 OK, now we'll have a talk by Francesco Bernini, who is not Leo Pandosaias. Hello? Is it better? OK, so in these lectures, I will talk about supersymmetrical localization. And you actually had already the first lecture by Leo that essentially explained the basic idea, although in a much simpler realization, the basic idea behind localization. So essentially, what I will do will be to spell out some of the details that you have in the full quantum filtering setup. But so supersymmetrical localization is a very powerful technique, which allows us to perform exact non-perturbatory computations in certain field theories. And while essentially it is an infinite dimensional version of the equivalent localization that Leo explained. So let me start with a little bit of motivation. So we're interested in quantum field theories. And at least in principle, all the information about a quantum field theory is contained into the Euclidean party integral. So roughly this is an integral where all field configurations in the theory e to the minus the action, which is some function of the field configurations. But by H bar, this is the Euclidean one. And so this object is an infinite dimensional integral because we integrate over all field configurations on some manifold, on some spacetime. And so even though formally this contains all the information we want, in practice it is too hard to solve in general. So we know what is the standard approach. So we can do some perturbative expansion of the action. We do perturbatory computations. These work perfectly fine at weak coupling, but it doesn't work at strong coupling, where the coefficients that we used to expand are order one. Because all terms in the expansion are equal important, so we should compute all perturbative computations. And even though if we do that, if you are able to do these and sum them, still in general, the series is only up-syntotic. We will need to compute non-perturbative corrections. And so the problem is hard. So what do we do? Well, of course, this is probably with a long history, so many different approaches have been developed. And one possible approach is to study some class of field theories for which maybe some part integral can actually be done. These will be some restricted class of theories. And in fact, for a long time, so localization has been known for a long time, probably 30 years or something like that. But for a long time, it was thought that this class of theories in which we can actually exactly perform computation of the part integral was some restricted class of theories, maybe some homological theories, some topological theories, but very specific theories. But somehow, the realization that it sparkled, the sort of mini-revolution in this context, was to understand, and this follows from work of Necrosof and Pestum, to understand in fact the class of theories in which we can do exactly some of these part integrals is much larger and includes standard quantum field theories, or almost standard quantum field theories that we might be interesting in. And in particular, these includes very physical supersymmetric gauge theories. So before going on, let me stress that in these theories that we will discuss, even though we are able to do some part integrals, we will not be able to do all possible part integrals. So that will mean that we can actually solve the theory because we can compute all possible observables. This would be awesome, but this is a dream. This is not how it is right now. So we will be able to compute some part integrals. In particular, we will be able to include some sources in the action, and this will give us access to some observables. But in fact, this would be a pretty interesting and large class of observables that we will be able to compute. And so this will give us quite some interesting information about those theories, even though we cannot yet solve completely the theories. So the objective of this lecture is to see how to compute objects like the following. So let me write this with a little bit more detail. So we want to compute some Euclidean partition functions. So this will be, once again, part integrals of the theory. So we look at the Euclidean theory. And we will place the Euclidean theory on some compact manifold M. So this will be some compact manifold. And here we want to compute, so once again, these are weighted by the classical action. We set h bar to 1. And in general, there will be some parameters that we call t for now. So these parameters might control either some sources that we turn on in the action, or there might be some coupling constants. And we want to depend on the coupling constants. Or there might control the compact manifold where we put these theories. There might control some background that we turn on. Now, we will choose the Euclidean theory and compact manifold because this will solve various type of divergences, in particular infrared divergences. And so in particular, this object will be truly functions of these parameters t. Or if you want, the part integral will give us numbers because we consider some compact manifolds. Now, in fact, it is a very profitable exercise to try to study. So if I give you some quantum field theory, it is a very good exercise to try to put this theory on very different and many different compact manifolds, possibly with different type of backgrounds, background gauge fields, or other type of background fields, because of a couple of reasons. So first of all, when we do this exercise on different types of manifolds and on backgrounds, and we apply the technique that we are going to describe, so localization, well, on some manifold and some background, this part integral reduces to something that can be more or less simple. So in some situation, this will be reduced to a simple problem, and still to a problem that is relatively hard, but in some cases, this is reduced to a very simple problem. And then we can actually carry out this computation in a very explicit way. So this is a good reason to explore different manifolds to see in which cases we can get very explicit answers. So if you wish, different m implies different levels of complexity. And more importantly, essentially studying the quantum field theory on different manifolds grants us access to different sectors of observables and data about the theory. And so even though on a simple manifold, we might get some information that we see, OK, this is very restricted information. But in fact, as we change m, we can get access to a relatively large set of data about the quantum field theory. We can learn a lot about the quantum field theory. Essentially, the second point is that we get different class of observables. And in particular, these class of observables can contain either holomorphic correlators. These observables have been known for a long time since the early days of localization, but also some holomorphic, non-holomorphic correlators, or conserve currents, and so on. So things that in the past were not known to be extractable from localization. So this will be our general program. Do we have questions so far? OK. So since we want to study this quantum field theory's own compact and in particular curved manifolds, so of course we could receive flat manifolds, but it would be a small set. And we want to apply the technique of supersymmetrical localization, which in particular, as the word suggests, requires supersymmetry, we need to understand how to preserve supersymmetry on curved manifolds, so on all these manifolds, and we also want to understand what type of supersymmetric backgrounds we can turn on. And so the first thing that I would like to address, although in a kind of brief way, is how do we preserve supersymmetry on a curved manifold, given supersymmetry on a flat space? And this turns out to be a non-trivial problem. OK. So in particular, this question overlaps quite a bit with Stefan lectures. Lecture. So let's start from the basics. So we start Lorentzian flat space. Then we know that supersymmetry algebra is an algebra that enlarges the Poincare algebra of symmetries with some fermionic generators. And for instance, we know very well if we consider four dimensional, minimal, and equal one supersymmetry, what we have are the generators Q, and the anti-commutator of the supercharges schematically gives translations that give us momentum, while the other ones, Q with Q and Q bar with Q bar, anti-commute. Now if we go in a local quantum field theory, the supercharges can be written as an integral of currents, which are called the supersymmetry currents. And this is true in any dimension for any amount of supersymmetry. So I'm not restricting now just only to this case. So the supersymmetry current contains a vector index and a spinor index. And as any other current, if we integrate it on a co-dimension one subspace, we generate the charges. So in particular, we take a slice of space at a constant time, and we construct the charges. So we take the zero component, and this gives us the supercharges. And, well, a theory is supersymmetric, if you wish, if these supersymmetry currents are conserved. Larger? OK. It's not going to change much. It's easy for me to write larger, but what you ask, I'm not so sure. OK, so if this supersymmetry current is conserved, it means that the theory is supersymmetric. Now, if you look at theories with a Lagrangian description, and this is the setup that we're restricting to, because we want to compute pat integrals, and so in particular we need a Lagrangian, or at least an action, then the action is invariant under the action of these supercharges, and so the Lagrangian is invariant up to total derivatives. Sorry. Yeah. And so the variation of the Lagrangian will be a total derivative of something. And this delta is an anti-commuting scalar operator, which comes from contracting the supercharges with some commuting and spinorial parameters, which are the supersymmetry parameters. So like in Stefan's lecture, but for now these are constant. So now, we start with a theory, a Neuclidean theory in flat space, which is supersymmetric, and we want to place this theory on a smooth-core manifold. And in particular, when we do that, the theory at short distances is not modified, because of course when we put the theory on a core manifold, we have to change it to be the theory, so we need to say what do we mean by putting the same theory on a core manifold, and our definition is that at short distances the theory is not modified, because if you take a smooth manifold and you go at very short distances, the manifold is flat, so we want to recover our original theory. And so in particular, we only allow the formations of the Lagrangian, which are relevant. OK, we also restrict to local deformations. Now, it turns out that this procedure is ambiguous. So given a theory in flat space, there is a unique way to say what is the theory on a flat space, essentially because if we give some way of putting the theory on a core manifold, we can always add some other couplings in which we use the curvature invariance of the manifold, or some scale of the manifold, and of course all these couplings disappear if you go on flat space. And so there is no unique answer in general. There are different ways to put it on a manifold. Yeah. Just counting the dimension of operators. Yes, yes, yes. Yes. Yes, so OK. So first of all, everything that I would say, well, it's classical. Of course, we compute pati integrals. And then, of course, there will be quantum, but these arguments on the action are classical. And moreover, yeah, I'm not doing some RG flow. So I'm literally saying, OK, we started some theory in flat space. We have dimension of the various couplings, which is a classical dimension. It's precisely this scale that you mentioned about. It's precisely what tells us that only the relevant ones are allowed to us. Because at short distances, we want this deformation to disappear. OK. Now, we also require that the theory preserves some supersymmetry. This is precisely what we want to achieve. And so in particular, suppose that we preserve some supercharges after putting the theory on some core manifold. Then these superalgebra that we obtain on a core manifold, if you go in the UV, it must be a subalgebra or the flat space supersymmetry algebra. So in particular, the supercharges that we will be able to preserve are always some subset of the supercharges that we are in flat space. However, the algebra that they preserve might be a deformation of the flat space supersymmetry algebra. And precisely, the deformation that comes from the various scales that we introduced when we put the theory on a manifold, but this deformation must, once again, disappear when we go in the UV. So it will be also relevant deformation of the supersymmetry algebra. Yet, as we will see, ambiguities survive. So even imposing supersymmetry is not going to fix these ambiguities. Still, there will be different ways of preserving supersymmetry on a core manifold. And on the other hand, it will not always be possible to preserve supersymmetry on a core manifold. So first thing, this problem of putting theories on a core manifold is really non-trivial. On some manifolds, there is no way. So this is not always possible. And so one has to answer to the question, what are the manifolds on which we can preserve supersymmetry? When it is possible, then still there will be ambiguities. OK, so how do we do that? Well, as a first attempt, well, we put the theory on a core manifold. So essentially, let's just substitute the Minkowski and invariant tensor with the metric and standard derivatives with the covariant derivatives. So at a mu nu, we substitute with the metric. So from this is a first attempt. And then standard derivatives go to covariant derivatives. And we do this both in the Lagrangian and in the operator, which is these supersymmetry variations. So this sounds reasonable, however it doesn't work. Because in general, if you compute this variation that contains the metric and the covariant derivatives of the Lagrangian that contains the metric and the covariant derivatives, this does give us a total covariant derivative of something plus other terms. And they do not integrate to 0. So supersymmetry is spoiled unless we can find on the manifold a covariantly constant spinor, so some epsilon, so that epsilon which satisfies some function that satisfies this condition. If this is the case, then there are no other terms, and we do preserve supersymmetry. However, it turns out that this is a very strong condition. On the manifold, in particular, in two and three dimensions, this implies that the manifold is flat. And in four dimensions, it doesn't have to be flat, but still it has to be a rich flat, which, if the manifold is compact, is a very strong condition. So this is too strong. I mean, it's OK, but it's too strong. We would like to do better. So there are two strategies that we can try to follow. So if you want, here we can write too strong for our test. OK, so there are two strategies that we can try to follow. So the first strategy is a trial and error procedure. And so we first of all introduce some scale by rescaling the manifold and the metric. So we write our metric. We replace this to a rescaled version of it. And then we try to expand both the Lagrangian and this supersymmetry operator in powers of this r. And we organize the terms, order by order in r. And we try to make things work. So we take delta. So the first term is, of course, the delta in which we put the metric in the covariant derivative. But if you say this is not enough, so we write down corrections, which are awaited by these powers of r. And we do the same thing with the Lagrangian. So we construct these terms, order by order, in such a way that each order cancels the problem that we had in the previous order. And the important thing here is that since we restrict to relevant deformations, particularly if you look here, there is only a finite number of terms that we can write down. So the average is a series, but in fact, it is a finite number of terms. Because at some point, then these corrections are irrelevant, but we exclude those. We want you to restrict to relevant deformations. So there's only a finite number of terms that we can write here. However, it doesn't mean that we are going to succeed. So when we have exhaust the possible terms here, either we got an answer that works or we didn't. So it's not guaranteed to work, but at least it's some finite procedure we can really do. And a finite amount of time. So in fact, this procedure is correct and has been used a lot in the literacy of localization, because I don't know, it's the most triforical thing to do. But it has a few drawbacks. So first of all, as I said, it's not guaranteed to work and we will understand why. But essentially, because not all manifolds admit supersymmetry, so in those cases, you just finish your terms and you just cannot cancel the variations. Well, it's also a bit tedious, because you have to do this order by order. And especially if you don't have much symmetry on the manifold, it might be complicated. But more importantly, the underlying structure is not clear. If you fail, it's not clear why you failed. You don't understand on which manifolds you are going to succeed on which manifold you are not. So the second method, instead, yes, of course, it's up to you to be able to write down all possible terms. And as I said, technically, it might be quite difficult, maybe impossible, because maybe if not, as long as it's too complicated. But if you have enough symmetry, this is easier to do. In fact, it's easier maybe than the general method where you have to work. Once you have the answer, of course, the general method is easier. But yeah, OK, this is just one method. What is the fact of what, sorry? Well, because this supercharges the Q in a given theory I realized and some function. Yes, well, so this delta is epsilon Q. And Q is, well, yes, it's an integral of s. Yeah, I mean, it's not, well, OK. But when you start, you have some l. So in flat space, you have some l. The supercharges have some expression in the fields. I don't know, the variation of the scalar is the derivative of the fermion. The variation of the, sorry, it's the fermion. Yes, you modify l. And then if you want that this one, and it's just, I mean, you can try to just modify l. But this is not going to work for sure. You'll see that you also have to correct how the supercharges act if you want some hope. But there's no l in the fact, is that not true? No, no, no. They are correlated, right, because this acts on l and you want the things cancel out. No, no, it's dimension full. Sorry, it's the, OK. So we can say that this G, this is a dimensionless metric, if you wish, and then you put the dimension in R. Or you can keep the dimension here, and then this R is dimensionless. But of course, it encodes the fact that it's encodes are scaling. OK, so the second, any other question? No, it's the one where you substitute the curve metric and the covariant derivative. So it doesn't have an explicit depends on R besides the curve metric and covariant derivative. But as we said, this is not enough, so you need extra things. Just to introduce an expansion parameter, because we have this, sorry. It's not going to change. I mean, here I have a totally, so this is a curve metric. This G zero is a arbitrary curve metric. I just want to organize my terms and keep into account this constraint that I put, which is a physical constraint, it's not a mathematical one, that I'm restricting to relevant deformations. I could remove that constraint, but then in some sense I'm not discussing the very same theory from flat space to a curve manifold. Then is another random theory, which is modifying the UV as well. So this is just to implement that. Yes, yes, so if you require on the manifold that there exists a spinor, so this is a function is commuting, but it's spinorial. So if you impose this, so first of all this epsilon is never vanishing just because it's conveniently constant. So you have this never vanishing spinorial function on it and these are strong conditions. So for instance, you can multiply by another covariant derivative and take a commutator and from this you can derive that R mu nu on the manifold is equal to zero. So the manifold has to be a Ricci flat. It's a very simple exercise, we multiply by R delta nu, you play with it, I mean I can give you this afterwards if you want. So you get this condition. Now this is very strong, because if you are in two and three dimensions, this tells you that the Riemann is zero. So the metric is flat. So in two and three D, this tells you that Riemann mu nu rho sigma is zero. So the manifold is flat, so it's just a torus. And in four dimensions, again it's not just flat because this is Ricci flat, but essentially give you only T4 and K3 as possible manifolds. So it's very restrictive. We can do, I'm saying that we can do, I mean we can stop here and say these are the manifolds where we can preserve supersymmetry. But we can do better than that by adding these extra terms. Yeah. Well in five dimensions they're not Calabi-Aus manifolds, but still this is a certain strong condition. Yeah, I don't know if this is, I do think about in five, well but we can do better than in three as well, right? We can turn on fluxes or other more complicated backgrounds. So even in that context, we can do better. Of course, okay, if you are in six dimensions you have all the Calabi-Aus. Yes, yes, okay. In two and three dimensions are trivial. I agree in six dimensions are Calabi-Aus, I'm not trivial, but still my only message is we can do better. Okay, so the other thing is to use some systematic method. And this is due to Festuche and Cybeck initially. Well, at least I explore this method in full generality. I mean this method is not giving you less than this. I mean this is correct. It's just that it's tedious, complicated. Maybe you're not technically able to find all solutions because you need to write this method, but I mean this in principle can give you all solutions. This is just, you know, every time that you have some problem if you can find a systematic method to do that, it's more elegant, it's better, right? Okay, so let's describe this method. And essentially this method in a word what it consists in is in coupling the theory to supergravity, to off-shell supergravity, and then take a rigid limit in which you send the Planck constant to, but the Newton constant to zero, the Planck mass to infinity. And so in this sense, we make connection with Stefan's, well lecture today and probably in the following days as well. Yes, so this method, I mentioned this because this came first, is the very most straightforward thing that you can think about. And so S4 partition function, S3 partition function, S2 partition function were done with this. And then Festus and Cyber came with their method and is allowed to construct many more interesting and complicated solutions. But people did a lot with this before. Yeah, I'm talking about the past computation, I mean past this computation without Festus and Cyber. Also they, yeah. Yeah, yeah, I mean this, I mean they were not the first. The first method. Oh yeah, yeah, yeah, sorry, yes, yes, yes, yes, yes. Okay, so let's describe this method. So what do we do? So okay, we want to put the theory on a core manifold, so let's proceed in the following way. So okay, let's first think without supersymmetry. So what we do, we take the theory, we couple to metric in a generally covariant way. So in particular, we include some new field, g mu nu, okay, this is our metric, this is our field. We couple to the theory in a generally covariant way. And then we give an expectation value to this field, as the expectation value is the metric on the method that we want to study. Now of course now we have one more field, so we have more equations of motion, in particular when we do the variation with respect to the metric, we have Einstein equations. And these Einstein equations put constraint on the metric, so would limit the type of manifolds where we can put the theory on. However, what we can do is we undertake this rigid limit in which we send the Newton constant to zero, or the Planck mass to infinity. And in this case, we don't impose, so these freezes, if you want these degrees of freedom, so we don't impose Einstein equation anymore, and we can consider any possible expectation values for these fields, okay? This allows us to consider any possible metric, okay? This is, I'm saying something trivial here, if you wish. However, now let's consider supersymmetry. So if we have supersymmetry, now we don't just couple to gravity, but we couple to supergravity. And it turns out that in order to, and we will see this in a moment, in order to make this method work, in fact, we should couple to off-shell supergravity. And in particular, as we saw in Stefan's lecture, in off-shell supergravity there are extra fields, extra auxiliary fields, such that the supersymmetry algebra closes off-shell, we don't need the equations of motion. So how do we couple the theory to off-shell supergravity? Well, first of all, so okay, we are discussing local theories, so first of all, we have a stress tensor, symmetric and conserved stress tensor. And the stress tensor, together with the supersymmetry current that we also have, and we discussed at the beginning, they sit in the same multiplet, in the same supersymmetry multiplet, which is called a supercurrent multiplet. And when it turns out that this is supercurrent multiplet, there are not just these two operators, but in fact, there are other operators of spin less or equal to one, okay? So here there are other operators, in particular this is a spin two, spin three halfs, but then there are other operators of spin one, spin one half, spin zero. And in fact, it turns out that there are different types of supercurrent multiplets that one can construct. So one multiplet that one can always construct is called the S-multiplet. And this multiplet always exists. So you give me some supersymmetric theory in any dimension with any amount of supersymmetry, this S-multiplet exists. However, this multiplet in general is pretty long. For instance, if we are in four dimensional, minimal supersymmetry, it contains 16 bosonic and 16 fermionic independent components. So it's pretty long, it's larger than what Stefan's described. However, if the theory has some special properties, then it might be possible to find a smaller multiplet that still contains these two guys, but that it contains less other operators. So another way to phrase it is that one can do some improvement transformation to set to zero some of the operators in this multiplet. And so what are other interesting cases? Well, if the theory contains a conserved r-symmetry and as Stefan stressed, a supersymmetric theory does not need to have an r-symmetry. This is an outer automorphism of the algebra, it's not part of the algebra, so it need not be there, but if it is there, one can reduce to a smaller multiplet which is called the r-multiplet. And in particular, this r-multiplet contains this conserved current. And then there is another multiplet that, roughly speaking, one can obtain if some target space for scalars does not have two cycles and there is no phi etiopoulos term, but okay, this will not be particularly important. Let me just mention it. This is the Ferrara-Zumino multiplet. And then if the theory is actually super conformal, it's not just supersymmetric, in fact one can find, it's just a super conformal multiplet contains these two guys. So these multiplets are simpler than the s-multiplet. Another connection with Stefan's lecture comes because, in fact, to each official formulation of supergravity correspond one of these supercurrent multiplets. So if you want to off-official supergravity, this corresponds to sum a supercurrent multiplet. And if you want, why is that? Well, because this supergravity has to couple to the quantum field theory, at the linearized level, it couples to operators in the field theory and so there is a pairing of fields in the supergravity multiplet with operators in the supercurrent multiplet. So the basic example, of course we are very familiar with, is so in the graviton, so in the graviton model, of course there is the metric. This is the first field that is there. And of course we know at the linearized level how the graviton couples to the field theory, it couples to through the stress tensor. So here we have the supercurrent multiplet. And in fact, at the linearized level, once again, so if we take the Lagrangian at the linearized level of supergravity, the coupling is the obvious one. The metric couples to the stress tensor. But then we are in supergravity, so there is also the gravitino here. And how does it couple to the theory at the linearized level through the supersymmetric current? In fact, here in the Lagrangian there will be another term where they are coupled linearly. And then as we said, so here in general in this multiplet there are other operators of spin less or equal to one. In the same way in the graviton multiplet there are other auxiliary fields of spin less or equal to one. That's Stefan described. And they are coupled, so for instance we can have, we can have some vector, or we could have a spinner, or we could have a scalar. And which one really depends on the particular offshore supergravity we are considering. And in the same way here we can have some current which is either conserved or not, depending on whether this is mass less or not. We can have some spinorial operator. We can have some scalar operator and so on. And they are coupled in the obvious way. So currents are coupled to gauge fields. Spinors to spinors, scalars to scalars, in the obvious way. And so this simple fact gives us a correspondence between offshore formulations and super current multiplets in the field theory. Any question? No, no, this is pretty general. Yes, in fact, notice that this is not a double arrow. Of course you might expect the opposite, but for every super current multiple there should be an offshore formulation. So I just learned that in four dimensions for the full S-multiple, in fact there is, I didn't know that, but there is an offshore formulation. If the supersymmetry is large enough, there might be problem with this. And in fact the statement I'm making here is in this direction. Okay, so now that we have this, what do we do? Well, we do the same thing that we do that was pretty trivial before. So now we take the bosonic fields in this supergravity action. And so this supergravity theory, which is coupled to our field theory, and we give an expectation value to the bosonic fields, in particular to the metric, but also to the other bosonic fields, for instance to vector fields or to scalar fields. And then we take a rigid limit in which we send Newton constant to zero in such a way that we get rid of the extra equations of motion that we get from varying the supergravity fields. And then these bosonic fields are not constrained. And in this limit we keep fixed the background. So we don't impose the equations of motion, but we still want supersymmetry. And so we do impose the supersymmetry variations are zero. And so in particular we impose the vanishing of the gravitino variation. Now this variation in general takes the following form. Okay, I'm not sure, I mean probably this too depends on the conventions. So this is not particularly important. But okay, so this is the covariant derivative of the parameter epsilon. And then in general there is some matrix that depends which is constructed of the supergravity fields. And this function of this matrix will have vector and spinor indices and axillary on epsilon gives us this. Okay, so since I'm discussing general case here I'm not specifying what we have here. So let's call this a generalized killing spinor equation. But the important point here is that because we are working with off shell supersymmetry, what appear in these gravitino variations are only the supergravity fields. If we were doing these with the on shell formulation here in general we have complicated function of all the other fields in the theory. And in particular what we have the particular function of the fields will depend in the Lagrangian. It will be very much theory dependent. However, if we do it with off shell supergravity, only the supergravity fields appear here. You see the matter fields of our supersymmetric theory do not appear anywhere. And so in particular the form of this equation is completely independent of what theory we are trying to put on a core manifold, okay? And so this is the crucial point because we can discuss the problem of supersymmetry on a core manifold in a way which is almost independent of the particular theory that we are discussing, okay? We just have to solve this equation but this is an independent of the theory. It's not completely independent because well if we want to couple to a certain supercurrent mass, so if we want to use some off shell formulation of gravity we need that the theory contains a certain supercurrent multiplet and which type of supercurrent multiplet the theory has does depend on the theory. So there is a certain dependence, a mild dependence on the theory but after that dependence, after this fact has been taken into account there is no further dependence on the theory, okay? And we can stop discussing the theory and just look at this equation. Is this point clear? Because this is a crucial point. So the dependence on... Sorry, your question was about what? Yeah, I'm sure. Yeah, then it's... Yeah, then it's very, I mean you can still do it but it's very complicated because then this equation strongly depends on your theory, what is the action of your supersymmetric theory and so on. And probably this is the reason why, I mean this was maybe one of the crucial observations of the two-chain side but that if you do it in off shell supergravity the problem, the couples, from discussing a particular theory and we can just discuss this equation. Yes. Yes. Yes, this is very general. I mean when you couple the, I mean the way in which the graviton multiplet couples to all the other fields which I will call my supersymmetric theory. So you know vector multiplets, hypermultiple, chiral multiplets and so on. At the linearized level it couples this way. And of course there are higher order terms but the linearized level is how it couples. Yes, yes. It's compatible with what he said. So the ambiguity is that, so at the level of gravities as Stefan said, so you can take an on shell supergravity, you take some supergravity and there are different off shell formulations of it. So it was off shell formulations? Yes. Yes. Yeah, but precisely the ambiguity, he talked about you can even use a chiral multiplet as a compensator or a tensor multiplet as a compensator. So those give rise to different off shell supergravities. Are these the ambiguities you're talking about? Yes, yes. Yes, yes, because the one that he described with the chiral multiplet gives you the Ferrara-Zumino, the old minimal supergravity that couples to the Ferrara-Zumino multiplet. And then he mentioned the other one where you couple to the tensor compensator that give you the new minimal supergravity that comes to the R multiplet. And I asked him after one hour ago and he told me that even the S multiplet, there is some off shell supergravity where you have some other compensator that I don't know what it is, but. Yeah, I'm not going through the, somehow I'm jumping to the end result after the compensator as you have gauged away the compensator and you are left with some off shell formulation of supergravity which is not super conformal gravity. I'm not sure, I've never seen it. It might be possible. Okay. So, okay, so we have to solve this equation, this generalized Keating-Spinoff equation. And when we solve it, we have to solve it both for these supergravity fields, such as, of course, the metric, but possibly also some vector fields, some scalars. What do I write here? So these extra fields, and in fact these extra fields are what I will call the background. So when I say that we put a theorem on a manifold with a certain background, I mean that there is not just the metric of the manifold but there is also these other fields which are classical, but still we have to specify them. They control couplings in the final Lagrangian. But we also have to solve for epsilon, of course, in this equation. And for each solution at fixed values of these fields for each solution in epsilon, we have one more supercharge. So depending on how many solutions we can preserve here, then we have as many preserved supercharges on the code manifold, okay? And this number can vary. In particular, if you don't find any solution, it means that we cannot preserve supersymmetry. Okay, so in particular, let me stress, this maybe we have already said, but let me stress it one more that given a given theory, we can couple this theory to different, so as long as we have supercurrent multiplets, so a given theory might admit different supercurrent multiplets. So as long as this is the case, we can couple the very same theory to different offshore formulations of supergravity. And for each different offshore formulation of supergravity, we have a different equation that depends on different type of background fields. And so this will give us a different way to preserve supersymmetry on a code manifold, okay? Even though it's the very same theory, but this, so we should explore the various cases to see the various type of ways that we have to preserve supersymmetry on a code manifold. Okay, so for instance, there are four dimensional theories that admit both ferrazumino multiplet and R-multiplet, and that gives us rise to different ways of preserving supersymmetry on different classes of code manifolds. So once we have done this, essentially we are done, so we have solved the problem because first of all, we have found our backgrounds that can preserve supersymmetry on, so we have found the class of manifolds and backgrounds that admit supersymmetry, but also then we can take these backgrounds and we can plug them back both in the algebra of variations of the supergravity, of course the supergravity comes with what are the variations of the various fields under supersymmetry transformations. So we also get the deformed supersymmetry algebra and we also read off what is the action because we just plug in the values of these background fields in the full supergravity action and we are left with the deformed action for the matter fields, so deformed quantum filter reaction. Okay, so essentially this, if we find solution to this, this solve our problem. And I guess I have to stop here, okay.