass, so the natural parameters dimensionless M times the radius of the sphere, then this answer in the box should be independent of the scheme. That's what I'm saying basically. And then in addition to the path integral, I can kind of insert a BPS Wilson loop which we heard about on Monday from various speakers. And this one is half BPS. It's compatible with a supercharge used for localization and it has an expectation value given over here. So these two things in the box, I should be able to calculate at large n and large lambda, which means in the supergravity limit. And that will be my goal for the next 10 minutes or so. Any questions or complaints? Good. So how do we do this? Well, it's a long and arduous road, but we know how to do this, that upshot. So what I'm gonna do is I'll construct a 5D solution of 5D supergravity, maximal SO6 gauged supergravity, which will be the bulk dual of this deformation of n equals four angles. So the reason I wanna be in 5D is that it's kind of easier than in type 2B. And one of the, how to say, justifications for this is that type 2B on S5, if you only keep the lowest KK modes, is consistent. So this 5D theory is a consistent truncation of the lowest KK modes of type 2B on S5, which was proven in 2014, but it was expected already in the mid-80s. And in addition, if I want to ask exactly this holographic question, but on R4, it was already answered by Pilham Warner in 2000, I think. And so both of these are motivations for me to work in five dimensions, instead of in ten dimensions, which is harder. Okay, so what I have to do is I have to carefully analyze the symmetries of this deformation of n equals four. And I have a u1 symmetry of this deformation in the Lagrangian, classically. In addition, there is an additional subtle u1y, which is kind of emerging at large n. It's a subgroup of the SL2R or SL2z duality group of n equals four. So it has been argued in 98 that it's an emergent large n symmetry of n equals four. And it's also present in this model. Okay, so if I impose all of this on 5D n equals 8 supergravity, what comes out is that I have to keep the metric and some scalar fields, three of them. Two of the scalars are due to two specific operators. So bosonic, bilinear, and a fermionic one. And these are the usual superpotential masses, which I'm adding on flat space. In addition, though, I have a third scalar, chi, which is due to another bosonic bilinear operator, which I only have if I'm on S4. It's an additional coupling which I have to add on S4. And so if I kill this last one, I'm back to flat space and I should recover the well-known 5D model of Pilkin Warner. Okay, so this is the basic setup. And so this is the model with the metric and these scalar fields in 5D. It comes out of the 5D n equals 8 supergravity after a long calculation. But it doesn't matter, that's the final model. And I parameterize them scalars as a real scalar and two complex scalars. But I'm reminding you that I'm on S4. And so these are really two kind of independent scalars. And that's why I call them tilde here instead of bar to remind you of that. Okay, so in this model, I'm after backgrounds which have an isometry of S4. Because I'm on S4, which is round. So in the metric, I can only allow for a function A of r. This is the radial of anti-sitter space. And then I have a bunch of scalars, which could also have profiles as a function of r. Okay, good. So what do I have to do? I have to put this ansatz into the BPS equations of the 5D supergravity of this model, and then I have to work a bit. And I think all of this is kind of standard and the only new or different ingredient as compared to flat space is that I have to use a different spinner on S4. It's not a constant spinner. It's a conformal spinning spinner of sort, this one here. And then with this at hand, I can derive BPS equations and they look like this. So the equations in red here are differential equations for the scalars and they form a closed system of first order ODE's. And if I have a solution of these, I can just put it on the right hand side over here and find the metric function. So this last one is, in a sense, automatically solved. It's telling me how the metric evolves as a function of r. So unfortunately, we couldn't analytically solve these equations. We are not very strong, it seems. But we can do numerics and we can also do kind of an expansion in the UV and in the IR. So we did all of this, I'm gonna spare you the details. The upshot is that in the UV, at large values of r, the BPS equations have two kind of integration constants, which is absolutely kind of expected in calligraphy. I have a mass or a source for all of these operators, which is one parameter mu, and then I have a veth. And so the veth is determining the state, so it's a free parameter. But in the IR, I'm on S4, so I should have determined in a sense the state. And so we kind of expect that this mu is due to MR, the dimensionless combination. And this v should somehow be determined by the condition in the IR on the geometry. So what we did is we kind of imposed that as you go into the IR, this S4 smoothly caps off into topologically r5. So it's metrically also r5. It's a smooth capping off of the 5E space. And if you do this, what happens is that this veth is determined by the mass or the mu in this way. So we have an analytic expression for how the veth is determined in terms of the source and comes out from imposing this regularity condition. In addition, we can go through the normalizations of various fields and so on and map to kind of operators on the boundary. And you find that mu is i times MR. So indeed, mu is connected to the source in the field theory. Okay, so this is essentially kind of input from the VPS equations that I need to calculate things holographically. So what I'm interested in is this function F, which is the free energy. And we know since 20 years ago how to do this. We just have to evaluate on shell action of this 5D solution. And this should be the path integral or the free energy of the dual QFT. And unfortunately, or fortunately, I don't know how to say this. But this works, of course, up to the vergences. Because at large R, I have divergences which I have to cancel, which are the usual divergences I have to cancel in a QFT. We know how to do this systematically due to costas and many other people. And the only issue here, or subtlety, I should say, is that in addition to this, in addition to canceling divergences, you can still have finite counter terms with coefficients which are free. So here I have a supersymmetric theories. So I should insist that I'm adding all of the finite counter terms which are necessary for supersymmetry. And if you do this, you should be able to compute unambiguously this quantity. And indeed, it is possible to do this analysis. If you're not willing to discuss these counter terms, meaning if you allow for an arbitrary coefficient, of course, I cannot compute this unambiguously. I have to go to more derivatives and get rid of the counter terms, which I don't know. This is just a technical point. So all of this is unfortunately also long and arduous. So I'm going to spare you the details, but the upshot is kind of simple. So I have this 5D model, the classical Lagrangian which I showed you, to which I have to add the counter terms, the usual one, the divergent ones that cost us instructed us how to compute. And then I have to add also a specific finite counter term, which we can discuss after the talk if you want. And then this action I should evaluate on shell, on the background which I calculated before. And if you do this, what happens is that the third derivative of this on shell action is actually simply the second derivative of the function V, of this VEF function as a function of mu. And so it turns out that all of this mass goes down to a simple answer, which is over here. And of course, if you put now mu is IMR, you can see that this answer here compares favorably with supergravity. So I get a match. So all of this amounts to just, holography just works in this example, which is good. Okay, any questions on this? Yes? Right, so physical meaning is a bit hard. So people have understood in some cases what's the physical meaning on R4 of a path integral on S4 as a function of marginal couplings. But over here, I have function of relevant couplings. So it's an integrated correlation function. But I don't have a clean answer of a physical meaning. What I know is that it's scheme independent and it's a good observable. What that observable is on R4? Yeah, I'll pass out on that, I don't know. Okay, so we have matched the path integral, the free energy with localization, but we can do a bit better. We can also calculate the Wilson line expectation value. Unfortunately, to do this requires two amazing collaborators. Fortunately, requires two amazing collaborators. Unfortunately for them, it's a lot of work. So what we have to do is to evaluate a non-shell action of a probe fundamental string in a type 2B background, which is an uplift of this 5D background, which I showed you. So the first exercise, which is long and painful, is to construct a type 2B solution, which is an uplift of this 5D solution. So over here, I don't want to scare you, but that's the formulas we had to deal with. This is the type 2B metric. So the first line is the 5D metric I showed you. Then when you uplift, you can get a warp factor and then you have a topological S5 over here written in God-awful coordinates with some squashings. All of these functions are determined in terms of the scalars in five dimensions. So this Z and Z tilde and eta are the 5D scalars. That's only the metric. In addition, we have all of them fluxes in 2B turned on. Axiom, dilatone, two-formed, five-formed. All of them are on and we have all of them. They fit on a couple of pages. So if I kill one of the scalars, so if I take Z tilde is equal to minus Z, this is the model that Pugh and Warner studied 18 years ago, and of course all of this collapses to a much nicer formula which they had in their paper. So I have a background. I have to do now the calculation of a probe fundamental string. That one is not so hard because the supersymmetric string wants to sit at a particular locus on S5 and it's wrapping the kind of equator on S4. And so the configuration of the string is kind of simple. So most of this mess collapses to something nice. And so what I have to do, which we did in the paper in detail, is to use the number to action calculated on shell on this profile of the string, regularize it again with a simple counter term, and then this must be the blog of the Wilson-Luth expectation value. And indeed it happens. So when you do all of this, you obtain this on shell action, regularized. And here is, so we have this numerically, right? So all of this is numerical. And so the numerics fit with this function in this way. So I don't even know which one is which. So one of them is the function, the other one is our numerics, one top of each other. So for all practical purposes, they're the same function. And so of course this function matches localization. And I want to emphasize that over here I'm assuming that mu is real. I don't have to assume that. We have arbitrary complex mu. These are complex saddle points in general. And so we have checked all over the complex plane that the two functions match. But this one is only real section, if you wish. Okay, any questions? Yes? Yes, very good. So what happens is that because I'm insisting on this special one-half BPS, Wilson line, it has to be on the big circle of S4. And in the bulk it just wraps all of the space. It goes down all the way to the pinching point, to the regular point. But indeed, generally, if I allow other profiles, it will be a mess. And I have to calculate it again. This one is obviously special. Yes? Well, I'm not sure if... Okay, the usual answer I can give is that I'm at large N. So I mean some kind of thermodynamic limit. So things could not be analytic, right? So the observables don't have to depend analytically on the parameters. But I'm not sure if I can give any more concrete answer. In the bulk? I don't know of any. We've tried that. So it will require identifying how these BPS equations know about mu. But as I've told you, the BPS equations in the UV depend on both mu and v. And it's only if I integrate them out to the IR that I see a relation between v and mu. And I don't know how to avoid this kind of... So it's hidden if you wish in this kind of integration into the IR, which I have to do from a supergravity perspective. Good. Any other questions? Yes? So you're... You know what to ask so that those are the subtle questions. So all of these subtle points in the bulk, I think that's my opinion, should be thought of as complex settles. And they should not be analytically continued in any way to Lorentzian signature. And so in this case, if you change mu to not be real or something like this, your fluxes could pick up ice. So the B2 and the C2 flux become imaginary. Because some of the 5D scalars become imaginary or complex in general. They're not, but each of them is complex. Nothing is real. It's only the metric is real by some magic and only the 5D metric is real. So the uplifted one is not. So it's a fully complex set of points in the bulk. Okay? So you should think of this geometry as a tool to evaluate on-shell actions to reproduce. I don't think it will be good for any gravitational analysis of scattering or gravitational waves or anything like this. This will fail, of course, because it's bad. Right. So yes, if I kind of orbifold N equals to 4 and make it into a necklace quiver, of course all of this goes through. I pick factors of K from the ZK orbifolds and it just works in a boring way, I'd say. So K, right. And I don't see that. So the differences of the couplings are B field modes on the topological cycles which I get from the orbifolds. All of these are kind of outside the supergravity limit, stringy modes. And so I don't see them here. N equals to 1 star. I don't have a lot. I mean, I have a lot, but I don't have a lot of time. So I'll just constrain myself to a few comments. So the first thing that you should ask is that it's not obvious that a partition function on S4 is a good observable. It might just as well be completely scheme-dependent. I don't have localization calculations which give me guidance here. I'll analyze it generally whether it's a good observable. So the first thing that people ask is whether the partition function on S4 is a good observable as a function of marginal couplings. So tau and tau bar are labels of the marginal couplings, of the gauge couplings if you wish. And it was pointed out in this paper that there's a complete freedom to add to the path integral an arbitrary counterterm with an arbitrary coefficient function of these couplings. So as a function of the margin deformations, the free energy is completely scheme-dependent. So it's a useless observable if you wish. However, what we ask is what happens if we add relevant couplings, which was the new piece in the analysis which we did. The calculations are more or less verbatim repetition of the calculations of these orders in some super space in all minimal rigid supergravity. And what you find is that if you have a mass, let's say, as a relevant deformation, there's only one counterterm that depends on the mass in this one. And so what you learn is that if you're interested in the free energy as a function of the masses, it's actually a good observable if you differentiate enough times to kill this counterterm. So this analysis is completely general. It's independent of the bulk dual. It's a QFT statement I'm making here. Now we applied all of this to N equals 4. And N equals 4, I have additional symmetries which restrict the dependence on the masses to be either in this quadratic way or in this cubic way. So whatever the function F is for the mass deformation of N equals 4, it should be a function of exactly these combinations. So for example, if you have an answer for F of S4 and if you want to expand it to a quarter-quarter in the mass, it should look like this. It must look like this according to this analysis above. And so the constants A and B, so I have only two quartic terms allowed by the symmetries, and there's a pre-factor which is essentially fixed by dimension and scaling with N. And so the numerical constants A and B are unambiguous observables in the QFT. I don't know what they mean in flat space, but they're good observables, I claim. So I should be able to compute them. And again, I'm emphasizing that I cannot compute these in any other way, at least as far as I know. So calligraphy is, if you wish, the only way currently that I know how to compute these two constants. And we did that, and this was even more work. So to capture all of these masses, M1, M2, M3, and allow for a Gejinoveth and all of that, I have to keep 18 scalars in 5D N equals 8 supergravity. And these 18 scalars kind of organize themselves into two vector multiplets and four hypers. And so this model is a consistent truncation of type 2B supergravity on S5. And so if you want, I can give you the model and go analyze it, but we are not that brave because it's a lot of scalars. So we focus on two limits that are somewhat special. The first one we've heard from costas already in the context of flat space, and this is when all of the masses are the same. This is also known as, in flat space, this is the GPPC flow holographically. Here, though, I need two more scalars, in addition to the ones we've heard from costas. The other one is the flow in which two of the masses vanish, and only one is known as zero. This, in flat space, goes to a fixed point in the IR, strongly coupled fixed point. It was found by Alice Trussler, and the bulk deal of that is this famous FGPW solution. And again, on S4, I need an extra scalar. So we've analyzed these two models in gory detail, numerically. And we've computed these two constants, and we've computed other things, but I don't have the time to talk about it. So if you wish, we have numerical estimates for the two constants, and they obey a nice sum rule, if you wish, which is imposed essentially by this minus equals to two supersymmetry. So we can do calculations in this model on S4 holographically. That's a bunch line. Okay, any questions? Right, so it's understood because it's essentially this over two. Right, so n equals to two supersymmetry gives you additional constraints between these coefficients, yes. So it's a special case. Good, so in the next half hour, joking, I will talk about maximum Young Mills on a sphere, supersymmetric Young Mills, and an alternative point of view of all of this is that you can think of this as some kind of a supergravity solutions describing the near horizon limit of a d-brain with a world volume which is a sphere. Because we know that maximum Young Mills lives on the world volume of a flat d-brain, so this is the spherical version of that. In the supergravity approximation, I want to emphasize. So I don't know microscopically how to do spherical d-brains, but I know in the supergravity limit how to do that. Good, so this is a brief overview of maximum Young Mills. So the Lagrangian is over here. I have a parameter which I call p, which is the dimension of space and time, and I keep p general. So this thing has 16 supercharges. There's a bunch of scalars in the adjoint and a bunch of fermions in the adjoint and a gauge field. This Lagrangian can be obtained as a dimensional reduction of 10 d Young Mills on a torus or on some Lorentzian space, and in our case, we insist that we're in Euclidean signature because the asymmetry is non-compact, this one here. And so only for p equals to 3, which is d3 brains, if you wish, I have conformal invariance, and this is n equals to 4 Young Mills, and I'm going to skip it as an example. I'm interested in non-conformal examples for this part of the talk. So I want to put this on a sphere, and as we know, I have to add extra couplings, and these were worked out by Blau in 2002, I think, or 2004, I'm not sure, and then it was recently, this model was revisited by Minahan and Maxime here in the context of localization. So these two couplings that I have to add, built out of the bosons and the fermions, break this SO1 8 minus p into a subgroup, and so if I want to construct a bulk dual of this, I have to mimic all of this, breaking and all of these operators in the gravity dual. And this construction works for p up to 6, so up to 7-dimensional space and time, because essentially, more or less speaking, the way supersymmetry works on the sphere is by some algebra, which is the same as the super conformal algebra in one dimension less, so I have an upper bound on the dimension in which I have super conformal algebras, which is responsible for this. Okay, and physically, what we're doing is we're taking Young Mills in flat space and we're by hand introducing a cut-off, which is the sphere, the length scale of the sphere, however, we're doing it in a very special way, compatible with all of the supercharges. So this is probably the unique IR cut-off compatible with all supersymmetries. Good, so I want to construct the bulk dual of this and so if you are naive, which of course none of you are, but if you were, you would begin in the following way. I know that I want a bulk dual of a theory on sp plus 1 with this much global symmetry, okay? And I should be in type 2a supergravity because I'm dealing with d-branes of some sort. And so this is an ansatz for a metric which kind of builds in all of these isometries. So here's the sphere I'm putting a theory on. Here's the radial of ADS, if you wish, and there's the sitter space here, which mimics this, and there's another sphere which mimics this. And if I do this, I see that I have another theta angle left over in addition to this R. So all of these functions which I can add in the metric are functions of two variables. And in addition, of course, I have to look at the fluxes and make an ansatz compatible with all of these isometries and this, of course, will lead you to PDEs. And in addition, you should impose that if you're in the UV, the radius of the sphere should not matter when you're back to the usual flat d-brain solutions. So if you want to do this, you will stumble upon PDEs necessarily, I claim. And, of course, that's hard. Okay, so there's a better way. And again, I want to emphasize Peter and Friedrich. So this was heroic efforts here. So what we did, actually, is to use the fact that in these maximally supersymmetric cases, we often, in fact, always have a consistent truncation to some lower-dimensional supergravity theory. In this case, it's a P plus two-dimensional gauge supergravity. And then we can work in there with a bunch of scalars and eventually uplift back to 10-dimension. So that will be the strategy here. So I just want to emphasize that we have, in the deformation of the Lagrangian, we have these couplings morally. We have these two kind of operators, which I had to add on the sphere. This one, of course, is there in flat space, but the coefficient is dimension-full. So it runs. So in the bulk, if you wish, I have to add a dilaton, capturing the running of the gauge coupling. So I'm looking for a model, which has scalars corresponding to each of these operators in the bulk, and which breaks SL18-P to this subgroup. So that's my goal. And so the question is, thank you, do I have supergravity theories which have this content? Can I do this? That's the question. That's the goal. And you can do it. It's kind of nice. So after a lot of work and digging through Baroque papers from 85, altered by the person who, this institute is named after, you can distill this model. So the model comes out of a consistent truncation of a maximum supergravity theory in P plus two dimensions for P, 6, 5, 4, 2, and 1. And it looks like this. It has two scalars, tau and tau tilde, and it has another one, lambda. And it has a potential which is determined in terms of a superpotential in the usual way, and that's the superpotential. It has a slight difference between these values of P, but basically that's the model that you have to play. Super simple. And I want to emphasize that when P is equal to 6, I actually don't have this scalar lambda at all in the Lagrange, and I have only two scalars. Okay, and the model when P is equal to 1, so d1, variance, is new. We just couldn't find it in the literature, but we postulated it and it works. So it must be true. Good. In the flat space limit, these two scalars are gone and I have the usual dilaton and metric, the one that you studied with Townsend. Yeah, it's exactly this action. So if you wish, I'm adding two more scalars to your story. Okay? Good. So with this model, I have an ansatz for all of the fused. Again, it's fairly symmetric ansatz, so I have a metric function A and all of the scalars are functions of the radium variable, and I get some DPS equations of this form, which I have to solve. Okay? I'm out of time, basically, but let me kind of explain what we did. So for all of these values of P, we can have a solution. For P1, 2, 4, and 6, it's analytic, and for P equals to 5 is numerical. We couldn't integrate the DPS equations analytically, but for all of the cases, the picture is this one. So in the IR, I have a smooth cupping off of the geometry because I'm on a sphere. This is if you wish, the manifestation of the cut-off in the field theory. In the UV, I'm... I don't have anti-deceptive space, of course. I'm not conformal, so in the UV, you kind of approach these backgrounds which describe D-brains in Type II string tip. Okay? And so I have a flow which kind of interpolates between these two. And all of these backgrounds which we have in the paper depend on one constant only. In the Lagrangian, I have only one dimension less constant, which is this combination of the Yang-Mills coupling, which is the dimension full, and the radius of the sphere. So in the bulk, I should see only one integration constant in my models, and indeed I do this lambda IR. Okay? It's the value of the scalar lambda at this point here, if you wish. That's how we parameterize it. Okay? So after this, this is showing the lower D supergravity. I have to uplift it. And again, there was some heroic effort because now I have to go back and dig out from the supergravity literature uplift formally, and you have to go back some years and so on, but that's the answer, basically. So I'm back to my naive expectation that I started with, except that now I have an explicit function for these p and q and lambda and so on. So we know these backgrounds kind of explicitly, with all of the fluxes. Okay? And it's a full one-half BPS background in 10D supergravity. And there's a curiosity with the six brains, but I don't have the time to talk about it. So if you have a background like this, you can aim at doing photography with it. But the background is not asymptotically ADS, so I cannot just open the books and read about it. I have to be a bit more inventive. But I have to do it in a sense because I know that there's localization results for this model on a sphere, which were worked out by Joe Minacan, Manton, Nedelin, and Maxim in various papers. And the point is that the partition function is a function only of this lambda, which is now dimension full, this coupling lambda, because I have an energy scale, right? I don't have a conformal theory. And you can aim at calculating it. And unfortunately now I have to understand how the energy scale in the QFT is related to the radial coordinate in the bulk, which is a subtle issue, and there could be constant coefficients, which I don't know about. And I have to learn how to regularize this on-shell action without having asymptotically ADS spaces and so on. So qualitatively the answer is over here. So we managed to avoid all of this issue and find the scaling of the free energy holographically with n and this lambda. It's a function of p, but we were not able, when we published the paper, to compute in the coefficient. Currently we're working with Anton and Joe on computing these coefficients, both in holography and in localization, and holographically what's useful is this paper cost us in America on how to do regularization if you wish on-shell actions in the context of non-conformal brains, which is what we're doing here. Excellent question. p equals to five is a mystery of the universe, right? Because we have d5 and ns5 brains, and in the UV I have a little string theory. So this divergence we really don't understand in QFT. The putative answer is that it has to do with a little string theory, but I don't know more. So obviously this is not valid for p equals five. Right, so the calculation which we did for non-shell action also breaks at p equals five. So this formula should best be viewed as breaking down for p equals five, and we need a new analysis, which currently we don't know how to do. Yeah, it's the usual linear dilatone stuff of, yeah, little string theory. Okay, so summary. What I've done is to construct solutions of type two supergravity, which are dual to massive deformations of n equals to four on S4, and we demonstrated agreements between localization and holography for the case of n equals to two. For n equals one, we don't have localization formulas, but we can compute on the supergravity side with a bit of work, and then I demonstrated also how to extend all of this for maximally supersymmetric in p plus one dimensions. Okay, so some future work that I think has to be done. So some of it we're doing here. We can extend this calculation of a Wilson line to our other representations of the gauge group, which means that you should be looking not at probe fundamental strings, but D-brain probes. As I mentioned, we're working on understanding this OSHA election for the case of maximum young mills, both in the holography and in localization, and one can do similar constructions for three-dimensional conformal theories and deformations thereof, and this is work I'm doing on Krzysztof. Okay, and then these are more ambitious questions, which I think you should think about, but I'm not gonna keep you any longer, so I'm gonna finish here. Thank you very much.