 Hello? Hello? Hello? It's not working. Hello? Test, test. Now it's working. Thank you. Thank you very much. So this is my last lecture already. Time goes fast. So what I want to do today is to discuss gauge supergravities and the black holes that arise. Of course, this is a whole topic by itself. And so I want to maybe be a little bit sketchy, but sketchy. I was also a little bit sketchy perhaps yesterday. And the whole point also of these lectures is to prepare for future lectures in particular Alberto and Samir, perhaps also Joao. And so I'm not going to give like a full analysis of gauge supergravities, but I want to discuss a little bit what are the main ingredients of such a theory. How does it look like? And then in a concrete example, discuss BPS black hole solutions. So the whole, let's set the stage a little bit. So we're interested in black hole solutions that are not living in asymptotically flat space decided yesterday. But I'm interested, we're interested in solutions, black hole solutions that are asymptotically living in ADS. We have seen in ADS4, we have seen that if you just do the simplest version of supergravity in ADS, that was just with no matter couplings, then it's essentially the original nurse term living in ADS4. That could not be made super symmetric without having negative singularities. And so we need to do something. In fact, the analysis was also done, I think in 1999, James Liu, where you see, I think a famous paper by Duffer and you, where one studies n equals eight gauge super gravities. And so if you focus there also on electric electrically charged black holes, that doesn't do the job either in the sense that these solutions become naked singularities. But that paper will still be relevant also for a later discussion because there's a lot of embedding of four dimensional supergravity into M theory. And then eventually you want to make connections with ABGM theory and explain the entropy of black holes microscopically using field theory techniques. But that's not the purpose of my lectures. So what is gauge supergravities? So well yesterday I brought you to the point where I discussed matter couplings to n equals two supergravity. And so I'm going to leave out the hyper multiplets for the moment and just discuss vector multiplets. And if we do all the gauge fixing for the present purposes, the off-shell version is not very relevant for us. I'm interested in constructing black hole solutions, just equation of motion, no fluctuations or localization around it. So I'm just interested in constructing solutions. So that is just an on-shell procedure is good enough for that. And after you do all the conformal calculus, you end up with supergravity theories for vector multiplets, which are of the following type. So there's the Einstein-Hilbert term and then we had vector multiplets. And in each vector multiplets, there is a complex scalar. And then there is, I didn't write that down yesterday, but today I will write down the kinetic energy or the kinetic terms for the gauge fields. Let me first write it down and then say a few words about it. In fact, I wrote down one particular model yesterday explicitly on the blackboard where I had one complex scalar and I had two vectors. So i runs over the number of vector multiplets and this index lambda, which I called i yesterday, capital I, now I call lambda, lambda sigma. So i runs from 1 to nv and lambda runs from 1 to nv plus 1. The reason why there's an additional vector is because there is the gravity photon. The gravity photon is part of the Poincare multiplet together with the metric. And of course there's also fermionic terms that I'm not writing down. And so here we see the kinetic energy for the scalar fields in the vector multiplets. There is a metric on that space. It's called special scalar metric. And I gave an explicit example yesterday where I only had one scalar and this thing was just 1 divided by imaginary part of z squared. And here we have the kinetic terms for the gauge field. And this i is a matrix that also depends on the scalar fields. I wrote down an example yesterday. Then this was just the imaginary part of z and this was the real part of z. But in general this is now a matrix. This would be theta angle like terms. And all these quantities here have geometric interpretations in terms of special scalar geometry. But for the present purposes the only thing that we need to know is that all these three quantities here are determined by a pre-potential f of x. And these x's then can be parametrized by coordinate z. The reason is that yesterday I wrote down these kind of constraints. Let's say this is constant. So NIJ was made out of the second derivatives of f. And so this is a constraint on all the fields. So the solution of this is a hypersurface which has co-dimension 1. And coordinates on that surface are the small z's. So these z's end up being here. Another way of saying this is you do the super conformal quotient from the off-shell formulation to the on-shell formulation. And then you exchange, you change the axis for the z's. You go down in one complex dimensions. So this is just one constraint. There's also the U1 constraint that allows you to go down. I'll give explicit examples. So important examples that are relevant kind of for our discussion is this model here. Let's say you have four vector multiplets or three in the four vectors and three scalars. So this is a known example. It's the STU model essentially. Or you can also go to another example of a pre-potential that is often studied is this one here. It had to be homogeneous of second degree. Both of them are homogeneous of second degree. And in fact, for those who know this, these two pre-potentials are related by electromagnetic duality. Sometimes one makes truncations. You set certain fields equal to each other. If you're off-shell, you can easily do that. And so for the purpose of my talk here, I will look at the case where number one, two, and three are really the same. And so I'm going to study this particular model, which is x0 and x1 cubed. It's still homogeneous degree. So of course, now I have only two vectors, vector multiplets. And I can do the same thing here. You can have f equals x1 cubed over x0. So that's kind of the choice of this pre-potential basically dictates and determines this entire Lagrangian. So we have also gravitini here and the gauge genie. And so a natural question that comes to your mind is, hey, we have gauge fields, but nothing seems to be charged under the gauge fields. All the fields are neutral. The scale of fields is just a flat derivative. There's no connection here involving this. And if you write down the kinetic terms for the gravitino or the gauge, you know, they're also not charged. They're neutral. So the question then is, well, can we make some of these fields charged under the U, under the U1s that I have available here? And the answer is yes. And in essence, that is what gauge supergravity is. You're charging certain fields under vector fields, under gauge fields that are already available in the multiplets that you have. And there's different choices that you can make. You can make different fields charged under these U1s. For instance, I can make these scalar fields to be charged under the U1s. But I will do that maybe when there's time left over. But inside the fermions, there's a gravitini. I can also make the fermions charged under the U1s. And that's a different choice. And there are different types of gaugings that you can do. I didn't write down the hypermultiplot. They also have scalars. And I can gauge them. I can give them charge. And so the gauging that I will look at most of this lecture today is gauge, give or gauge the gravitini. I have two of them. Give them charge, in other words. Yeah. Now, the way you do that is just by using the ordinary procedure of gauging. You just covariantize derivatives. And you couple the fermions to the gauge field by minimal coupling. And that works here as well, at least that's part of the procedure. But now you see that we have many gauge fields. And I can take the gravitini and I can give them charge under any of these gauge fields. And so or the most general thing I can do is to take a linear combination with arbitrary parameters and take that gauge field and use it to charge the gravitini. So use the combination xi lambda a mu lambda to do this. Where xi lambda are arbitrary parameters, they're called, in this context, they are called faeiliopoulos terms. Actually, I'm not doing this in the canonical way. If you read the textbook on gauge supergravity, you won't hear the presentation in this way. You will start reading about potentials and superpotentials and then terms in there. I always refuse to do it in the canonical way. So I think of my own way. And so you will see later what is the connection to terms in the potential. So I can pick a general combination and I call them parameters and whether they are called faeiliopoulos terms or not is not very relevant. I pick a combination. I use that gauge field to charge the gravitini. And if you want to use a different, you can just adapt the choice of these parameters. You can set some of them to zero and then others to one. Then you just make a particular choice which gauge field that you are using to charge the gravitini. Now, once you do that, then what you have to do is change the kinetic, the covariant derivative. You have to change the covariant derivative and add, of course, a term. Well, there's the ordinary derivative, then there's the spin connection. I don't write down all these terms here. I just write the additional terms that are used in this. And here is my gauge field, at least the combination of it. And then if you go through the details, you also find that you have to, there's two gravitini and they have opposite charge. That's why there is a sigma 3. And so this is a standard term that arises when you gauge, when you give the gravitini charge. This is also sometimes called gauging the r symmetry. I don't like to use the terminology that much, but well, whatever. It's not very important. Good. So this is what you do. This is called an electric gauging because the electric charges of the gravitino are equal to g times psi lambda. These are the electric charges. Well, I have two of them. I have plus or minus e lambda because the plus sits in one eigenvalue here and the minus in the other one. And so these are the electric charges of the gravitino, gravitini. And so of course we're going to be looking for black hole solutions. And these black holes turn out to have magnetic charges. So not electric charges. The electric charges that has been tried before, they always turn out to have, I don't know if this is really a theorem that holds in the most general class of gauge super gravities. But electric charged black holes will not appear here. So now we have a situation where we have electrically charged particles and we have a magnetically charged black hole. And so you can go through the usual Dirac arguments for the quantization of charges. And so if you have a magnetically charged object somewhere and an electrically charged particle here, the Dirac quantization condition in the conventions that we're using is something like e lambda p lambda is an integer. And in fact you can also do something slightly more complicated and in fact it is done in the literature and it's also important in the black hole story. You can also do magnetic gaugings. And you can do magnetic gaugings by introducing magnetic gauge fields and parameters. This is just a side remark to make you realize that there exists such a thing as magnetically charged gauge super gravities. So with magnetic gaugings there's quite a bit of technology to understand from the super gravity point of view what these magnetic gaugings, how they look like, how that works. But you will get something like combinations of gauge fields where these are the magnetic gauge fields. There are different gauge fields from these ones. They are the magnetic the magnetic one. So let me put a tilde here. And these are new parameters here, the magnetic phi in Leopold's terms if you want. And so you can use gauge fields like this. You have to double the amount of gauge fields if you want and then you can make gravities that have both electric and magnetic charge. And then if you go through the black hole with magnetic gaugings you can also make black holes to be both charged under electric and magnetic charges. But if you're an electric frame then the gravitino are electrically charged and the black hole is magnetically charged. That's the only possibility if you impose super symmetry. Yes. Yeah, yeah, yeah. I have to complete that argument. This was just a side remark. So the first step in the procedure is what we always do when we make symmetries local. We gauge something and you do the of course then the gauge transformations come into the game and the gauge transformations they form certain commutation relations with supersymmetry. And so it turns out if you want to obey the supersymmetry algebra you also have to change the supersymmetry transformations. And for instance this would be a magnetic gauge field. Yes, you can make it part if I introduce some more indices. So suppose this lambda I have to let it run for right from the start over twice the amount of values. Then half of them I consider to be electric once and half of them there's a people who know me a bit better. And the other one are going to be the magnetic gauge fields. Of course everything is subject to electromagnetic duality transformations. So you sometimes can go to a frame where it's purely electric or purely magnetic or something in between. And this in between leads then to combinations of this type once I start doing the gauging. In an ungauge theory you don't see the difference that much because you can always do rotations and it is semantics what you call electric and magnetic. And well here of course with the gauging that makes a little bit that makes the theory different. And so well the field strengths will look differently for the electric and the magnetic potentials etc. I'm going to explain more in detail perhaps later. I'm not going to use the magnetic gauging I just want to mention this but I suspect that maybe Alberto will use them. No you won't forget about it. Yes well that's why I precisely introduce new fields. You cannot have that with the same field there's a whole formalism to it and well this is going to lead to a too complicated discussion. You cannot do that that's why this is an A tilde if you would if you would if you'd really introduce one gauging you could not do that locally. But there's also some choice of selective form when you write it like this so that's the magnetic. I'm afraid I can only explain as well if I take a half an hour and I don't really want to do that but it is important if you want to discuss black holes that are dionic but I'm not going to do this here. No no no this is not in the book anymore so this is so the solutions that I'm going to discuss today are basically found by well there's three papers it started with Kachatory Klem and then Njaki and Dalagata wrote a paper especially also about the magnetic gaugings and then Kiril Histov who arrived today and myself we also constructed these solutions and embedded them in m-theory computed the masses and stuff like that and the entropy. Very good these all these three papers are closely related and Kachatory Klem was the first the first paper. Excuse me can you gauge more than you want? You mean like non-Abelian groups? Yes this you can do that also becomes a story by itself like what is the story about black hole physics when there's non-Abelian gauge fields present that's already a complicated story when you just do gravity and Young Mills theory. I don't know what the state of the art is in this but there are numerical solutions where where this works but it's it's all you run into problems with the no hair theorem because a black hole cannot have non-Abelian charges that's forbidden by the non no so the question is then what do the non-Abelian sectors really do on the other hand well black holes presumably are constructed by the standard model particles so yeah I can give you here this is I'm only going to consider a billion charges of this type but there is there is literature about also in super gravity about non-Abelian but the story becomes much more complicated this is already complicated are there any more questions right so forget about the magnetic gaugings we won't do it so when you do this here you also have to adapt to super symmetry transformation rules and so what you have to do is a few things but I'll focus on the Gravitino so this is d mu epsilon i it involves the spin connection there's some other fields that are already present in the ungauge theory and then there is a modification of the following type sigma 3 ij psi lambda l lambda and then gamma mu epsilon j so what do we see here and there's one new symbol here l lambda is by definition e to the k over 2 times x lambda k is the scalar potential of which this is the special scalar metric here and so you deform also the super symmetry transformation rules if you don't your algebra doesn't work anymore as a consequence of this modification here and well now you've meant now you've changed the well i've changed the action i've changed the super symmetry we'll have to recheck whether my action is invariant and the answer is the action is no longer invariant but you can make it invariant if you also add a potential and the potential looks successful i'm just going to write it down what the potential looks like and of course all these things were derived with well a lot of effort there were basically two schools or there are two schools on on on gauge super gravities perhaps one is the sort of like belgium dutch school the width and from the width and from poin and then the the italian the italian school that started with and many other people there's lots of references many people in the audience have worked also on this and so i'm just going to write down what is the result the result is that you have to also add a potential to the theory and that potential looks as follows g i j bar let me first write it down okay here's the potential well it's good that there is a potential because if we ever want to have a ds black holes in a ds there better be a potential that generates a cosmological constant so i have not explained all the symbols here here are the phyliopoulos terms here are these else there are essentially the sections x i multiplied by the kilo potential here's the inverse metric g i j and then there's this f this f is defined where can i write it perhaps here f i lambda equals e to the k over 2 di x lambda lambda lambda and this di is the derivative with respect to z the coordinate over here the x is dependent on the z and you have to covariantize this with a so right equals e to the k over 2 di plus the k log potential acting on x lambda hope you can still read this there's a lot of definitions and abbreviations all these objects have geometrical interpretations but i won't have time to explain all the beauty of the special geometry so now the model is completely defined here this is my action that's still true in the bosonic case nothing has changed because this we left uncharged so the only modifications to the Lagrangian arise because of these kind of terms and because a potential needs to be added and if you add the potential then you can check that the whole thing is invariant on the local n equals 2 supersymmetry so now we can discuss a black hole solution so so that is my short resume of gauge supergravity and one can make that more complicated if i have time left over i will say a few things by also gauging these scalars or adding hyper multiplets and and if you do that there are still the same procedures you make derivatives covariant you change the supersymmetry rules and you add more terms to the potential and that's all well understood and well under control good i wanted to stick to this particular theory with the phyliopolis terms this is also called phyliopolis gaugings and construct black hole solutions because these were the first models where bps solutions were written down and they preserved not half of the supersymmetries but only one quarter one quarter out of eight means two supercharges and so that is something slightly particular because that doesn't happen in asymptotically flat space there it's always one half and one half out of eight is four supercharges and n equals two so the models the example that was the first example of a bps black hole in ads4 with a regular horizon no naked singularity is the one quarter bps solution black hole with nv equals one you can easily generalize that and so one vector multiplet besides the gravity photon and f to be the pre-potential i wrote down before you can easily extend this by taking four vector multiplets and just write here x one x two x three x four probably that's also what alberto will be doing so keep that in mind i just do it here for having simplified notice so basically what i'm doing is i will set some charges equal to each other and so that's a truncated version from the version that you will get that you will see in alberto's talk so to make very explicit here what this coordinate z is in this case the coordinate z is simply given by x one over x zero so this is the relation how the z coordinate arises as often you also see x one is zero is z and x zero equals one not always is the coordinate z found in this way or given in this way there's choices you can do different symplectic parametrizations but this is a particular choice that i will use so before you write down the black hole solution you have to find the minimal of this or the extreme of this potential and find that there is an ads vacuum because there was no ads vacuum there's no point in finding a black hole solution but the that's rather easy and so first you find an ads for vacuum so how does that work so you have the pre potential so you can construct all these quantities here l and f that's all dictated by the pre potential and you can just write out this potential in all its gory detail it's a function of z and z bar and these size and you just extremize it and you find an ads vacuum for a constant scalar z star equals three psi zero psi one and then the value of the potential at this extrema is the cosmological constant and it's given by a minus two g squared there has to be a well there's a g squared here or i can also put it in front of g that's the coupling constant right minus two g squared over square root of three psi zero psi one cubed and if you want to go to the model that has four vector multiplets you just write here psi one psi two and psi three so this is an ads vacuum it preserves all the supersymmetries and so that's that's step step number one then um we're going to find write down the black hole solution it's going to be magnetically charged yes there's a question a okay k is the scalar potential so the scalar potential um so i wrote down this metric here g ij bar that's of course the scalar potential of the special scalar geometry and then this this and then the question is how do you construct this out of the pre-potential there is a formula that goes something like this maybe up to factors of i's and minus i'm not entirely sure minus x bar lambda f lambda so this is a function of z and z bar i don't think i wrote down this formula yesterday so okay so we're going to look for magnetically charged black holes because the attempts to find electrically charged black hole had sort of failed because they produce negative singularities at least when you do an electric gauging and so we're going to look for we're going to make an ansatz in which the in which there is no electric field and so the ansatz so the black hole well first you write the metric in a particular ansatz you write this here the s square equals yeah we use also conventions in a in a plus minus minus minus metric forgive me it's Kirill's fault but he was very good at computing with this signature i mean so we make the ansatz here i'll specify and we make the ansatz that there's no electric field and the magnetic field theta phi lambda equals p lambda over two sin theta and so we have two magnetic charges here lambda runs over one and two there's a zero and a one so we have magnetic charges p zero and p one these are the two magnetic charges there's no electric charges because i put it in there and then essentially you turn the crank and you find that well how explicit should i be u of r is equal to e to the k over two g r plus some constant that i will specify in a moment h of r is given by r e to the minus k over two so these are the functions that you plug in here u and h it's again determined in terms of the scalar potential so you have to know what is the value for the scalar field and the scalar fields are given by harmonic function x zero equals alpha zero plus beta zero over r and x one is alpha one plus beta one over r and so um so we have a bunch of constants here namely alpha zero beta zero alpha one beta one and we have this constant c they are all if you work out the the bps conditions they're all determined by the two phi a eliopolis terms and the two magnetic charges so they are all so alpha beta and this coefficient c they're all determined by xi lambda xi zero and xi one and p zero and p one i didn't write down the explicit expressions here but they're in uh in the papers that i uh quoted okay so um that's it everything is specified here well to make it very concrete you have to plug in some value because you have to compute the scalar potential you have to find the coefficient c here but uh i have not given the formula it's crucial to have a horizon because uh the the c should be negative because if c is not negative you will never find a zero of of of this and then there will be no horizon but things work out in the way that c is negative and if c is negative then everybody can solve this uh find a zero of this question then you find the value of the horizon and the value of the horizon or the location of the horizon is given by our horizon equals well it's a it's a it's actually a rather complicated formula well rather in some sense the easiest way to write it we found was um minus two g squared but i still have to eliminate this beta in terms of the phyliopolis terms and the charges and then you get a um value for the horizon and with the horizon you can compute compute the entropy and s equals a over four um i think alberto will also write that formula in a more elegant way so you can plug it in uh by you have to be it's a little bit subtle because you have to compute the function h at the value of the horizon because and that you have to integrate so it's not just r h squared yeah it's h squared at the horizon squared and that multiplied by four pi and so um that's um so that is the that is the result um some remarks well the near horizon geometry just like for asymptotically flat black holes is again given by an ads two cross s two factor near horizon that's of course very important that we have uh in the infrared also conformal symmetry dictated by the ads two factor the only difference with asymptotically flat space time is that the radii of of these two spaces are not equal to each other but that doesn't bother us too much so another remark is that we can also construct i mentioned this very quickly in that we can construct a black hole with an electric and a magnetic charge not in the theory and in the gauging that i wrote down um with let's say a magnetic charge with respect to the first gauge field and an electric charge with respect to the second um but then you have to use electromagnetic or symplectic transformations i didn't explain that well but apparently it's also not going to be needed what you do then is you have to act with that on the full theory then also the pre-potential that's not invariant under these transformations if you choose equals to x one cubed over x zero then of course also the gravitini are going to have different charges they're going to have one electric charge and one magnetic charge engage with gravity you cannot there's no invariance on the disimplific transformation because i is there's an object that i can see this is the absorption um there's there's covariance but not invariance um well i i wrote down the potential for instance uh and this potential has phi a heliopolis term and so that is not invariant these these phi a heliopolis terms they they they also have some index lambda if you want and on that index lambda electromagnetic duality transformation or symplectic transformation they act on anything that has a lambda index but then you see already that there is a there is a its partner is missing so we have symplectic pairs this is a little bit of an aside x lambda and f lambda this is a symplectic pair but also the phi a heliopolis terms xi lambda um there will be xi lambda and xi lambda and these will these will generate electric gaugings and these will generate magnetic gaugings and so um one of them was missing uh i only had in my potential i only had since i was doing electric gauging the other one was not there and so the theory is not invariant under these transformers there's many ways of saying the same thing also the kinetic terms for the scalar fields without gauging it's not invariant it's only invariant if you also change the pre-potential therefore it's not invariant but it has some covariance properties if you also change the pre-potential then the Lagrangian state has still the same form except that you have to change uh take a different expression for the pre-potential so in the potential you clearly see that some terms are missing and in fact if you people constructed the magnetic gaugings in a way such that this covariance properties become um become uh well actually have the expression on the next slide um no it's not scalar transformations it's electromagnetic transformations so you basically rotate also so um so another symplectic pair is f mu nu lambda and then we have g mu nu lambda which you can think of as the magnetic field strength think of it as just the star of f and so symplectic transformations also act on this pair and so there you see that you're basically changing electric and magnetic fields so that's that's the way to think of it physically yes yes yeah yeah yes that's right yes it's a supersymmetric solution by its own and I think um you'll help me if I'm wrong but I think if you just the near horizon the solution is is again maximum is is sorry has augmented supersymmetry and that is one half bps uh is that correct yes yeah yeah yeah so so that we've seen also in flat space so if you have a black hole asymptotically flat space maximal supersymmetry then along the way to the horizon you break half of it but at the horizon you have again maximal supersymmetry with ads it's a little bit different there also at asymptotically flat so you have maximal supersymmetry along the flow it has one quarter and at the horizon it is bps but it's one half not not maximal but one half very good yes uh can you speak a bit louder yes yes yes and there's also an attractor mechanism going on it's a little bit more complicated but in this example it's very easy because we see that z the coordinate z is the ratio of this two and it's the same story since b well now uh now the the attractor mechanism and the no hair theorem says that the black hole horizon should be determined in terms of the charges only and it's going to be the charges of the black hole but also the phi a heliopolis terms but these are charges because there are electric charges of the of the gravitini and so there's an attractor equation as well the only thing you have to be careful with is that if there is flat directions in your potential there are some scalar fields that don't see the black hole those of course are not fixed by attractor that they're just you can set them to constant at any constant very good um there's an m theory lift that's important of course um that's important um to make the connection with ads cft so the story is that um this can be embedded by uh in m theory using the spherical compactification of course you get a more complicated n equals h supergravity the details are in the dove u paper um already from uh 99 i think huh yeah so or something like that um that is you huh isn't it yeah that is you you yes yeah so very good um so the details of it uh are important for uh making contact with ads cft and so here you get of course so eight gaugings here you see an example where you have non-habilian gauging but what you do is you make a constructed black hole uh the so eight has a carton which is you want to the fourth there you see your four vector multiplets coming out all the non-habilian stuff you you you truncate or that by this i mean you construct a black hole solution that's only charged with respect to the you want to the fourth and so you will get uh charges p zero p one p two and p three um and um um well of course i would need to write it out and exactly go through the truncation but uh i want to have time for that um very good so let me just mention since i have 10 minutes still time about uh the the broader class of gaugings such that you get a little bit of a total overview so after the kachatory clem and our papers um people investigated extensions generalizations um to include also hyper multiplets or other gaugings and um that is still very much ongoing there are some results some of them are numerical many people in the audience here have worked on it i'm not going to give all the names but um and sometimes people could also only construct the near horizon solution so the eddy has two crosses two solutions and in only very particular cases one could find analytic solutions uh where more general gaugings were considered so let me say a few words because i still have 10 minutes about these more uh complicated gauging so so we can also so one was a gauging of the gravitini or called also called r symmetry gauging and so we can gauge gauge the or charge the vector multiplet scalars and so well for this every time you gauge something you must have a global symmetry first now the scalar fields in the Lagrangian have the following form kinetic term so you must have a global symmetry first but here you have a non-linear sigma model so that means that this this you will never have a global symmetry if there is no killing vectors or symmetries associated to this metric so you need what is called a killing a killing vector and or or a bunch of killing vectors and then you can have a global gauge symmetry acting on the scalar fields alpha lambda k lambda i which is a function of z and these are killing vectors vectors of of the metric g i j bar so they satisfy the killing equation once you have this you have a global symmetry you can gauge it and so you change d mu z i into d a covariant derivative which is just and of course there is a well the gauge transformation now there is a covariant and so you've gauged this symmetry but that doesn't that's not enough you have to also change the supersymmetry transformation rules and you also have to then change the potential i will write down the potential in a moment because well now that this is kind of a clear procedure you can also of course gauge the hypermultiplet scalars and of course you can do a combination of everything yes the most general gauging and the most general gauge supergravity gauges everything the gravitini the vector multiplet scalars and the hypermultiplet scalars so here we have a kinetic term which is something like well how did i write it down yesterday g u or it's called h u v in the notation of some of our papers d mu q u so they're called q u u equals 1 to 4 n this is quaternionic geometry this is the metric on the quaternionic geometry you again need same story as there you need well what shall we call it well you can use the same parameters of course because there's only one type of gauge so lambda we can have killing vectors in the quaternionic geometry and so call them q tilde so these are killing vectors of h u v and so then of course you do the same thing you construct a covariant derivative d mu q u consisting of the ordinary derivative plus g a mu lambda and k tilde lambda u function of q so they're the same vectors we use the vector we use the gauge fields from the vector multiplets to charge either the scalars that sits in their own in their own multiplet or you use them to charge the scalars that sit in another multiplet so this amu and that amu is just the same there's only one set of gauge fields that are present so then you have to adapt to supersymmetry transformation rules and to close the or to make the algebra closing on shell here we're on shell and once the supersymmetry transformation rules are modified you have to start varying the action you find is not invariant unless you also add a potential and the potential i'm going to write it down is a function of z z bar and q i'm only doing electric gauges here and so here's g square g i j bar k i lambda k j bar sigma plus four h u v that's the last formula i will write down so okay let me so let's say a few things about it so in the previous case where i only had the the r symmetry gauges there was no killing vectors here so this this term was not here i don't know hyper multiplet this term was completely absent this term was completely absent the only thing that i had basically was this term and these p's were then the phyliopolis terms now generically these p's are called moment maps i would need to tell a little bit of geometry to explain what moment maps are but in some sense the simplest choice of these moment maps are setting them to a constant and those are then and those are two vectors and you can wrote it put them in one direction and that these are called the phyliopolis terms so that reproduces the reproduces the potential that i wrote down before this term was absent and this term you see there's a computation between positive terms and negative terms because here we have the minus sign so that allows for vacua either decider anti-decider minkowski there's a balancing between positive and negative terms in contrast to supersymmetric field theories where potentials are always positive and so well that's it so the the the the the the the story here about this potential is that it's it is a completely geometrical interpretation it's a function of all the scalar fields both in the vector multiplets and the hyper multiplets and it's determined by geometrical data the pre-potential on the Keeler potential it's killing vectors so it's symmetries same for the hyper multiplets it's the metric and and it's killing vectors these else i wrote down before it was e to the k over two times x and x where this is all determined by the pre-potential same story here determined by the pre-potential and these are moment maps moment maps are basically constructed out of the out of the killing vectors on the hyper Keeler space so i'm not going to bother you with the details of these moment maps but it's something explicitly constructed in terms of killing of these killing vectors so then in principle that you can start playing the same game you can look for bps for bps bps black hole solution and that story is very much going on there's not too many examples of black hole solution maybe maybe there's even only one i think where there's an analytic solution known with with hyper multiplets present and so that's still under development i i'm also not up to date with the very latest so people can inform me here if i'm overlooking something clearly it's the the story is to find more general black hole solutions just rather than just having that singular example i just found find a generic structure of the solutions find the embedding in string theory or in amp theory they must asymptote to ads4 well we want them that's that's the the goal of this whole exercise because we want to explain the entropy of black holes using the dual field theory which was successfully done in the example that i showed in the paper by francesco alberto kiril uh so i think we're going to hear more about it in alberto's lecture so i will stop here the timing seems to be perfect so thank you for