 Does the mic work? Maybe I'll speak loud enough as well. OK, very good. So yesterday, I spent some time in introducing you to this wild multiplat and these super conformal methods to actually construct supergravity theories that can potentially also be coupled to matter, not potentially, but today I will do the couplings to matter. And so whether you're on shell or off shell, even this conformal calculus, as it is called, is useful to construct Lagrangians for matter couplings to supergravity. And historically, also this is how it was constructed by David Lowers and van Poeren. So today, I want to discuss a bit matter couplings to n equals 2 supergravity. And you will see here that how these conformal calculus, these conformal methods, actually are used in practice. This is not the only method to construct supergravity theories. There is other frameworks. Here in Italy, there is a whole school on supergravity, of course, that essentially originated by Ferrara and Pietro Frey and Doria and company. And so these methods are very similar, but not identical. At the end of the day, you produce the same kind of results. So I will stick to the conformal approach here because that's what I learned myself and also because the conformal techniques and conformal algebras, of course, are in fact also useful for superconformal field theories, et cetera. So very good. There is two multiplets that I will discuss in this lecture. And this is the n equals 2 vector multiplet. And such a multiplet, this is an off-shell multiplet, contains, of course, a vector field. And I will introduce a bunch of them, labeled by capital index i. It contains a complex scalar field, xi. And then there is per-multiplet. There is three auxiliary fields. I denote them by y with a vector notation. And then there is two per-multiplet, basically one Dirac fermion or two Majoranas, if you want. And this is an off-shell multiplet. And it's studied, of course, a lot also in n equals 2 supersymmetric field theories and Cyberg-Witton theory, where you also learn that the dynamics of these multiplets are governed by a pre-potential, governed by, I mean, if I want to construct actions, you can write down a holomorphic superspace integral. And that you integrate a holomorphic function, which is called the pre-potential of x. It's holomorphic in x. It does not depend on x-bar. And of course, I'm going to be imposing that this whole theory is going to be superconformal, classically at least. And not all pre-potentials allow for superconformal couplings. I'm going to later couple this to the well-multiplet. And in order for this to be a superconformal theory, that one of the constraints that come out is that this is homogeneous of second degree. Mogeneous of second degree means, in this particular case, that this is equal to, if I take the derivative, this is what I call fi. So fi is the first derivative of f with respect to x. I contract the game with x. I get two times the pre-potential. So it's homogeneous of second degree as a consequence of imposing dilatation symmetry. So now I'm going to write down the kinetic terms. I'm going to focus mostly on the scale. Everybody knows supergravity Lagrangians are long. If you write down all the fermions, I just want to give you all the crucial conceptual ingredients in this conformal calculus method. So I will focus on the scalar fields. I could also do the vector fields, but I chose to do the scalar fields here. So in the scalar field, in the scalar sector, we get terms that look like follows. I will introduce the notation or define the notation in a moment. This is essentially the most important or the kinetic terms or the quadratic terms in the kinetic terms for the scalars. Of course, these are auxiliary fields. They are only algebraic. And so what do we have here? We have this nij is minus i fij plus i f bar ij, where these are just the second derivatives of the pre-potential. So that's the notation. f is the pre-potential. fi is the first derivative of fij. The second derivative. This is complex conjugate. And so here, d mu x are covariant derivatives. And here, we see the coupling to the while-multiplet arising. Because we'll have to make the theory super conformal. I need to covariantize and introduce the gauge fields. And the relevant gauge fields here are the gauge field for dilatation. That's b mu. And then there was a u on r symmetry that in the super conformal approach is a local gauge symmetry. And the gauge field is called a mu or curly a mu. And so in this Lagrangian here, in this approach, well, later on, we can gauge fix this b mu to 0. So that is not going to play a role in the action because the special conformal transformations are allowed to set b mu to 0. So here, there is a mu. And one important fact is to notice that the a mu itself will not have a kinetic term. It just appears in these covariant derivatives. And if it has no kinetic term, I can later eliminate it algebraically using its own equation of motion. I will do that in a moment. These are just the auxiliary fields. And this is, in fact, invariant. Well, if I would also write down the other terms, I will be invariant under the full super conformal algebra. So these are the terms that I want to focus on for the purpose of this lecture. Are there any questions? Very good. The second multiplet is going to be the hypermultiplet. And a hypermultiplet has four scalars for each hypermultiplet. And here, we don't have an off-shell formulation for generic hypermultiplets. So unless they have special properties such that you can dualize them into tensor multiplets, they won't have off-shell descriptions. So the construction there will eventually lead to on-shell super gravities. So n equals 2 hypermultiplets. Hm is hypermultiplet. We're going to denote them. Well, yesterday I called them q. Today I will call them phi. And a runs from 1 to 4 times n. n equals 2 supersymmetry requires already whether you couple to supergravity or not. The number of scalars has to be a multiple of 4. And if you couple to a rigid supersymmetry, you get nonlinear sigma models. And they should be parametrizing or co-ordinateizing hypercalium manifolds, whereas in supergravity, quaternion scalars. These are all geometrical structures that are not so important for the study of black hole physics because the hypermultiplets will not play that important role. But for the purpose of the conformal calculus, it is useful to introduce them to see everything at work. So we have Lagrangians for these hypermultiplet fields. Of course, there's also the fermions, but I'm suppressing them. Minus 1 half, my notation. And we have d mu phi A. I should also call it maybe d mu phi B. And so this is a nonlinear sigma model with a metric GAB. And we have covariant derivatives here. And then the covariant derivative here is let me first write it down and then explain the notation d mu phi A minus chi A B mu plus 1 fourth k dot A dot V mu. So let me talk you through this. We have a besides supersymmetry here that enforces this metric to be of the hypercaler type. We also want to impose dilatations. And also the SU2 symmetries. Now for a metric to have, so for the vector multiplets, the imposing the dilatation symmetry forced the pre-potential to be homogenous of second degree. Here there will be a similar constraint, but it's all more phrased geometrically. It forces GAB to have a homothetic killing vector. That means it's a killing vector that is not an isometry, but it should be an isometry up to rescaling. So it's a special type of a conformal killing vector. And that's conformal killing vector is denoted by chi A. So this chi A satisfies dA chi B equals delta AB. So if you lower an index with this GAB, you get dA chi B equals symmetric GAB. So this is a particular type of a conformal killing vector where it's already symmetric in its indices when you take, without writing the dB chi A term. And this means if you take the anti-symmetric part that, in fact, this chi A should be the derivative of some function chi, which is called a hypercalic potential. Not super important for what I want to say about black holes, but if you want to understand this conformal calculus, you'll need to understand a little bit more about the geometry of hypercalic manifolds with scaling symmetries. Essentially, it means it's a cone, and if you move along the cone, you act with the dilatation. So there's a radial variable along the cone. And that radial coordinate can be seen as this function chi. This is all on-shell hypermultiplets, unless you have also triholomorphic isometries and stuff like this. You can dualize to tensor multiplet in that case. And then there is an off-shell formulation. You can also think about doing harmonic superspace or projected superspace. And then you can do more than just have off-shell descriptions with or you can relax the assumption of having isometries. But I will suppress all these things for today's talk. For black hole physics, all the hypermultiplets do not play a role. I'm going to freeze them to constant values. What I want to do here is just show how that elimination of the well-multiplet fields, how that works. So this is the first term. So you see the scale transformations, b mu, they're not linearly realized in general. So they have a conformal killing vector chi a. The second term arises when you need to impose the SU2R symmetry. So the R symmetry is SU2 cross U1. The U1 acts on the vector-multiplet fields. Here's the U1. And the SU2 acts on the hypermultiplet fields. Now, that SU2 is also not generically a symmetry of the Lagrangian unless this metric has SU2 isometries. So you're going to impose that this metric has both scaling properties and SU2 isometries. The killing vectors of this isometry are denoted by ka. I have three of them, SU2. And the gauge field is the gauge field of the well-multiplet. Yesterday, I used ij indices for, well, it's a matter of taste. SU2R doublet indices. You can also use the triplet notation. OK, so this is the covariate derivative. This is the metric. This metric has a special property on top of being hypercaler. It's a cone. And it's a cone with SU2 isometries. That's called a hypercaler cone. And I can give you a separate lecture on that. This is where I worked on in the past myself. Very good. So now, the super conformal tensor calculus is actually quite easy in the sense that what you do now is you gauge fix this B mu to 0. Well, that's easy. So you just delete it. And you integrate out. You eliminate the A mu field by its own field equation. And similarly, you integrate out the V mu by its own field equation. What happens then is the following. So then we get A mu equals 1 over 2 x bar nx f bar i, the left-right derivative. This x bar nx is just x bar i, nij, and then x of this in a product. So this is the solution of its own field equation. And similarly for V mu, it is minus 2 d mu phi A. And then something that is called V A. This is essentially function of phi. If you go through the equations of motion, you can explicitly write it down. It has a geometrical meaning. It is the SP1 connection that eventually will be present in all quaternionic manifolds because that defines the holonomy group. The explicit expression is not super important here. So you see when you plug back this A mu into these equations, then, well, you have to plug it in here. But it appears quadratically. You see that because of these derivatives, you will get, again, d mu x, d mu x terms. So you basically will get corrections. If you group that together with the nij terms and the flat derivative, you will get corrections to this metric here nij. And that is precisely what a quotient is. What you are doing here is nothing geometrically. It's nothing else but a quotient. And we're quotienting with respect to the super conformal group. In general, for those who know what quotients are, quotients in physics can be understood as you gauge a certain symmetry without introducing kinetic term for the gauge fields. And then there appear algebraically. You integrate them out. And then you reduce the theory to one of lower dimensions. And that also arises here. So now I'm going to write down the result of the action after this exercise. It's plugging in. You can all do that. That's just computing. And so the action that we get is, in fact, well, it's the determinant of the metric I'll bring to the. So I only focus on the kinetic terms, and not the vector fields kinetic term, but just the there's also, of course, the gauge fields of the vector multiplet. I'm not including them in the discussion here. So it's the kinetic terms from the scalar fields for both hypers and vectors, but also the metric I'm going to write down. And so what you get is something like this here. Let me first write it down. So I'll discuss it term by term. So this is what you get after you've plugged in the value for the gauge field here. This Mij was the old Nij, but there's now correction terms. You can compute these correction terms. The whole thing I denote Mij. This Mij has, again, a geometrical meaning. It is the metric on a special scalar manifold. And it differs a bit from the rigid special scalar geometry. But it's still determined in terms of a pre-potential. The second term is the kinetic term for the hypercaler fields or the hypermultiplets. And so here we see this hypercaler potential. And here we see the metric. It's not the old metric, which I denoted by little g. There's correction terms to it. And I put them all together. I call it capital G. And capital G will eventually be the metric on a quaternion scalar space instead of a hypercaler space. Then come the more interesting terms, perhaps. So here we see the rigid scalar appearing. I have not told you exactly where that comes from. In fact, when you go through the wild multiplet calculus, then at some point you learn that the gauge field for the special conformal transformations, it's not an independent field. You can write it in terms of the other fields, the field bind, and so on. And when you go through the details, it's precisely that term that generates the rigid scalar. It generates other terms here. And the totality of that is captured by these two lines here. Say again. For sure there should be a bracket here. But I don't hear you very well. A bar? Logarithm. Teamwork. Log. Elend is log. Oh, Elend is log. Yeah, log. Yes. Yes, it's the logarithm. So small GAB is a hypercaler metric. Yes. Capital GAB is a quaternion. Yes. It's a little bit more subtle because you see that I still have not reduced the number of fields. This GAB is still the same dimensions. But it will have a couple of zeros. It's a degenerate metric. I cannot invert it. If you cut off these zeros, you remain with a smaller space. That's what happens after you do the quotient, essentially. And that one is then a quaternion scalar metric. So this is special scalar. For those who know it, it's just a word. It's the word for the thing that you get. This is how it was discovered, this geometry. It's the word for what you get if you do this conformal approach. Later, this has been understood more mathematically in terms of scalar hodge manifolds and bundles. But this is how it was discovered also historically. And then by the Wittem van Poorn. And this is a quaternion metric. Actually, it's called quaternion scalar. But it's not a scalar manifold. And so then we get here these terms. And here, there's this d field. Remember that yesterday in the while multiplet, I gave all these fields and we did some counting. And there were some auxiliary fields. One was called T mu nu, or TAB, an anti-self dual tensor. And that tensor is not here on the blackboard. It talks to the gauge fields. It talks to the vectors and the vector multiplet. It's not written on the blackboard here. But d, there was another auxiliary field, a real scalar. d is actually coupling to the scalar fields. And it only arises linearly. Only arises linearly. And so that's the field that is very easy to eliminate. So if you put here the equation of motion of d, then you see that x bar i and ij, xj, should be equal to 1 half chi, the hypercal potential. And so what you do then is you take that constraint. You plug it back into the action. And then you see that this term and this term, they become actually equal. And this term and this term, they also become equal. But of course, the signs are such that everything works out well. They don't cancel. That would be a disaster. They just combine. And what you then do is also what you see, for instance, you get this combination times the Ricci scalar. Now you see the role of the compensator, not just as a single field, but it's composed of all the fields in the matter multiplets, hypermultiplets and vector multiplets. They all are multiplying. It's all part of the compensator. So the d equation of motion relates them. And I'm going to now also, of course, fix the dilatation gauge and set this to a constant, essentially just one of our nuisance constant. This is dilatation gauge. It's one constraint or one gauge fixing condition. Then you get Einstein-Hilbert. When it's a constant, then these terms luckily also drop, because what to do with this term? They drop out and you get Einstein-Hilbert coupled to matter multiplets. And that's it. That's it as far as the scalar fields are concerned. Then there is the vector multiple or the vector fields. It's a similar story. And the fermions, it's also a similar story. And so what we then get is we get Poincare's supergravity. And I've eliminated also the auxiliary fields of the vector multiplets. They only play a more important role if you consider gaugings. So I've not written them down here. That was a question. Of course, you have to take care of everything. And this has been done properly, of course. And you see another consequence of this. Maybe it's saying the same thing as what you're saying in a different way. If you put this to a constant, you see that you put a constraint on these scalar fields. So you can basically eliminate one of the scalar fields. And if you do the same with the u1, you can eliminate another one. So for instance, what you can do is you can take, let's say, the 0th component of, not the 0th component. If you take the 0th multiplet, x0, you can set x0 to 1, for instance. That's right, yes. And of course, then your manifold of scalar fields become of lower dimensions. That's precisely what a quotient does. And so you have to do that consistently, that your gauge choices are admissible, but people have done that, of course. And similarly, with this thing here, the quaternion-calor manifold that you get here is a four-dimension lower than the hypercalor one. That's because you can impose a gauge, the SU2 gauge, to eliminate three scalar fields. And that's precisely reflected in the property that this GAB, although the indices run over the same, it has four zero eigenvalues. So you have to gauge, fix them away, or these rows and column away. And what you left over is a manifold of lower dimensions. And that all works out consistently. So you got n equals 2.0, coupled to vector multiplets and hypermultiplets. And these models, they are very important, of course, also in string theory, because they arise in Calabio compactifications of type 2a and type 2b strings. And the number of vector multiplets and hypermultiplets is determined by the algebraic properties of the Calabio manifold. The Hodge numbers of a Calabio manifold determine, are in one-to-one correspondence, to the number of vector multiplets and the number of hypermultiplets. So in a Calabio 3-fold, the two independent Hodge numbers are h11 and h12, for those who understand this terminology. The number of vector multiplets is essentially h11. And the number of hypermultiplets is h12. If you're in type 2a and if you're in type 2b, it's precisely the other way around. That's, of course, important to have that connection, because when we study black hole physics using string theory, we want to have the microscopic explanation or description. And then you need to know the brains that wrap the cycles and how many multiplets are there, and so on. I will not do the microscopics in these lectures, perhaps Joao and Samir will say more about this. Very good. Perfect. Are there any questions about this? So I'm going to illustrate now this further, or I'm going to discuss a particular example, because now I'm done with the metric couplings. Now I want to go back to black hole physics and study black hole solutions in these kind of theories. To do that, I first want to essentially start with a very particular example where there is no hypermultiplets, except the compensator, of course. But after you've eliminated the compensator, that's gone. So in the Poincare theory, there will be no hypermultiplets, but only vector multiplets. The reason why hypermultiplets are not so important for black hole physics, at least in ungaged theory, is because the hypermultiplets, they are not charged. And typically, BPS black hole solutions, they are charged, like Reissner-Nurse term. So what gives the black hole charge are the vectors of the vector multiplet, of course. And so what is going to play a role are the scalar fields in the vector multiplet and the vectors themselves, because they charge the black hole. But the hypermultiplets, they're decoupled from the theory. There's also no interactions between vectors and hypers. So in any solution, you can take phi to be constant, and then this disappears from the equation of motion. Good. So in the example, it's actually one of the first, I think it's the first example where also the notion of the attractor mechanism was explained. The attractor mechanism discovered by Ferrara Kalosz, Sturminger Gibbons, there's a few people involved. And so let me discuss this. So example, we take one vector multiplet and zero hypermultiplets. Remember, when you do the conformal calculus, you start with two vector multiplets, one is compensating, and one vector becomes a gravity photon. And we start with one hypermultiplet here, but it's eliminated through the calculus. So this is in the Poincare theory. No hypermultiplets, just one factor. So that means that the bosonic fields are g mu nu, m u 0, m u 1. This is the one from the first vector multiplet. This is the second. You can call this a gravity photon if you want, but that's just semantics. And we will have one complex scalar. You can call that x 0, but usually you take ratios between the x's and a bit of a story. Let me call it just z. And then I will call this a mu and a mu prime because I want to follow the notation in the book here. And then the action, I have to make a choice for a pre-potential f of x equals i or minus i over kappa squared x 0, x 1. And so again, I start with two vector multiplets, x 0 and x 1. And out of these two, I construct the downstairs after gauge fixing one complex scalar. And so the action for the insiders, I'm cheating a little bit here. There's a pre-potential, but the action that I'm going to write down now, there is a symplectic rotation such that I'm in a basis in which there is no pre-potential. Don't worry. Everything is under control. It's well understood. But I'm skipping these important details here. And so I'll write here plus symplectic rotation. And the action, if you work everything out with this conformal calculus, or this example is so simple that you can even do it without a conformal calculus effect. Very good. So what do we have here? Yes. I wish I could say yes. No, it's not quite that. You have to define these symplectic sections. And then there, I'll explain you this in privately. You would like to think so, but that has to do with the fact that it's not quite that because I've done an electromagnetic dualization. And then it's no longer the case of you. I wanted to suppress this, but it's a good question. Good. So what do we have here? We have one. We have this special Kepler geometry. And it's, well, of course, there's a sigma model, this Mij from the blackboard, the previous blackboard. Well, it's just one of the imaginary part of Z. And here, now I have included the vectors. And here it is. And these vectors, they couple to the scalars. And here are theta angle-like terms. But the theta angle is not constant. It depends also on the scalars. So these are not total derivative terms. If there would be total derivative terms, then I could have also dropped them, of course, in the equation of motion. And so this term is a little bit more difficult. And when we look for black hole solutions, in particular, we're going to look for BPS black hole solutions. We will look for solutions where the real part of Z is going to be 0. And I think I'm not sure I have to double-check, but maybe the experts in the audience can correct me. If you impose BPS on this model here, it will force to set the real part of Z to 0. Correct me if I'm wrong, but I think so. Certainly, the case for non-rotating ones, that I'm sure about. So I will look for solutions, BPS also, with the real part of Z equals 0. And the imaginary part of Z I will call e to the minus 2 phi. Well, there is one thing here. So you see that this is the metric here on this nonlinear sigma model. I want the kinetic terms to be positive definite. So the imaginary part of Z, well, here's the square anyway, so that's not so much of a problem. So I will require this to be for the kinetic terms of the vectors. This is the answer. I want to hear positive definite kinetic terms, and that forces the imaginary part of Z to be positive. That's why I write it as e to the something. Good. So now, well, now we can just write down the solution for you. Well, the gauge field will be solved, and there will be no trivial. Yeah, I will write down the solution now. Yes, yeah. So the rest is to be solved from the equations of motion and the BPS conditions. So one half BPS solution is governed by a metric of the form. In fact, the BPS condition already forces you to put it in this way. These are isotropic coordinates. And so what one finds, this is Ferrara-Gibbon's Kalosch stormwinter, is that this is determined in terms of two harmonic functions on R3. This is essentially R3. And e to the minus 2 phi is equal to h1 over h2. And so these can be harmonic functions can have multiple centers. But I will basically look at the simplest solution where this is e to the minus phi phi 0 phi infinity plus q 4 by R and h2 is e to the plus phi 0 plus p prime 4 by R. So what is here? Here is the asymptotic value of the dilaton. We can see that because when R is infinity, it's just this. And when R is infinity, it's just this. And then we take e to the minus 2 phi is h1 over h2. That becomes e to the minus 2 phi 0. So phi 0 is the asymptotic value of the scalar field at infinity. It's a constant. And it's an arbitrary constant that is not determined here by the BPS equations. So p and q or q and p prime are charges. q is the electric charge that belongs to f. This f here, we have two gauge fields, f and f prime. And p prime is essentially the magnetic charge, not of the gauge fields, a mu, but of a mu prime. So this is f dual or f star prime. I didn't write down the explicit expression, but they are essentially similar to what I wrote yesterday, 1 for f and 1 for f prime. So this is a solution. If I would spell out also the form of f, they are in the book also. And so the point I want to stress here is the simplest example of the attractor equation. That can be made explicit once we look at the value of the dilaton at the horizon. If we compute the value at the horizon, yeah. Let me just stress that the horizon is located at r equals 0. So you see at r equals 0, these harmonic functions they diverge. And so if they diverge here, then this is e to the minus 2u. If they diverge, then e to the plus 2u, that is one of the harmonic functions, this becomes 0 here. So that is the non-rotating black holes. This is the location of the horizon is at r equals 0. So at the horizon, then e to the minus 2u, you can compute it from here. At the horizon, you get h1 over h2. So these constants can be ignored. 4 by r cancels. We just get q over p prime. I've assumed q and p to be positive. And this is what the attractor mechanism is in the simplest case. We have that asymptotically. These scalar fields, in this case there is only one, they have some undetermined value. But then there is the radial variable that allows you to close to the black hole, or the black hole horizon at r equals 0. At the value or at the location of the horizon, the scalar field that started out being completely arbitrary, it is fixed in terms of the charges of the black hole. And in between, there is some attractor flow, it is called. It has a particular profile. There's even a differential equation that you can write down for the attractor flow. I'm not going to do this in these lectures here. But basically, there is a radial equation that tells you that, whereas you can start with an arbitrary value of the scalar field at infinity, once you get close to the attractor, then they are determined by the charges of the black hole, in this case, q and p prime. That's consistent also with no-hair theorems that says that there is no additional degrees of freedom characterizing the black hole horizon other than the mass of the charge. This mass can be computed, and the mass is given in the conventions of the book, 8mg equals e to the minus phi infinity, well, for phi 0, p prime plus e to the phi 0, q. So this is the mass. It's also determined in terms of the charges. But now in the mass, there arises these asymptotic values. Excuse me. And so the BPS bound doesn't say, like for Reisner-Nurseum, that the mass should be equal to the charge. The BPS bound says that the mass should be equal to the central charge, the central charge arising in the n equals 2 supersymmetry algebra. And when you compute the central charge here for the case at hand, then you will find that this is precisely the central charge in the Susie algebra. Yes? Well, there are essentially the BPS equations where you plug in these ansatz for the metric, and you plug in that everything only depends on the radial variable. The BPS equations, they essentially turn the second order equations of motions into a first order differential equation of motion. All the time dependence is fixed. All the angular dependence is fixed. So it's essentially the equations of motions, or the BPS equations, reduced to a differential equation in a single variable. That's what the attractor flow is. It's not a new equation or something. It's the equations that just come out of these ansatz. Other questions? Yeah? Is there a more general solution where you have both electric and magnetic charge for each of them? Can that apply? Not in this model. I can double check, but not in this model now. I think they should be mutually non-local. You guys know the answer? Whether the black holes can have both Q and P for a particular gauge field. So they're dionic with respect to a single, because now this is electric and this is magnetic. Well, you can do symplectic rotations, and then you start mingling them, but there's still the condition that you can always go to a frame in which this is the case. That's why I'm saying in this model you will not find this. But if you do again the symplectic work, it's all related maybe again to your first question. If you start doing electromagnetic dualizations or transformations, then you start to mix f and f prime. But there will always be a condition or a statement that there exists a frame in which this is pure electric and this way. I will double check that because I'm confused. Good. So this is just a simple example, a relatively simple example. And you see that the harmonic functions, they are playing an important role. Harmonic functions basically characterize both the solution for the scalar fields and for the metric, and also for the gauge fields. You can write it in terms of the harmonic functions because the electric and magnetic charges are sitting inside the harmonic functions. So here we have just one vector multiplet, but we can generalize it to arbitrary number of vector multiplets. So for generic number of vector multiplets, nv. So I will do this. I will sketch it only here. I'll give the properties. So for generic number of vector multiplets, what we do is we first define these combinations, h lambda, as a definition. So now I have my scalar fields in the vector multiplet, x. I called it i, I think. This is not in the book, so I took it from somewhere else. Let me try to be consistent here, hi. These are just definitions, but it wouldn't surprise you that I call them h. If you impose the equation of motion, they're going to be harmonic functions. And I just had one set. I just had an h1 and an h2. Generically, I will have a symplectic pair of h1 and h2. There will be harmonic functions. So hi and hi are harmonic functions. That will include the charges of the black hole. And you see by doing symplectic transformations on the x's and the f's, you can act with a symplectic matrix on this here. You see how you are reshuffling the charges as well. So the metric can be written like this here. ds squared equals e to the k. I'll explain it in a moment. dt plus omega phi d phi squared minus e to the minus k dx dot dx. So what have I written here? This is the spacetime metric. Previously, I called it e to the minus u, or e to the minus 2u. Now this u is written as k. And this k is actually the scalar potential of this special scalar metric. So this is the scalar potential. Calor manifolds and particular special scalar manifolds have scalar potentials. Eventually, you can compute everything from the pre-potential. The pre-potential determines also the scalar potential. And then now we can have terms like this here in the generic case. This will also allow for rotating black holes if you have at least multi-centered solutions. You have a single centered black hole that is supersymmetric in asymptotically flat spacetime. It cannot have rotation. In the multi-centered case, we can have rotating BPS solution. And so this omega is determined by, well, an equation that you still have to solve in general. Let me write down the equation. There's a theta part omega phi equals h i d r h i minus h i d r h i. That's for the theta derivative. In principle, you have to integrate this. And then we have minus 1 over sine theta d r omega phi. I'm not sure here whether I'm using Russian conventions or the sine theta or cosine theta d theta h. And so the field strength, which I didn't write down in the previous case, I can now write it down. We have f i r and phi is equal to minus r squared sine theta over 2 d theta h i and f theta phi. Yes, now you actually see that you can actually have the charges. And then there is expressions for the dual field strength. They are determined by the harmonic functions with lower index i. So this is the sketch of the solution. Of course, it needs to provide a little bit more, a few more details to fully understand all this. So what are the main characteristics of the solution? We can have multi-centered solution. I already said that because the harmonic functions can have multiple centers. And once they have multiple centers, then they can also be rotating solutions. So there are spinning multi-centered solutions in four dimensions. But I cannot be spinning four-dimensional black hole solution. In five dimensions, you can have single centered rotating BPS solutions, but not in four dimensions. And then the metric is computed from the scalar potential. The scalar potential is determined by the pre-potential. The pre-potential is basically fixed by axis. And these axes are governed by the h's. You have to solve this set of equations to get the pre-potential and therefore also the scalar potential. That's just plugging in and computing. So this is specified by harmonic functions. This is specified by harmonic functions. And this is specified by harmonic functions. So the entire BPS sector of black hole solutions is, in fact, fixed by harmonic functions. The near horizon geometry, I didn't really derive this. But you can easily see that already in the Reissner-Nurstam case for BPS solutions. Around each center, you will locally find the product of ADS2 cross S2 with equal radii. So the size of the S2 and the ADS2 is equal. I stress this a little bit because in black holes and asymptotically ADS spaces, you also get ADS2 cross S2 near horizon geometry, but with unequal radii. And so they all have attractor behavior in the sense that the scalar fields asymptotically are determined by the asymptotic values of the constant parts of the harmonic functions. They are arbitrary. But once you float to the horizon, then the physical scalar fields will only depend on the charges. So a model that is studied a lot is the STU model. I just mentioned, I'm not going to work out this model. I just mentioned it because it's studied so much. It's based on a pre-potential that has the following form. It has three vector multiplets, ST, and U are the scalars. So you have to start in a conformal calculus with four vector multiplets, x0 up to x3. And these models arise naturally from string compactification. And the constants here, basically, are determined by the triple intersection numbers of the Calabio, and so forth, and so on. So this is an example, which is well studied in the literature. Very good. That's all I have for today. But time is up. Maybe there are still a few questions, so thank you.