 OK, welcome to the second day of the workshop. And the first speaker is Costa Scanderis, and he will tell us about holography for an equal one. Super quantum field theory. Thank you very much. Thank you for organizing this nice conference, and also for giving me the opportunity to give this talk. It's the first time I talk in this amphitheater. I do have fond memories from my graduate students in one of the high CTP summer schools, and it's nice to be from this side of the audience. I also like to encourage discussion during the talk. It's very early in the morning, and if we start discussing, everybody is going to wake up. So let's start. So as everybody here knows, the best to distribute holographic dualities between N equals 4 to 3 equals N, 8 years 5 versus 5. Since the very early days of ADS-CFT, it was a significant amount of effort was devoted into obtaining dualities with less symmetry and not conformance symmetry. So what I will do today is I will try to address some of the same issues that were discussed 25 years ago, but not a bit more systematically. So I will address these issues systematically for N equals 1 deformations of N equals 4 super Young Mills, and many of the things I will discuss have generalization in other dimensions or with theories with less symmetry, but I will try to stay with the cleanest possible setup, so see how far we can go. And then towards the end, I will analyze the simplest case of this class of theories, which is the N equals 1 star super Young Mills theory. So before I move on to the discussion, what the references are, the first part would be based on a paper with my student Stanislaw Smith that should appear, hopefully, this month. And the last part, which is about the N equals 1 star and the uplifted 10 dimensions, will be based on the paper that appeared last month plus ongoing work. And I should also say that it was worth it to appear simultaneously by Nikolai and his collaborators, Nikolai sitting there. And I encourage you also to read both papers and discuss the both of us. I should also say that many of these things is based on work done many, many years ago. And I would have needed many slides to put all the references right here. I just list some names. So many of the things we will discuss is based on things already done 20 years ago. But there is something that could not have been done 20 years ago. And this is the last part of the talk because at the time, while people were expecting that maximally supergravity theory of five dimensions is a consistent irrigation of type 2B, it was not known whether this is actually true. And also, the actual formulas was not known. This problem was finally solved a few years ago. And we're going to use the formulas that they obtain in this paper. So I also like to encourage people to kind of look at this paper and use this because a lot of interesting physics can be extracted using these uplifts as I will argue in this talk. OK, so the outline of this talk is I will kind of give a longer introduction first. Then I will discuss how to reverse the logic. So usually, at least let's say in the beginning, one would start from the supergravity in five dimensions and try to learn things about quantum field theory. And often, these computations would be very challenging. At least some of those you can actually circumvent by, if you understand well enough the quantum field theory, you can actually go the other way around and get information about the supergravity directly from quantum field theory. In particular, we'll see that we can obtain the potentials, the relevant potentials from the supergravity theory with a very simple argument. Then after that, we'll pause and then start discussing this classification of N equals one deformations of N equals four swimming young mills. And finally, we'll go to the simplest case, which is the N equals one star theory, the GPP solution, it's uplift, and I'm gonna give a summary of the properties and a few words about the outlook. And then we'll conclude. OK, so any question on the generalities before I move to my introduction? OK, so let's move to the introduction. So let me briefly summarize the basics. This should be very well known to the people that grew up in that period, but some things are forgotten as people move on to discuss other things. So how do you go about obtaining non-conformant dualities if you know maximally symmetric conformal ones? So first of all, knowing that there is an ADS-CFT dual, means in particular you understand how to map the operators from one side to the other. And the operators that we understand best are the half-PPS operators. And this half-PPS operators, the holographic duals are bulk supergravity fields of specific mass, which depend on the dimension of the operator. So this formula here is for scalars, but the discussion is more general. So if you start, if you only wanna do supergravity, the only thing you have access to is this half-PPS operators. Or the other stuff you have to understand by extending the duality to string theory, which is still in its infancy. So now suppose you have an ADS-CFT dual like N-quash-wash-bearing-mills and ADS-5-quash-S5, then the next step is to go to a quantum field theory and to do that is you can just deform the Lagrangian of your theory by adding, deforming it by one of these operators. And to do that, what the thing that you require to do is you require to turn on a field with specific boundary conditions. So if this is a scalar field of dimension delta, then you require that the boundary condition should be of this form over here. And then the coefficient of this term is just the coupling, the new coupling in your Lagrangian. Now once you have that set up, so you have your supergravity theory that describes gravity coupled with scalar field, this specific scalar field, then the next step is to find supergravity solutions with that admit this boundary conditions. And now this gives you the uniform theory. So that's the strategy. So the strategy was very clear kind of shortly afterwards, the ADS-CFT conjecture. Okay, a few remarks. So first of all, in general, the normalization group flow, so once you put this operator here, then the theory will start running because it's not formal anymore. And generically, this will reduce further operators, making the problem pretty much uncontrollable, at least in generality. So to avoid such issues, one often considers single scalar truncations on the supergravity side. Now truncations, consistent truncations on the supergravity side means that on the quantum field theory side, this operator is closed under a piece, so that it lists a subsector where the dynamics is controlled just by the fields that you kept. So now once, so now we need to have single scalar truncations of supergravity, and then starting from the full theory, the maximum of the supergravity theory, you need to find a way to extract what is the dynamics of that subsector. And this means, in particular, extracting the potential for such truncations. So this is really an art on its own and often requires curriculum efforts that only a few people on a planet can do. And this restricts a lot of what you can do because it's very difficult to extract the potentials and therefore find the solutions. So people from there on kind of move to kind of, most people move to bottom up approaches. You just postulate the potential and then you see what you can get. Now suppose you found the potential and the next step is to find supergravity solutions with these boundary conditions. This will give you some information about the system, not all information. Ideally, one should uplift this back to 10 dimensions because it's the 10 dimension solution that encodes information about the entire system. Good. So what we're gonna do in this stock is we're gonna try to focus on simple enough cases so that most of the steps are tractable and we can understand them fully. So we'll start from non-normalization theorems in supersymmetric guanouville theories. This will give us exact results about specific kinomization group flaws and then we will use gauge gravity duality to geometize these results and finally use the framework of say fake supergravity which I will explain to obtain the corresponding potential. So instead of kind of doing the hard work on the supergravity side, here we're gonna use the fact that we have a control about the specific kinomization group flaw and this will allow us to extract the potential and then go back to find the full dual, okay? Yes. Yes. Yes. They're not, actually, they're not general rules. It's just, it's case by case. You saw how, in some cases, they're global symmetries that tells you that certain coefficients are zero and most of the cases that were consistent truncations were found using this kind of symmetry arguments. There is some synclet under some symmetry group and you know that this synclet is not gonna mix with something else and then I can use the same argument on the other side. So you have some operators that transform under some symmetry groups and then the symmetry can allow you to kind of select which operators can appear on the right hand side. I think in all non-consistent truncations there is an underlying symmetry behind it. Yes, and I don't know any counter example. I don't know if anybody in the audience know a counter example to that. Yeah, that's a little bit more general. Yes, yes. Yeah, some underlying structure that. But I think in general, let's say from the QFT to gravity hasn't been used so much. Kind of, you see what you know from, I mean we used it in one paper where we started massive truncations and then the guiding principle was indeed, what happens on the QFT and therefore some massive field. You would see here, I'm gonna have a few more examples where you expect new truncations that doesn't look possible a priori but the dynamics suggest that it should be possible. Any other question? So what is off-shell potential? Off-shell means I don't evaluate it on the solution. The actual potential, yes, the full potential. Includes all dynamics. Would be, let's say, from the way I derive it, it would look like I have only the on-shell potential. So I put the comment there to discriminate between that case. Okay, so how do we go from corner theory to supergravity? So let me now first kind of give you a little bit of a longer introduction about this holographic G-Flows. Now first on general grounds, you expect that the solution should have the form indicate over here which is also called domain world type. So you expect this because you know the corner field theory should only have Poincare symmetry. So this part here should be flat. And then as you go to infinity where the conformal boundary is, this factor here should approach R so that this metric becomes a yes. And then this scalar field, as I discussed earlier, should have the boundary condition indicate over here, encoding the deformation. Now there is a similar case where if the RG-Flows is not driven by deformation, but you are in a different vacuum, you have a condensate. So for instance, in N-quash-Forsch-Pilger-Mülsch, you can see it in the Coulomb branch. And then in that case, there is a similar discussion where now the scalar field instead of having this fall-off behavior, it has that one. And then in this case, describe security with condensate where the coefficient, this coefficient there is now proportional to the condensate of the corresponding operator. Yeah, that's the setup. Now suppose we have a holographic G-Flom which means the solution of this type exists. Now there is a simple argument that leads to first-order formulation. So if it happens that this field is invertible, then this implies there is a first-order formulation given by these two equations. This equation here is actually what defines this what is called a FHP potential. So if you again, suppose you have, you know the existence, you assume the existence of a holographic G-Flow, therefore this function exists. Then if you take its derivative and you evaluate it, then you invert the scalar field and you put it here. This gives you this potential and then automatically the scalar field satisfies the second one. Okay, so now that's a very simple diagnostic. Suppose you obtain your solution numerically, then if it is monotonic, then that would exist and then this would imply the corresponding first-order equations. And then that automatically implies that the potential is given in terms of this W which is obtained via this root, via this formula. Now, this discussion here has been running in many different directions in many different ways. So the same system you can also get by Hamilton Jacobi theory. And the same system also follows if you have supersymmetry and then W is the actual superpotential that governs the supersymmetry of the system. But this discussion does not assumes any supersymmetry. Now, why and then also some people use this as a differential equation to find W, which is of course, it's a very hard job because this is now a partial differential in general, is a complicated nonlinear differential equation. But yeah, this is a very simple construction that only uses kind of the fact that this is invertible. Now, if you have the theory, gravity theory with such a potential, there are a lot of nice things are happening. So for example, this theory is to draw a positive energy theorem. And now it turns out that the solutions that saturate, that have the lowest energy solutions of the system are precisely the solutions that satisfy this equation. So these domain wall solutions or the lowest energy solutions are allowed among all solutions with the same boundary conditions. And then this implies that this solution here is stable at the perturbations, both at the small perturbations, and under some additional assumptions also non-preturbatively. So you kind of gain a lot of mileage out of this. Now, but we're gonna use this formalism in a kind of inverted way to try to get the potential from the actual solution. So let's see how this goes. Now in holographic or G-floss, the beta function is related also to the space potential via a formula, via this formula over here, where R is now associated to the energy scale of the quantum field theory. So post now, we know the beta function exactly. Somebody gave it to us, and we'll see examples in a minute. Then we can just combine the ingredients we just had. These two equations together with this one. And then I get an order in differential equation for the superpotential as a function of R, which can be very easily integrated. And here's the formula. And again, combining the equations, now you get an equation that relates phi with W. And now we're ready to solve for this. And then we can obtain, we can just solve for the scale of field. And now if we invert this relation, now we solve it, we invert it, plug it back here, and then we get the superpotential, and then we plug it back here, and we get the potential. This argument could have been written down 20 years ago, and I don't know why nobody has written it, but it's very simple. You just combine the equations, you have three equations in hand, and just combine them, and then you quickly get the potential that governs the dynamics. To summarize, yes. You mean this one? Yes. Well, this one is again comes from intuition that the TT component of the metric with the appropriate redshift measures the, it's giving you the energy scale of the theory, and then you just combine the ingredients. You can view it, I view it as a postulate rather than, and then what I'm gonna discuss from here on, and provide support to this postulate. It depends on which dimension it is. I will discuss many examples as we move on. So this here again, you have if you look back in my, so you start with a scalar operator that has this mass, then you know there is an operator of dimension delta, and then you run this argument, and then what would be running is the running of this coupling. Now it's not the running of the gauge coupling which sits in here, it's the running of the deforming operator. Okay, so to summarize this part, given the beta function, we can obtain the potential as that's the slogan. So one, do we actually know the beta function? So that's the next question. So before I move on, any more questions, okay? So now suppose you have spesometric theories, then spesometric theories, I mean the oldest of the, we know that the oldest of the non-normalization theorems tell us that there are no corrections to the superpotential. So suppose we modify the superpotential by just deforming it with a scalar superfilt. Okay, then the beta function for this operator will be given by this expression. So there are some parts because this is the classical dimension of the deforming operator. And in general, they can be the operator can have an anomalous dimension. So that will happen if there's wave functionalization of the fundamental fields. So there will not, there are not gonna be any corrections to the superpotential. But if the fundamental fields, if you need to do wave functionalization, that would use some anomalous dimension to the operator and then we'd modify the beta function. Now suppose in addition, that this operator actually has anomalous, has protected dimension. Then this part goes away, that's zero. And now this equation can be readily integrated. And this is a solution. Again, associating the radius with the normalization group scale. And we see that this is the beta function. So here's an example. Or we actually know the beta function exactly. So what happens if we now use this beta function in the formulas I gave you earlier? Okay, that's what you get. This is the superpotential and this is the potential. So then one can ask, has this superpotential and potential appear before? The answer is yes, there are many, many cases and I will review one of the cases as we move on. Our explicit realization, one of this happens. So considering, again, line equals four should be young mills. It's half BPS operators have no anomalous dimension. Now any course for should be young mills can be expressed in n equals one language. And some of the half BPS operators are actually f terms from the perspective of the n equals one theory. So now the point is to deform n equals four using any of these operators. Yes, and that's what we're gonna do. Good, any questions? Yes. In this case, nothing is running. So there's no scheme dependence. Yes, so there is scheme. So once you start moving to more and more complicated cases, there are questions of scheme. There's also questions, what is the precise identification between the radial coordinate and the enumeration groups? Okay, all of this are there. And I think the literature is a little bit kind of not very clear. I would kind of start from the best understood cases and move progressively to more and more complicated because then you're gonna meet kind of less kind of complications and resolve them as you move on. I think that there would be issues to be resolved. Any other questions? Okay, so now this leads us to kind of the main part of the talk. So we need to understand n equals one deformations of n equals four speaking in mills. Okay, so first let me remind you that the falcon of n equals four speaking in mills in n equals one language is three car or multiple z sub i and one vector multiple it. Once you write the theory in n equals one language, the r symmetry group of the theory, which is as you four, as you four is now broken to as you three cross you one. And then the action take the form I indicate over here. So as in all n equals one systems, in the sense that there are three functions to give. One is the scalar potential, which for n equals four is given by kind of a trivial kinetic term. Then we have the gage kinetic function, which in this case is also trivial. It's just the gage coupling with that angle. And then we have the user's potential. So now the question we want to ask is how do we deform each of these three cases using half DPS operators is again. So if we want to study holographically, n equals one theories, then we have to do the only thing we can do is use half DPS operators in our disposal. There are other deformations, which are not one cannot study using holography. But if you want to use the usual tools, then that's what we have. So we need to identify which parts of half DPS operators to form the super potential, the gage kinetic term and the scalar potential. Now once you go, you block the symmetry, you go to n equals one, then the bottom components of carus super fields may condense. So if you want to study those holographically, we also need to identify where this component sits in a half DPS operators of n equals four super meals. Because this would give you a holographic access to this information. So I think the main message of this talk is that we have classified all of these cases. Now if I had maybe two hour talk, I would have gone through it slowly in all detail. So I'm not gonna give you all the details. I'm gonna give you a sketch and a feeling of how it goes. But the actual paper would contain all cases. Yes. So if you want, so basically this program, now I can go and do for Clevon of Whith and the Lagrangian. So you're gonna call the field theory Lagrangian. I know the field theory. I know what happens in the field theory. And I know exactly how to identify which fields are turned on on the super gravity side. And I will try to make this a little bit more clear in the next few slides. Okay, so let me, first a quick reminder about half DPS operators of n equals four super meals. So the super performer primaries are given by these metric traceless combinations of the traces of the six scalars. So phi i are the six scalars. And then the rest of the half DPS operators are obtained as a super conformal descendants. If you put P equals two here, then this is the energy momentum multiplied and this corresponds to the GH super gravity multiplied. If you look for P greater than two, then this corresponds to the Kaluzha Klein modes or the five sphere. Now the structure of this multipliers has been fully understood. So here I gave it for the P equals two. So in general, for general P, so P equals two, P equals three are special. For P equals four, they form this kind of diamond shape. Then you start from the top, from the super performer primaries. So for P equals two, this is again phi i, phi j, and then you move down by acting with supercharges. So here in the boxes, I denote the SU-4 representations and then in the brackets, it gives you the Lawrence indices. So this is the scalar, this is the fermion, this is the soft dual form, a scalar and so on. And you fill up all the multipliers. Again, P equals two and P equals three are special for P equals four, a generic. Now the way the mapping is happening in the bulk is like putting them, stacking them on top of each other. So you start with P equals two, which is, so this is the multipliers I just described. Then P equals three has this shape, P equals four, then it's the full diamond. Then you stack them on and then corresponding to each of them, if you go vertically up, you have the kaluzak line modes from the various 10 dimensional fields. So then you know exactly where things appear. So if one tells you that you want, let's say, you're in P equals three and you want the mode, then you have to look at the bottom component of the under symmetric tensor in 10 dimensions and so on. Okay, so we know how to go from the SU four for the kind of super conformal primaries to supergravity. But now what we're interested in is, again, we're interested in doing the n equals one problem. So the next thing we need to do is we need to kind of break the representations from the SU four to SU three equals one. And here I give you the corresponding kind of branching ratios. So the 20 is decomposes into six bar eight and six and so on and how they act. So now we understand and assess how they decompose. We still now need to organize them into n equals one multiplets. And we know how to do this. We know how to cover all of this with n equals one multiplets. But in particular, we're interested in this problem of deformation. So we need to identify f terms, d terms, kind of the gen gena multiplets and the condensates. So for that, you need to do then the composition with in chiral superfields. For just chiral superfields, it's very simple. You just need to go kind of to the left. So this is a chiral superfield starting from z square. And this is the top component. And you can do this now with all the multiplets. So now let's now analyze in a little bit more detail this case because it's the simplest case where everything sits within supergravity. So in this case, we start with the chiral primary. So first I go to this complex basis. So this is z i z j. And then I supersymmetrize. And then I turn this into superfields. So the superfield is has this is the bottom component where we have a fermion and then the f term. So now what we want to do is again, we want to deform the superpotential. So we're interested in turning on a term of that type. Okay, so now if we put it in, now if we do any cos one, it's off shell. So there are auxiliary fields and then you need to integrate them out. So that's an additional complication. So once you integrate the auxiliary fields, you get the Lagrangian of that form. So now I deform the theory. I have the n equals four Lagrangian plus these two terms. Here there again. So this first term in red is what we wanted to do. So we wanted to add a mass term for the scalars. But now because we integrate out the auxiliary fields, we automatically generated an additional term. And this additional term in terms of as you see representations, it has a component which is in the eight and component which is in the one. And actually the eight is part of the 20. So here if you see the decomposition I have over here, so the 20 decompose to six and eight and six. So the eight that we get here is this eight. So if I just do this, do that, then I need to include two bulk scalar fields. So I also have to turn on the scalar field that corresponds to the eight of the 20. Then this is gonna be a two scalar case. I want to be as simple as possible. So we need to project out this. And one can project this out by considering this matrix M, which in general is symmetric traceless. It's a symmetric. Just consider the diagonal elements. And if the diagonal, this projects out the eight. Now there's still one, and this one is the Konishi operator. Now in the supergravity limit, the Konishi operator decouples. So it's not gonna be visible. So we now have to worry about the Konishi. Although this may be related to the fact that we see later that the actual solution is gonna turn out to have a naked singularity. So to really describe the system, in principle, we have to have access to the Konishi. Okay, so now to summarize, if we're interested in the N equals one star theory, which is the theory with three equal masses, then the only thing we need to do is we need to turn on this operator where the M is diagonal. So now we have a case with a single scalar, and this is consistent of its own because from the quantum field theory answer. So now I can use what I did earlier because everything I said earlier now exactly applies to this case. And Planck delta equals three, D equals four to the formulas I had earlier, and this is the formula for the superpotential and this is the formula for the potential. That's precisely the superpotential and the potential which we worked out by TPPZ with a lot, a lot of work. We actually spend a lot of time reproducing the potential the way they've done it. I mean, this would require, I would think, a two-hour lecture to explain it fully. And then here comes for free. It also tells you what is the physics behind it. So the physics behind it is, that says there's only classical running. Now the general P generalizes, so the story is very similar, so you start with now the full diamond, and if you want the formations of that form, so this is superpotential deformations, and then this is the corresponding superfield. All of this are Kaluzak line modes which are associated with the under-symmetric tensor with the likes on the S5. The details are very similar to the P equals two case, and like in the case of the P equals two case, when you integrate out the auxiliary fields, you get a lot of extra terms. In this case, it was just one term, but you get similarly many other terms. And in all cases, there is precisely one of those is Kyra primary, so the generic case requires two scalars, but you can always arrange so that you project that additional scalar out, and then you left just with the one that you want to deform the theory. So this argument suggests that there exists massive single scalar consistent locations of type to be supergravity with the potential that they gave earlier. So that's a challenge for the supergravity community. So now let me briefly discuss the other cases. So now if you want to deform the gauge kinetic function, then the corresponding F term comes from here, so now you start from this nut and you move down. So if I now start with, so this is gonna describe this kind of deformation, so that deforms the function from just the gauge coupling to z to the P minus two, and then you can discuss all of them. So that would be very interesting to kind of analyze combining with kind of diagram of our technology from 15 years ago, understand the better functions and so on, but we haven't done that. And finally a few terms about D term deformations. Clearly D terms are a lot more possibilities. D term deformations appear only for P greater or equal to four, one that we have the full diamond in the representation, and one result that I would like to point out is that there is a class of deformations which is characterized by a harmonic function, namely function satisfying this condition given by this expression. So that's a general result. So if you do a D term deformation of N equals one, then equals one, it would be characterized by this general function. And again this, I think there's a lot more room to be analyzed, a lot more room for analysis. Okay, so now let's move on to condensates. Now N equals four speaking mills does not have accurate that's potentially break N equals four to N equals one. However, if we first deform the theory to N equals one, then there would be condensates. In particular for the N equals one star case, we know that the condensates have the form I give over here. So this M here is the deformation I discussed earlier. So the condensate is related to the mass of the theory. And there is this exponential factor here that comes from non-perceptive effects. But now in the large tough limit, if you take strictly lambda goes to infinity, this factor just goes to one. And we left with W square proportional to M cubed. Then this implies that there is an RG type equation for the condensate which is given over here. And it's the analog of the RG equation so we were discussing earlier. And then one can run again the same argument. I had earlier when I derived the potential from the RG equation. In this case, the fixed potential is this one. And then you can combine it with the fixed potential we obtained earlier to get that one. And from here you can get the potential. And that agrees exactly with the GPP0 result where they derived the potential for the entire system, the M and the sigma field. And again this clarifies what does the underlying physics behind this potentials. And here you get both of them with almost back of the envelope computation. Okay, so what I'm gonna do next is now I'm gonna focus on the GPP solution and discuss its uplift. Any questions? Sigma I got from, so sigma is the field which is due to the operator W squared. And so now if you go back to the difference between the two RG plus is in this kind of different behavior. And that translates to, you can get the same formula as you had earlier but you take D minus delta to minus delta and that gives you the formula. So if you do this replacement in this formula that gives you the formula for the condensate. It is really completely analogous. You follow the same steps. The only difference is you trade delta for D minus delta. Yes? I don't know, I haven't really, I mean it's clearly a lot of things to analyze in this cases. And now we have access to this holographically because some would be sick. Or they were, I don't know, I haven't analyzed those. Any other questions? Okay, five minutes for the uplift. Okay, first let's go back to the GPP solution. So the questions are actually, this first of the questions are very simple and they can be solved by terms of elementary functions. Here is the solution. So M and sigma are the norms of the complex sources. Actually that's one of the questions I had for many years because from the point of field theory it was clear that you need M and sigma to be complex but the access to gravity had the real fields. And it wasn't clear for long time, at least to me, it wasn't clear. Is it the real part, is it the norm and so on? It is actually the norm of the complex sources. But now this solution actually has an naked singularity. Iphone uses groups as criterion then the singularity should be acceptable if we have C1 and C2 to integration constants and if they satisfy this relation then they're acceptable according to groups as criteria. But of course, okay, this is more diagnostic. I think one needs to kind of go deeper to understand whether the singularities are physical or not. And the solution was proposed to be dual to one of the confining vacua of N equals one star. I mean N equals one star has a rich vacuum structure which I don't have the time to review here. In particular there are confining vacua and then this author's here, one of which is our chair, will propose that this should be dual to one of the confining vacua. I can feel here as I said, the complex are complex. So as a first step and it's clear this faces do play a role. So we need to generalize the solution to include the faces. Secondly, I can feel theory as I reviewed earlier in the limit of infinite tough coupling. The condensate should be proportional to the mass cubed, where M is again the deformation parameter. Now engage gravity duality, we expect such relations to come from a regularity of the solution. However, for no choice of the integration constants the solution is regular. So that points out towards that perhaps solutions and physical. And there is also a computation from 2000 where they actually find a massless state which contradicts the claim that this is describes confining solutions or perhaps the solution is dual to a Coulomb vacuum of the theory. Finally, there was a proposal for a 10 dimensional dual of any Coulomb sphere by Puczynski Strassler which looked quite different from what we have over there. So for a long time there was a question what is the relation to GPBZ if any. And now to address some of these issues one really needs the 10 dimensional solution and that was not possible for a long time because we didn't know how to do that. Although I should say Kristof was sitting there that had an enormous progress around 2000 they almost got it. So sometimes the singularities in lower dimensions are resolved if we go to higher dimensions. And furthermore the 10 dimensional solution condense a lot more information than the five dimensional one. In particular one can extract the best of kind of primaries due to collusion client fields and that will help understand the physics of the solution. And also having directly a 10 dimensional solution should allow us to kind of compare with what Puczynski Strassler did. As I said earlier, the D equals five maximum gauge of the gravity is a consistent location of type to be should be gravity with very explicit formulas for uplift provided here. So that's what we did. So first we generalize the GPPZ solution to complex M and Sigma and then uplifted the general solution to 10 dimensions. Now it turns out that the phases of this complex fields in the beginning looked surprising. I don't think it's as surprising anymore but in any case here it is. So these phases are accounted by combination of rotation in S5 which corresponds to an r-symmetry transformation from the quantum field theory side and a rotation of a U1 inside the SO2 of type to be from the quantum field theory side this U1 is what the relegator called bonus U1 which is only present in the infinite tough limit. So the 10 dimensional metric and action in a lot of fields agree exactly with what Christoph had with one and two thousand. Now we have also all p-forms. All p-forms are turned on and then that solution is not that complicated. It's always given in terms of elementary functions. In our archive submission we also provided a mathematical file with a solution. So if you wanna play with it, it's all there or if you wanna check again it satisfies Einstein's equations. You can look at the mathematical file so I'm not gonna flash the solution. I'm just gonna describe some of its features. First singularity, so the singularity is exactly as it was described in this 2000 paper. So first of all the solution is still singular in 10 dimensions but the singularity is milder in 10 dimensions than in five. So in particular there is either a ring singularity in the internal space on the S5 if C2 is less than C1 or there is both a radial singularity and an angular singularity if these are equal. So I think this looks to me more singular so I tend to think the actual physical case is this one. But there is a curious observation that we made in the paper that if one considers different conformal frames so we have scalar fields which are non-trivial so a dilaton and an axion. So in general you can consider a new metric which is rescaled by these fields. Then I can make the metric almost completely regular. So for the case C2 less than C1 one can arrange so there is a singularity only on one point and it's five and it's regular everywhere else. And then in the case of C1 equal to C2 one can arrange but there is only singularity in the radial direction. Now what is the physics of this scaling? I don't know, I think at this point it's just an observation. So we also looked at the near boundary behavior. This is the beginning of understanding how to extract the vevs for the Calusa Klein modes. So we find agreement with an earlier work of Friedman and Minahan from 2000 where they did the special case. The boundary conditions that we're using, namely the non-normalizable modes are exactly the same with what Pochinsky's stressor use. But the subletting terms are different from those of Pochinsky's stressor and this implies that what they give as a solution actually doesn't solve the equations of motion. So I think that puts a big question mark about the validity of that solution. Okay, so what is next? So now we have the 10 dimensional solution. We explicitly check the type to be equations hold at generic points. But they can be delta function sources. A solution may be supported by brains. And checking for those is clearly much harder. To diagnose for possible delta function sources one may integrate the field equations against test functions and integrate overall space. So if there would be no delta function sources that should give zero. If the delta function sources one should get something which is non-zero. Because we're currently doing this. This is the results we have up to this point. First, they're definitely now saving brains into the system. So I think Christoph and Warner speculated about the solutions may involve seven brains but there's definitely not seven brains. There are no sources for seven brains. There are also no five brains localized away from the position of the 5D singularity. But there may be five brains localized in the position of the 5D singularity. That we have and we still have to do the complete analysis here. But they're still open at this point. And there's no, it doesn't seem also there are any other brains on the ones that I list over here. Yes, so the outlook. Okay, so we need to compute further variables to decide what the solution represent. Perhaps it's unphysical or perhaps represent some of the other vacuum of n equals one star. It's not clear at this point. And one way to do this is to use this program of Kaluzakline calligraphy that allows to extract the vacuum expectation values of the color primaries which are due to Kaluzakline fields. And then you can compare with one filter expectations depending on what vacuum you have to see whether you get the same or something different. So that's the outlook of that. Okay, so I need questions on this. So next I'm gonna have one slide of conclusions. Yes, yeah. So let me, so I summarized the start of the last part. So now let me briefly summarize the first part. So what I discussed today is I discussed how to classify n equals one deformations of n equals four should be in meals. We also discuss how to obtain supergravity potential for single scalar sectors directly from the quantum field theory. And this relies on knowing the beta function exactly. So clearly it would be interesting to generalize that to other cases where we understand the quantum field theory dynamics. This results suggest single scalar trangations of massive modes which is surprising from the supergravity perspective and it should be checked. And also novel trangations that involve just a few fields, let's say a couple of fields that should also be tractable. And I focused this discussion for d equals four but most of the discussion extend to other dimensions and other theories. It would be very interesting to do the same for AB and GM and three dimensions. For example, the potential for d equals three and delta equals one or two already appeared in the literature. Okay, thank you.