 Okay. Microphone works. Very good. Yes. Okay. So this morning I started talking about supergravity. It's an on-shell version. It's off-shell version. And of course the off-shell version a little bit more complicated. And so now I want to go to n equals 2 or extended supergravity where we have two gravitini. And I will discuss a few aspects of black hole physics already here. And of course there's also the off-shell version that I will discuss. But first part of this lecture is about the on-shell. So let me start with writing down models or actions. And before, well, the lunch, I already gave the counting argument, the on-shell counting argument, why in n equals 2 supergravity we need a graviton but also a gravifolton and we need two gravitini. And so the action can be written very simple. I count the gravitini. I run from 1 to 2. And so this is of course not the most general supergravity theory. It is, in fact, it's quite the opposite. It's the minimal supergravity theory because it doesn't couple to any matter fields. There is no, there's of course a gauge but it's part of the Poincare multiplet, the multiplet that is needed to make the general coordinate transformations symmetries. And so later on, or maybe probably even tomorrow, we're going to couple this to vector multiplets and hyper multiplets. And these constructions are actually much more involved. And to do that properly, you need that off-shell and superconformal methods to write down and to find these metacoupled supergravities. And so, but for today's purposes, I'm going to just work with the minimal supergravity here. So this is minimal n equals 2 d equals 4 supergravity. So the minimal refers not to the minimal amount of supersymmetry, it refers to the minimal that there's no coupling to other fields. So these lectures are also supposed to be about black holes. I want to quickly discuss the black hole solutions. And so, you can write down the equations of motions. Now we have, of course, the possibility to have a rational nurse term black hole solution, which is electrically, possibly also magnetically charged. This is Einstein Maxwell theory. And so you can write down ds squared. This is the metric over r plus q squared g 4 pi r squared dt squared. Well, the relation between kappa and g newton is kappa squared equals 8 pi g n. I'm using the conventions of the book of Ferrara and Van Pruyne, Friedman and Van Pruyne. Apologies. You can erase this from the camera. Here's the same function. And I guess everybody has seen this. These are spherically symmetric black holes, which have electric charge. The electric charge is given by q. The mass is given by m. And there is also a field strength. Of course, you have to solve for the field strength as well. And there is one component, FTR, for an electrically charged black hole minus q over 4 pi r squared. The standard electric field that belongs to a particle with charge q. So this is the Reissner nurse term solution that lives happily in n equals 2, minimal supergravity. The gravitino is set to zero in this solution. And so this is a black hole that has an inner and an outer horizon. So pictorially, this is the minus and r plus. These horizons can be found by studying the zeroes of this function here. And so there's two horizons, r plus and r minus equals mg plus or minus square root 4 pi. And so you see that something special happens when this square root is equal to zero. This happens essentially in the appropriate units when the mass is equal to the charge. Then the square root is zero and the outer and the inner horizon coincide. That is the solution also that is called the extremal solution. And if you embed this in supergravity, which is the case here, it preserves half of the supercharges. So you can look at the variations of the gravitini. That depends. There's a covariant derivative that both contains F mu nu and the spin connection. You have to compute and compute. And then you find one solution for the killing spinner. That killing spinner is actually explicitly written in the book by Friedmann and von Brunnen. And now it's interesting to make jokes. So the book by Friedmann underwits. So good. So in this case here, in this model of supergravity, it happens to be so that the extremality conditions coinciding horizons is the same thing as preserving some supersymmetry. So only at that point when these coincide that you have a BPS solution. That happens in many cases in supergravity but it's not always the case. There are also examples where you have extremal black holes, especially in metacoupled and especially in ADS, where you have extremal black holes that are not supersymmetric. And so one has to specify a little bit the assumptions under which this is correct and not correct, but in these simple models that is the case. Good. So this is the electrically charged black hole solution. There's also a black hole solution that has both an electric and a magnetic solution. And so, well, I'll write that here that has a solution FTR equals minus q over 4 pi r squared and F theta phi equals minus i over 4 pi p sine theta. And so this is dual magnetic field strength if you want. And so then the metric here, in the metric, you can just change the q over here or the q squared goes into q squared plus p squared. And in fact, you can also see that the charge cannot be too large. If the charge is going to be too large, this term always dominates over this one. And then something dangerous happens at r equals 0. You get naked singularities. And so to avoid naked singularities, we must have that q squared plus p squared must be smaller than 4 pi m squared over g. And when the equation is satisfied or when the inequality is satisfied, that's precisely the case of extramality, which is demonstrated here for p equals 0. And p is equal to 0. q equals m squared is precisely when this is vanishing. But in general, the charge cannot be too large because then you create naked singularities. You want to avoid that, but also if you try to make such black holes, cosmic censorship prevents you from doing that by physical processes. Although these physical processes depend on these theorems, they depend on certain assumptions. But for this type of matter, these naked singularities can always be avoided. So this looks like a bound of the charges in terms of the mass. If you combine this with a superalgebra, you figure out that this is precisely also a bound on the central charge of the supersymmetry algebra. I won't go much into detail, but I might explain it a little bit more later or tomorrow. Yes. What is written here? Minus i, imagine a unit. Oh, yes. It's also not in my notes, but I'm sure there's some power of r. That's what you say, right? Why the i is there? That cannot be correct. That is to do with the conventions in the book. Here it's called a dual, and that involves an epsilon tensor, the epsilon tensor. Yeah. Let's do it this way. Yeah, well, that could be. Okay. I'll give you the phone number of... Yes, yeah. I'll give you the phone number of... Very good. So this is the solution. This is the bound, and so these are not the... When the bound is satisfied, it's supersymmetric, but these are not the only supersymmetric solutions. These are the only supersymmetric solutions for spherically symmetric objects. There are also objects that describe multi-centered black holes, and they are called the Papa Petru Majumdar solutions. So these are metrics of the following form, minus e to the 2u of x. I'll just slide right down this form for u in a moment, because e to the minus t u of x, and then we have dx dot dx, and then e to the minus u has a particular form. It is a harmonic function on R3. These are the coordinates that involve the radial direction and the two angles, and then an a of t is equal to 1 over square root 4 pi g e to the u. So these are BPS solutions, and by BPS solutions, I already mean that... Well, they preserve supersymmetry, but also here you see there's no q here. It's already assumed that the mass of the black hole is related to the charge in the same way for each center. So we have multiple centers. We have n centers here, and each of them carries a black hole with mass mi and charge qi, but qi is related to mi via this relation with the equality here. Yes, there is. In fact, it takes exactly this form here where you should have field strength around each center, and then the mi is related to 1 over 4 pi g. So what you switch on... Well, of course it depends on the gauge here, but you switch on a phi, and it looks... I didn't write down the expression in my notes, but it can be written down, and this is the relation between mass and charges. They are BPS solutions, so they are 1 half BPS, and this is a reflection of the fact that they are BPS. Otherwise, you would already see that in the case of one center, it's not the most general solution because the mass and the charge are related here. Okay. Are there any more questions about this? Good. Then before I go to off-shell stuff, I'll quickly repeat this program in the presence of a cosmological constant. So also this morning when I did n equals 1 supergravity, then I added a small deformation, and the deformation was adding a cosmological constant, and in such lagrangians for supergravity, anti-de-sitter is a solution, and this we call anti-de-sitter supergravity. And there's also a simple version of anti-de-sitter supergravity for n equals 2, and let me discuss that briefly. Maybe anticipating on what I'm going to do tomorrow and Wednesday, we're going to discuss gauge supergravities, and so what is there to be gauged? Well, what is there to be gauged? In this case, for instances, well, we have here a vector field and we have fermions, and you can ask the question, is the fermion neutral under this u1 or not? In a normal supergravity, it's not. In this covariant derivative, there's not an a mu here, and if you do that, you can make a theory in which this is a local symmetry, and that is called gauge supergravity. So essentially what you do is you take this Lagrangian, you do the following manipulator or substitutions, you substitute into this covariant derivative already contains a spin connection, but now on top of it, you add a gauge field like this, and that's right, or plus in my notes, I'm not entirely sure. I'm using here different conventions perhaps in the book, because this is not part of the book. And so you do that here in the action, and then you also add a term here times, well, r of course, and then there is a term 6g squared, and I have to double check whether there's other fermion bilinears in the action appearing. Well, the answer is yes, but you can usually put that in a 1.5 formalism in the spin connection. So it's not very relevant for what I'm going to say. So there's a cosmological constant, and the cosmological constant is given by minus 3g. I think in the notation of this morning, g was 1 over l, the AdS radius, and now I call it a coupling constant because I want to make it look like it's a gauging. I'm gauging something here. There's a covariant derivative with coupling constant g. So this is the theory. You can write down the supersymmetry transformation rules. Yes. So why do you need to gauge here? Well, in the n equals 1 case, we didn't have a gauge field that was not part of the Poincare multiplet. You can still give it an interpretation in terms of gauging, but then you have to explain a few more things that you are gauging. You're using an additional compensating multiplet, which contains a vector multiplet, and then you can use that gauge field to do a gauging. Here I have it already present, and you can sensibly ask the question, can you give the Gravitino a charge under this gauge field? The answer is yes. If you do that, it is... Well, you can give it whatever name you want, gauge supergravity or deform supergravity. You can understand it as a gauging because I'm covariantizing this action here, and I could not do that this morning because there was no gauge field present. No, they're not. Yes. This is part of a bigger framework in which the whole gauging procedure becomes something very simple, namely this. So ADS supergravities and gauge supergravities. ADS supergravity is just an example of a simple example of a gauge supergravity. Yes. Yes. Very good. Thank you. Yes. So the gauge supergravity does have flat solutions? Flat solutions? No, Minkowski's space is not a solution here unless you switch off the gauging, G equals 0. So flat space time is not a solution of this theory. Well, it is a solution of this theory, but not of this theory. So all the space times here, so onto the sitter in four dimensions is a solution of this equation. And what I'm going to write down now is I'm going to put a black hole into onto the sitter and that asymptotes to ADS. So Minkowski's space is not a solution of this theory unless you put G equals 0, but then go back to the ungauge theory. So we can write down metrics. dS squared equals, I'm going to discuss the Reissner-Nursturm type solutions, u squared of r dt squared plus u minus 2 of r. Yes, sometimes I use e to the u, sometimes u. Forgive me, my notations. That is because this part here is not covered in the book, so I took this from another paper. I might want to call this u tilde or something. So the Reissner-Nursturm in ADS 4 looks as follows. So it is u squared of r equals 1 minus 2 m over r plus q electric squared plus, now I'm also putting G Newton to 1. You like this? I switch from Russian to Italian. Very good. So this is a solution to the equations of motion here and you see, for instance, if you put all these charges to 0 and the mass to 0, then it's just 1 plus G squared r squared. That's just onto the sitter space with nothing in it. And then you can put an object in onto the sitter space. You can give it mass and charge, m q, q electric and q magnetic and here are the expressions for the gauge field. You can ask yourself which of these solutions is supersymmetric. And this problem was analyzed a long time ago by Romans. So the BPS solutions in a paper by Larry Romans in 92, he took this class of solutions and plugged it into the variations of the Gravitino and set it to 0 and see for which constraints on the parameters can you get a supersymmetric BPS solution. So the BPS solutions are, there's two classes. The first class is the class where the magnetic charge is equal to 0 and the mass is equal to q electric. And then you get the mass and that one is 1 half BPS. And what we have then is u squared is equal to 1 minus q e squared over r squared plus G squared r squared. So that follows from imposing supersymmetric right? Yes, of course. It's also wrong in my notes. Very good. So you see that u is always a positive, it's a sum of two positive terms and in the absence of a cosmological constant there was no such thing and this function had zeros. If this function has zeros is precisely where the horizon is. The horizon would be then at q, well, at r equals q, I'm sorry. But now in the presence of this term you see that there is no zero of this function. That means there is no horizon. And so the point r equals zero is still where the metric shows singularities and that's a naked singularity. So this is a BPS solution of an electrically charged object if you want but it has a naked singularity. So here you see that an anti-decider or engaged supergravity, the notion of the BPS, well, the cosmic censorship and the BPS condition, they are two different concepts in the sense that imposing BPS is not imposing satisfying the BPS bound and avoiding naked singularities. So no magnetic charge and mass equal to electric charge. The second class of solutions is called the cosmic diam. It has mass equal to zero. We have to carefully define what we mean by mass and ADS. But if we just see there's a parameter here, it's m equals zero and then q, m is equal to plus or minus one over 2g and the electric charge can be anything. If you take this choice here, the function u then becomes u squared is gr plus one over 2 gr squared plus q electric squared over r squared. And you see again that this is the sum of positive terms. It has no zero. So that means no horizon. And so also a naked singularity, both this one has naked singularity and this one has naked singularity at r equal zero. This cosmic diam solution here preserves not one-quarter BPS, one-half BPS, but one-quarter BPS. That's a feature that we're going to see again in anti-decider space times. You can have one-quarter BPS solutions in n equals 2 supergravities. So they preserve two supercharges out of eight. These are eight supercharges, two Majoranas and one Majoranas, four real degrees of freedom. You have two of them. So that's eight. One-half BPS means it preserves four supercharges, one-quarter means it preserves two supercharges. In un-gauge supergravity or minimal supergravity, there's no one-quarter BPS solutions. Very good. So actually for a long time it was thought that in anti-decider there's no interesting black hole physics to be done with BPS solutions, sort of partly because of this reason. There were no honors to God at least in those days. And of course later on, various people have contributed to it, a black hole solution was, a spherically symmetric black hole solution was found by Clement Cacciatori, which has magnetic charge, but you need to add scalar fields. You need to go to matter-coupled supergravity. So this is also what I will do. And I think that that's also one of the black hole solutions that will be discussed in Alberto's talk, I suppose. And so we'll get to that point. Okay. Are there any questions? Yes? In previous lecture, in the minimal case, we don't have dynamic count for A mu. How do we get what? Sir John from non-dynamic. So, I'm not entirely sure. I understood the question. So this morning I did n equals one supergravity. There was no A mu. And now with n equals two, with extended supersymmetry, I have an A mu because I need to match the on-shell, I need to match the degrees of freedom. But that's maybe not your question. Yes, that's right. That's an n equals one. There, A mu is an auxiliary field. Here it's not an auxiliary field. It propagates and it has a standard kinetic term. I will now go to the off-shell formulation of n equals two supergravity. And lots of stuff happens there and it becomes complicated. But I want to talk you through this. Yes, please. Any questions? But here, why we cannot just turn off the electromagnetic field? The electromagnetic field. And then you just put a massive term to it. Yeah, but this is a math for the gravitinos. So if you want to look for bosonic solutions, of course this includes also you can switch off the electromagnetic field and then you have a Schwarzschild black hole in onto the sitter which contains just this term, one minus two m plus g squared r squared. That's a solution of both n equals two supergravity. Well, you can then also switch on or off g. If you also switch that off, then you have the Schwarzschild solution. If you switch it on. But not A ds. So I said wrong, find A ds solution equals to two without talking about it. Yeah, but this master, this master was a math, well, it's a math like, it's a, it's a, it's a bilinear term in the fermions. Maybe it was a bit confusing to call that mass. Maybe not. But we're looking for black hole solutions where the gravitino are set to zero. All these solutions have psi equals zero. My question is about A ds. It's not about black hole. Yes. I would like to know if it's possible to find A ds solutions for any question to solve it. Put in a master as we did in question. Sure. Just put m to zero and all the q's to zero. Just take this term and this term. Then A ds is a solution. And that's a maximally supersymmetric solution. It preserves all the supercharges. Yeah, yeah. So that master, what you call master that sits somewhere in here, but that uses 1.5 system. I didn't want to go in there. Yes. There was here a dot, dot, dot. But these terms are there. Yes. Yes. When you have a master, static, 1.5 bps, elliptically drive black hole? No. They will, they will have negative singularities. Is that a problem or is it just an experimental fact? If you put a rotation, you can have those, right? You should talk more about rotation. Because there was a singularity for the magnetism. Yes, yes, yes. I always have to be careful with calling, with, with theorems. I, I don't know of a theorem or where that would be written. But I don't know any example either. But perhaps somebody in the audience can correct me if I'm wrong. So, yeah, I don't, I don't, I don't think so. Good. Good. Now we go to, so this is all I have to say about minimal, minimal supergravity in its shell version. I will now discuss a few elements of off-shell supergravity. And so we'll have to go through the whole business of the wild multiplet. And so this wild multiplet and, and the coupling to supergravity that was developed in the 80s by the Wittem-Vampurian. And so I had to learn this stuff as well at some point. Very good. So off-shell n equals 2 supergravity. You see for the purposes of localization, this off-shell business is, is important. Or at least there's lots of advantages. It's not compulsory. But when you work on shell, you have to study case by case, model by model. Because the field equations vary from model to model. So you're going to have off-shell formulations. The whole program of localization in field theory. But also, in fact, if you want to do something more than just explaining the leading order entropy of black holes, then, well, Samir will discuss that. Artish also knows much more about it than I do. Very good. So let me give you some basics on this. So we do the same kind of tricks. Conceptually I'm not doing anything different from this morning. What we want to do is we want to construct supergravity by first consider or constructing a bigger theory that has more symmetries, namely the symmetries realized by the superconformal algebra. And then we gauge fix. We gauge fix those generators in the superconformal algebra that are not part of the Poincare. That's always the same idea. And that trick allows us to go off-shell because the wild multiplet in the superconformal algebra, they're actually, it's actually possible to construct that off-shell in a not too difficult way. So we need the superconformal methods. And so we need the n equals 2 wild multiplet. And I'm going to just, for the moment, just give you, again, a counting device how to see that things can be realized off-shell. So the superconformal algebra is now SU 2.2 slash 2. It contains bosonic and fermionic generators. And the bosonic subalgebra of this is SO 4.2 or SO 2.2. But now the r symmetries are larger. In n equals 2, the r symmetry group is SU 2 cross U 1. And in the superconformal theory, this r symmetry is really realized as a symmetry of the Lagrangian. Otherwise, because it's part of the superconformal algebra, the Lagrangian must be invariant under this full group. And on top of it, we have, of course, the q supersymmetries and the S supersymmetries. Let me repeat, the S supersymmetries are basically the superpartners of the special conformal transformations that sit in here. So when we construct this wild multiplet, we have to introduce gauge fields for all these generators. And so let me here make a list of these gauge fields. So we have emu a, that's basically the metric, for general coordinate transfer or for translations. We have b mu. For the dilatations, we have v mu ij. These are gauge fields for the r symmetry group. So this is a non-Abelian gauge field, if you want, v mu ij. And we have curly a mu, which is the gauge field for this u1r symmetry. Now, we can do accounting again without introducing these compensating multiplets. And if we count here, we counted here that off-shell, we have six degrees of freedom. But you can use the dilatation, the scale symmetry, to reduce this to five. So this is five. And what I've gauge fixed then are the general coordinate transformations plus the dilatation. It's the same counting as this morning. v mu, same story. It has no off-shell degrees of freedom. The reason is that I'm using special conformal transformations. These are precisely d generators or four generators. I can use them to set v mu equal to zero. So there's no degrees of freedom in here. Now, v mu, this is an SU2. SU2 has three generators, but I have four components. So I would have four times three. That's 12 component fields. But, of course, there is gauge symmetry. There's three generators here. So you get 12 minus three. That's nine degrees of freedom. And a mu, we have four fields. One generator. That's three. Now, so if you can, well, the notation here is just i and v mu. ij run from one to two. And so if we're going to covariantize here, there's going to be a gauge field acting on the gravitinos with SU2 labels, v mu ij. Good. So then we have in the fermionic sector, we have psi mu, i, alpha. And then we have the fermionic and that's, remember that off-shell, we would have 12 degrees of freedom. But now there is S supersymmetry. So we can reduce that from 12 to 8. But I have two gravitinos. So that's two times eight. That's 16. And what is gauge fixed here is q and S. So if you now edit up, so we see here that's 12 and that's five. That's 17. That's 16. So that doesn't quite work. And so we need some more auxiliary fields to make the counting work between bolsons and fermions. Now, there's a little bit of a story how people figure this out, this multiplet. But I'm just going to give the answer. The answer is that you have to add here another anti-symmetric self-dual tensor. It's complex and a single scalar d. So this t is a field. You can also call it t mu nu. I go from flat indices to curved indices with field binds. So this is a complex anti-symmetric field that would have six degrees of freedom. But it's anti-self-dual. That's what I mean by this minus. So that reduces it again to three complex or six real. And d is a real field. So that's one. And in the fermionic sector, we need an auxiliary field, which is a spinner, a chi, a alpha. And so this is four components times two. So that's eight. So if we add here, 16 plus eight is 24. And if we add here, one plus nine, that's 10, 19 plus five is also 24. So the wild multiplet consists of these bosonic fields and these fermionic fields. There's more gauge fields. But these are the dependent gauge fields. We had this slightly confusing discussion this morning. There's gauge fields that are just like the spin connection that can be eliminated because they depend on the other fields. They can be written in terms of the other fields. That can be understood in different ways, either via constraints or by equations of motion. I was not allowed to go in the equation of motion which is off-shell, but that's the difference between second order and first order formalism. It would be still off-shell in a second order formalism. And so I didn't want to go into this constraint. There's also another way of saying we know that if we gauge translations or the Lorentz symmetries, eventually it's all part of the same local coordinate transformation. So surely the spin connection is not an independent field, et cetera. So 24 plus 24 is the counting of the, and this is the field content. And this field content is going to be important also in future lectures. So just to get a little bit of an intuition how this multiplet is realized because I've just written down fields. And so these fields transform under all these transformations. I'm not going to write down everything, but let me just give some examples of transformations under S supersymmetry because you might be least familiar with S supersymmetry. Then for instance we have that the dilatation gauge fields transforms like eta bar, i psi ui plus Hermitian conjugate. That gauge field a mu i eta bar i. We have the same combination here, but this is the imaginary part. We have the gravitino, how does the gravitino or the gravitini transform under S supersymmetry by minus gamma mu eta i, et cetera, et cetera. So just to get, well, to make it a little bit more explicit. These are, this is how the S supersymmetry looks like. And there's also formulas for the variation of v and the variation of chi, et cetera. Let me just write down one more delta chi a, alpha, sorry, equals one over 12 gamma a b. And here you see this t, this anti-self-dual t field, epsilon ij eta j. Is there another transformation you need, Samir? Or you haven't prepared your lectures? Very good. I can give more details if needed. This is just some, to, to, to, so those close off shell, if you compute the commutator, everything is off shell. This whole 24 plus 24 dimensional multiplet is, is an off shell realization of the super conformal algebra. Yes. Yes. Does it mean that I can get rid of it or you can get rid of it? You can get rid of it, yes. And in fact, that's also, but it's a gauge fixing condition. So when we go from the super conformal Lagrangian, there is a b mu. But when I go to Poincare supergravity, then I just set by hand b mu to zero. It's an admissible gauge choice. And I'm fixing now the special conformal transformations. And then it's gone. Yeah. Well, all the other, well, yes, well, all the other ones you have to keep. Well, these will be composite connections later on. We will see that you have to eliminate them as well, not by gauge or by, by solving their equations of motion. They will, they will appear in the Lagrangian algebraically. And so they will become also, if you go on shell, they will become functions of the matter fields that I haven't introduced yet. Very good. So now the compensator, because I have here a multiplet, but I cannot write down an action, at least not a two derivative action. I cannot write down a two derivative action based on the while multiplet itself. The real, well, there's maybe again different ways of explaining this, but in the simple example of just gravity, I needed to have a scalar field in the, in the model in order to compensate for the lacking, compensate. Now, here there is a, a scalar field D, but that's not the one that does the job. That's harder to see perhaps or not, not, not possible with the information I've given you. But we need compensating multiplets and, and we need one vector multiplet and a vector multiplet contains an AMU. It contains a complex scalar X and it contains, well, a Dirac fermion or two Myranas or two Wiles, if you want. And we also need one hyper-multiplet. And a hyper-multiplet contains four real scalars, QU, Q is a field, a scalar field, and U runs from one to four and it contains fermions, zeta i, i is again, runs from one to two. And so these are the, that's a hyper-multiplet. This is an on-shell multiplet, but that's sufficient for the purposes of constructing this supergravity. So important is that a while multiplet is off-shell. So this AMU is becoming, is going to become the gravity photon. That's why I call this curly AMU and not capital AMU. The gravity photon can be understood as part of the compensator in a vector multiplet and n equals two. This one X here, I'm going to gauge it away. I'm going to gauge it away by using dilatations and sorry, not by dilatations. I'm going to gauge it away by U1. I have a U1 gauge symmetry here. And it turns out also that there is an equation of motion of the, that I will explain that perhaps tomorrow, that allows us to get rid of this. So this can be gauge the way or eliminate X is a complex scalar field and n equals two vector multiplet has a gauge field, a complex scalar and two chiral spinners. This is the complex scalar X. So there's no ij. Oh, you're quite right that this is, I did something wrong. There is a triplet of, yes, a y ij that is missing here. So very good. Otherwise, this is not an off-shell multiplet. Yes, you're right. So this one here is going to be eliminated by U1. I can gauge fix. I have local symmetries here. This U1 and the equation of motion of D. How the equation of motion D comes about. I have not explained to you. I'm just giving here some counting arguments how, how that all fits. And then this one is still back. This one is going to be eliminated by the equation of motion of chi i. We have here chi i's and they have their equation of motion. They should not be part eventually of the Poincare multiplet. And so it's equation of motion will in fact also eliminate the fermionic part of the compensator. So these q's are gauged away dilatations plus SU2R, because a hypermultiple transforms under SU2R. And these here are gauged away by S supersymmetry. Because I really wanted to have gravitinos that have 12 degrees of freedom. So I will augment this here. This 16 is 2 times 8. This 8 becomes 12 again. So we got 2 times 12 is 24. And these ones are eliminated. So I basically reshuffle how the counting goes. I don't impose S supersymmetry here on my gravitini. I'm going to use S supersymmetry to gauge away this compensating. And then I have here my 2 times 12 degrees of freedom of a gravitino. So what is left over? Let's see what is left over. Well, we have emu A and we have emu. That's this gravifold on here. And all the rest here in the bosonic sector is either gauged away or can be eliminated by equations of motion. So also this here is equation of motion. A mu and this one can also be eliminated by equation of motion. So we are left over only with these fields and with the fermions psi mu i. So this is my 2 plus 2 being 4 bosonic degrees of freedom. And so this is 2 plus, well, 4 plus 4 on shell. And of course I have not given you all the details of how to eliminate these V and A. But I want to do that, I want to do that tomorrow. And so this is the wire multiplet. It's an off shell multiplet. There's two compensator. One of these compensator contains the gravifold on and the rest is stuff that you get rid of in the gauge fixing procedure. But if you stay on shell then these fields, auxiliary fields, remain in the game. And the hypermultiplet is completely gauged away and eliminated. So the coupling to matter and super graph, the reason why I'm not doing this here in detail is that I want to do it tomorrow but in more general details. I want to just change these numbers right here. Number of vector multiplets plus 1. And I want to introduce here number of hypermultiplets plus 1. Make it conformal. We know how to make conformal, at least classically conformal vector multiplets. I will show you also how it goes with hypermultiplets. And then we do this gauge fixing procedure again. And then we end up with non-minimal supergravity containing the Poincare multiplet. But now it's coupled to physical vector multiplets and physical hypermultiplets. The compensator is gone. So I end up with NH hypermultiplets. This compensator is also gone. The only thing is left over is the gravifold. So you have always one additional vector field on top of, which is a gravifold, on top of the NV vector multiplets that you have in the theory. And so I will sketch a little bit tomorrow how you do the construction of the action directly in more general terms by writing down the while-multiplet couplings to all these multiplets combined together. And then I will show how the equation of motion for D and so on is constructed. And then I will do black holes. And I will do gauge supergravities. And then time. Thank you.