 OK, so first of all, I want to thank the organizers for the kind invitation. It's a pleasure to be here. So well, this is a talk on the gravity side of the gauge gravity duality. And it's based on mainly on these papers that were done with my students or former students. And, well, Pedro Ramirez is a postdoc, INFN postdoc, and Sergio Cacciatore is in Como. This is actually already an older paper, but these things have become fashionable only in the last few months. OK, so after a short motivation, I will try to give an overview of known black hole solutions, analytic solutions, in four-dimensional, N equals 2, Fajelopoulos gauge supergravity. And I will show you for which class of black holes an exact microstate computation has been done. And after that, I will still focus on the fi-gaged case and I will show you that there exists solution-generating techniques. And then I will give an application of this to generate, to add, actually, rotation to a given seed solution. And after that, I will come to the coupling of hypermultiplads. And this is, again, a topic that has become fashionable recently. If you consider black holes in gauge supergravity, they are typically, but not always, in fact, at the end of the talk, we will see a counter example. They are typically asymptotically ADS, so they are relevant for the gauge gravity duality. And they have many interesting applications, complexity equals action, applications in condensed metaphysics, quantum phase transitions, quarkluon, plasma, and so on and so on. But in this workshop, we are mainly interested in using exact results in three-dimensional super conformity field series obtained by localization techniques to compute microscopically the entropy of black holes. So that's the main, at least as far as this workshop is concerned, the main motivation to study black holes in gauge supergravity. What are the known black hole solutions in N equals 2? Abelian FI gauge supergravity in four dimensions. The Poissonic Lagrangian is this. So you have here the sets are the complex scalar fields that parametrize a special column manifold. This is the Fs are the gauge fields. There's a theta term, and they couple to the scalars. And then in gauge supergravity, you have also a potential for the scalar fields. And here these capital Greek indices go from 0 to nv, which is the number of vector multiplets, so this is a theory of gravity coupled only to vector multiplets. And i, j, and so on go from 1 to nv. And well, you have here a matrix that is called a period matrix. And its imaginary and real part are denoted by i and r that appear in the action here. And the period matrix as well as the metric on the scalar moduli space is g. And also the scalar potential, they depend on the particular model. And the model is completely specified once you give a pre-potential. So the complex scalars, they are written in terms of this holomorphic symplectic sections, the x lambda, f lambda. And all the quantities that I mentioned can be specified uniquely with a single holomorphic function. The pre-potential is f of x. And well, the only constraint is that f of x must be a homogeneous function of degree 2. So as I said, specifying the pre-potential is equivalent to defining the full Lagrangian. So this determines completely the model. And now we are interested in the most, well, in black hole solutions in this model. And the most general black hole solution would be specified by a mass, the angular momentum. So we are on four dimensions. We have only one angular momentum. And that charge, n, which is, if you wish, it's a gravitational dual of the mass. And then we have nv plus 1 electric charges and nv plus 1 magnetic charges, the q lambda and p lambda. And if you want, you can also have an acceleration parameter and possible scalar charges. But this is not what we want to consider here. So if you know the so-called C metric, the C metric is a metric that contains this acceleration parameter. OK, and well, another interesting point that appears in gauge supergravity is, well, in the engaged case, in the asymptotically flat case, at least in four dimensions, the horizon is topologically always a sphere. So you don't have many possibilities. But in gauge supergravity, it can be spherical, flat, or hyperbolic, or actually even more exotic. So you can have a horizon which is a non-compact manifold of finite area. So it's something like this. And it goes up here and down up to infinity. So it's non-compact, but nevertheless, it has finite area. So it has finite Beckenstein-Hawking entropy. So you can also have this. I have many possibilities. Now let us look at some specific models. So in the first column, you have the pre-potential, which you see is always the homogeneous function of degree 2. So one of the most simple ones that you can write down is this, x0, x1, which is a model with just one vector multiplet. And the most general solution is known, which has mass, angular momentum, nut charge, electric, and magnetic charges. And it was obtained in this paper by Chau-Compair and also in this written with my collaborators. Then a more, slightly more sophisticated model is the so-called STU model, which is this one. And this has three vector multiplets. So this, well, disappears in compactifications of, if you compactify 11-dimensional supergravity on a seventh sphere, and then you truncate again, you can obtain this model here. And, well, one solution that was obtained already in 2009 is this with, that has only magnetic charges and the other parameters are 0. So this is 1 quarter BPS. And this was constructed using the recipe of this paper here, where all supersymmetric solutions of FI gauge supergravity in four dimensions were classified. So this gives you a recipe and you can take the equations and then let our first order and then try to solve. And then another solution is, well, for 0, angular momentum and 0, nut charge, but all the other parameters are non-zero. So you have mass, electric, and magnetic charges, which, again, is in this paper by Charles Comperre. Well, on this model, actually, there was a lot of literature. Another important paper is by James Liu, who is in the audience, and collaborators who constructed solutions here with mass, so non-extremality parameter, and electric charges. Then you have the cubic models. So they define what is called a very special scalar geometry. And here you have the most general solution, almost the most general solution, but only for 0 angular momentum, which is in this paper here by Erbin and Thalmagi. But they have a constraint, so they have the solution only if the very special scalar manifold is symmetric. And then, in addition, there's a constraint on the charges of the black holes. Then another, well, this is still a cubic model, which is a sub-case of what I have just shown that has one vector multiplet. And here you can write down almost the most general solution, well, without nut parameter, but with mass, angular momentum, and charges, which is in this paper here that will appear soon. Something more complicated is this cubic model with this correction here. So this is, again, a model with three vector multiplets. And if you put this constant a here equal to minus 1, then this is related to Calabi-Hau string compactifications in the ungaged case. In the ungaged case. So this is related to quantum corrections. I think it's not known how to obtain this model in the gaged case by string compactifications. And also here you can, the most general known solution has one magnetic charge, three electric charges, and the other parameters are 0. And the last is this. This is still a quadratic model, where this eta is the Minkowski metric. So this is what is called the CPN bar model. This bar means that this is a scalar manifold. It's a non-compact version of CPN. And here you can write down the most general solution. So in this paper, that will appear soon. Well, also, just note that models of this type here, where n is an arbitrary number, have been considered in the literature. These are maybe more academic. And well, I do not bore you with details on how the black holes are made by writing down the metric and the other fields. But just note that in the quadratic models, and only in the quadratic models, typically the metric is of what is called Carter-Blibansky type, which is quite simple. So the coordinates are t, phi, p, and q. And the metric is completely specified by these structure functions that is p and q, the quadratic polynomials, and r, s, and w, the quadratic polynomials. And the coefficients of these polynomials are then related to the black hole parameters, mass, angular momentum, and so on. And also the gauge fields, we have the a lambda and the scalar fields, they are completely given in terms of these five functions here. So this is how the metric looks like. And well, if you want to make contact with the usual angular coordinates, so well, q is, you can think of q as a radial coordinate and p as an angular coordinate. And if you set p equals to j cosine theta, then, well, j is the rotation parameter, then you get the usual parameterization. OK, so of all these models, as far as I know, a microscopic computation of the entropy was done only for a class of black holes in this model with this pre-potential here, for the STU model. And this class of black holes is dual to a twisted and mass-deformed ABJM theory whose partition function, the topologically twisted index, can be computed exactly in the large n limit via localization. And it was shown that this reproduces correctly the Bickenstein-Hawking entropy of these black holes. And a list of references is, I think, the first paper on this. Is this here? And these authors, they considered magnetically charged the PPS black holes with spherical horizons found in this paper here. And this was generalized later in 2016 to have also electric charges, so to dionic BPS black holes, but still with spherical horizons. And then in 2017, also hyperpolyic horizons were considered, so BPS black holes with hyperpolyic horizons and magnetic charges. So for this class of black holes, also symmetric, a microscopic computation of the black hole entropy is available. OK, so this was the overview of existing solutions in these models and microstate computation. And the next topic I want to talk about is the solution-generating technique still in this FI-gaged case. Well, solution-generating techniques are a very powerful tool to generate new solutions from a given seed, right? Because, I mean, the equations of motion are highly non-linear partial differential equations coupled. So it's, well, you can either do some guesswork, or it's very difficult to construct analytically a solution, at least in the case where you have also rotation, so it's not static. So this is a very powerful tool. And at the base of this, there is the idea to reduce the action if you have sufficiently many computing-killing vector fields to three dimensions. And in three dimensions, you can dualize all the vector fields to scalars. So what you get is, in three dimensions, a non-linear sigma model coupled to gravity. And roughly speaking, you use then the target space symmetries of this non-linear sigma model to obtain new solutions to the field equations by starting from a given seed. So for instance, a very simple theory for the Einstein-Maxwell gravity. And if you reduce this to three dimensions, you get a sigma model whose scalars parametrized this Berkman space. This has already been done in 1988 by Biden, Loner, Meissen, and Gibbons. And well, you can try to do something similar in gauged supergravity. But engaged supergravity supersymmetry requires a potential for the moduli, for the scalar fields. So this generically breaks the target space symmetries, unless you consider flat gaugings where you have no potential. So typically, you have a potential, and then this breaks the target space symmetries. And because then in three dimensions, what you get is gravity coupled to a non-linear sigma model plus a potential for the sigma model fields. So for instance, in the example that I just quoted, so try to add a cosmological constant to 40 Einstein-Maxwell gravity at the lambda. And what happens is that of the 8 SU 2.1 generators, three are broken by the potential for the scalar fields. And these three, they correspond to what are called generalized ALOS, the generalized ALOS transformation, and two Harrison transformations, which are instrumental in generating new solutions from a given seed. And the residual symmetry is just a semi-direct product of a one-dimensional Heisenberg group and the translation group, R2. And this residual symmetry, unfortunately, cannot be used to generate new solutions. So nevertheless, in n equals to FI gauge supergravity, there is a solution-generating technique. And it involves essentially the stabilization of the symplectic vector of gauge couplings, the so-called FI parameters, under the action of the U-duality symmetry of the ungaged theory. So the original idea goes back already to 2013 to this paper of Hall-Margie and Vanell and was then elaborated by us in 2016. So what is the basic idea? Well, the U-duality is just the global symmetry group of the ungaged theory. And this consists of the isometries of the special Kehler nonlinear sigma model. And these isometries act linearly also on the field strength that you have in your action. And if you have purely electric gaichings, then, as I said, the scalar potential spoils this invariance. But if you allow also for dionic gaichings, so what dionic gaiching means that in order to gauge the symmetry, you don't use just the gauge fields that you have in your action, but also their duals. This is dionic gaiching. And if you allow also for dionic gaiching, you can recover the whole U-duality invariance. But the price that you pay is you change the vector of gauge couplings and sort of physical theory. So where you see this, well, you have here the scalar potential in the FI gaiched case written in a symbolically covariant way. So this, well, G is the special calorimetric. These dis are calorimvariant derivatives. And this L is a symplectic product of G and this curly V, where G is here the symplectic vector of FI parameters. And the curly V is the symplectic section. So you see in the electrically, in the case with just electric gaichings, you have only the cheese with the lower index. But if you also have magnetic gaichings, then you have also the cheese with the upper index. So this is the potential can be written in this symplecticly covariant way. And well, let us call this group, this U-duality group of the ungaged case UFI, where FI stands for fake internal. So this is the fake internal symmetry group. And this will act on a solution by mapping it to another solution of another theory, because it acts also on the cheese. So it changes the FI parameters. So you go to another theory, right? But then, OK, if you can fix a generic choice of the coupling constant G, and then what will be the true internal symmetry group that we call UI? The true internal symmetry group of the gaiched supergravity theory is then just the stabilizer of this symplectic vector of gauge couplings under the action of the U-duality group of the ungaged theory. And this can be non-trivial. For example, if you have this cubic pre-potential here, then you'll find that the U-duality of the ungaged theory is SL2R to the third. And if you choose your FI parameters such as to have only electric gaichings, then you'll find that the symmetry group of the gaiched theory is U1 to the third. So there is something non-trivial surviving. And let me give you an application on how to use this in order to add rotation to a given seed. So we start from a magnetically charged BPS black string in the five-dimensional FI-gaged STU model, theory with three real scalars. And this string has also some momentum along the string. So this is the metric. Well, the horizon is hyperbolic. And here the constant L is just the asymptotic ADS-5 curvature radius. The function H is given by this row is the radial coordinate. The function H0 is just a wave profile. So you have a wave along the string. And the wave profile is this H0, which is this function here. And Q0, C, and H0 are just constants. So I call this Q because if you compactified the solution down to four dimensions, then this Q0 will be an electric charge. And OK, you have also fluxes turned on. So magnetic, you have magnetic fluxes on this hyperbolic space. And the scalars are constant. But this was found already in 2007, this solution. And well, now you can dimensionally reduce along the string coordinate U. So this coordinate U here dimensionally reduce. And what you get is then a BPS black hole in the four-dimensional FI-gaged model with this cubic pre-potential that has one electric and three magnetic charges. So the magnetic charges come from the magnetic charges in five dimensions. And the electric charge comes from the momentum along the string in five dimensions. And now you can apply a 4D duality transformation. So remember, the surviving symmetry group was U1 to the third, so you can use this. And then after that, you can relive the solution to five dimensions. And what you get is then a dionic black string in five dimensions. And with dionic, I mean you just have an electric charge density along the string. And now you continue and you dimensionally reduce now not along the string coordinate, but along the angular coordinate, this phi, and apply again a duality transformation in four dimensions and then relive to five dimensions. And then you get in five dimensions a black string that has both rotation and momentum. So you see, you can use this surviving symmetry group, this surviving duality invariance of gauge supergravity to add rotation to a given seat. So well, this is, of course, a well-known procedure for ungauge theories. This is usually how you create solutions. But in the gaged case, I think this is the first instance of using solution-generating techniques in order to add rotation to a given seat. And the solution, well, I do not write a metric which is quite complicated, but it interpolates between what is called magnetic ADS-5 at infinity. I mean, magnetic means that the metric is ADS-5, but in addition, you have magnetic fluxes on the H2 on hyperbolic space. And in the near horizon regime, it becomes a deformation of ADS-3 cross H2. In fact, you can write down the IR geometry, which is this one. And you see, well, OK, I set all the magnetic charges equals. So P is just P1 equals P2 equals P3. Q0 must be negative. Omega is the rotation parameter. And you see that this is a deformation of ADS-3 cross H2 I mean, for omega equals to 0, this would be just H2. And this is ADS-3, OK? So this is the deformation of ADS-3 cross H2. And note that the ADS-3 factor, which is what you see in the first line, it is not the usual metric on ADS-3, but it is. But you have to apply a coordinate transformation. It is ADS-3, written as a hop-flag vibration over ADS-2, in which dt is a null direction. So this part here is just ADS-2. And then you have a fiber over ADS-2. So this is, if you want, an analytic continuation of the usual hop vibration, where you write S3 as a hop vibration over S2. And yeah, well, this comes from the momentum along the string, actually. OK, so in the last part, I want to talk about coupling to hypermultiplads. So what is this good for? Well, DPS black holes in the four-dimensional model with this pre-potential, the sort of STU model, coupled to the universal hypermultiplad, they have recently been attracting much interest in the context of microscopic computations of their entropy using localization techniques. In fact, what was done was to consider BPS black holes in ADS-4 cross S6 in massive type 2a. And it is known that they are dual to three-dimensional Hichon-Simons metagate series whose topologically twisted index can again be computed exactly in the large end limit, three papers here. And well, this theory is massive type 2a on ADS-4 cross S6 if you dimensionally reduce on S6. So this admits a consistent rankation to iso7-dionically-gaged four-dimensional N equals 8 supergravity. And this theory here can still be truncated. So this can be further consistently truncated to a four-dimensional N equals 2-gaged supergravity model coupled to three-vector-multiplads, precisely with this pre-potential, and the universal hypermultiplad. So this is still another further consistent truncation. And what is dionically-gaged is a group R cross U1 of the quaternionic hyperscalar manifold, universal hypermultiplad, where this R corresponds to the axionic shift symmetry I will show you in a moment. And I said that you get a theory that has dionic gauging, so electric and magnetic. And where does the magnetic gauging come from? Well, it is induced by the romance mass in the massive type 2a uplift. So this gives you the magnetic gauging in four dimensions. And well, the metric on the hypermultiplad and the modulized space, the universal hypermultiplad, is this. So you have to devise the dilatone, A to axion, and then you have other two scalar fields. So you see what your gauge is. You have here an axionic shift symmetry. And then the U1 is a rotation in the xi upper 0 and xi lower 0 plane. This is what is gauged. So the gauging involves these killing vectors here, dA, and this rotational killing vector in this plane. And well, the black hole is actually whose entropy was counted microscopically in the papers that I showed you. They are not known analytically, but one can obtain the near-horizon geometry using the symbolically covariant attractor equations of this paper here. So what is known is only the near-horizon geometry, but this is sufficient. And well, to the theory with hypermultiplads turned on, there are no numerical solutions. For instance, here by Halmagi-Petrin in Saffaroni, they obtained a numerical solution. But question, can we also get some analytical results? I mean, after all, OK, if you do a microstate computation and you know only the near-horizon geometry, I mean, actually, you are not sure if there is really a flow from this IR geometry to an ADS4 in the UV. In principle, you have to check at least numerically if they are connected. So can we get some analytical results? Well, we can try to use this paper by Mason and Ortin. So they classified all supersymmetric solutions of four-dimensional n equals 2 gauge supergravity coupled to both vector and hypermultiplads. So they classified. They obtained some first-order equations. And then you can try to solve these. And unfortunately, this was done for electric gaugings only. But we can still go ahead and see if we find something interesting. So for completeness, I give you the Poissonic Lagrangian. So what you have here, again, the kinetic term for the scalars in the vector multiplads, the kinetic term for the scalars in the hypermultiplads, and these gauge field terms you saw already before. And then the scalar potential depends now also on the hyperscalars. And it's given by this expression here where the l sigma are the upper part of the symplectic section. The k lambda u are the killing vectors of the hyperscalar target space that we use to gauge. And the p lambda x are called the moment maps where x goes from 1 to 3. And you have here covariant derivatives of the hyperscalars that are given by this expression here that involve this linear combination of the killing vectors and the gauge fields that you have. And what we are gauging is our only abelian isometries of the quaternionic scalar manifold. So we do not gauge any isometries of the special scalar manifolds parameterized by the sets. So that's why you have here ordinary derivatives. Good. And to keep things simple, but as we will see yet non-trivial, let us choose a very simple model, which is this pre-potential here, x0, x1, which has it has just one vector multiplet. And by the way, this is a truncation of the other one. The other one was this. And put some x is equal, and then you get this model here. And we coupled this to the universal hyper-multiplet. And what we gauge are the isometries generated by these killing vectors, where the K lambda and the C are constants. And this is precisely, well, apart from the fact that in the 2A theory that dimensionally reduced, 2A theory that I mentioned before, you have also magnetic gaugings. But this is, apart from that, precisely the same. So you gauge here the axionic shift symmetry and the rotation in the xi lower 0, xi upper 0 plane. Good. So with this, you get the following scalar potential. It's a function of phi, the xi's, and the z that sits in the vector multiplet. And this has a critical point for the xi's r0. Z is given by this value here, the phi minus c over k0. And the value of the potential at a critical point is this. So you see, well, z must be positive, in order not to have ghost modes in the action. So you need k0 over k1 to be negative. e to the 2 phi must also be positive. So c over k0 must also be negative. And then v critical will be negative. So this corresponds then to a negative cosmological constant. So you have an ADS vacuum. OK, so using the recipe of this paper here, where all the solutions were classified, we obtained the following solution. This is the metric. You still have a hyperbolic horizon. These are the gauge fields, the two gauge fields. So you see they are magnetic. So you have magnetic flux on the H2. The dilatone goes logarithmically with r, the radial coordinate. The axiom is constant. The x-sides are 0. And the scalar field in the vector multiplied goes like r squared. So first of all, note that this solution has no free parameters. Because all the constants that appear in the solution, the k's and the c, they are already determined by the choice of gauging. So they are, well, constants that appear in the action. They are no free parameters of the solution. You have a curvature singularity now equals 0. And you have a killing horizon in the square root of minus k0 over c. So this is a true BPS black hole with non-trivial pi that sits in the universal hypermultiplet and also running set that sits in the vector multiplet. You have these magnetic charge densities that are given in terms of these constants that determine the gauging. And in terms of these, you can write the entropy density in this way. And you can easily see that the near-horizon geometry is AdS2 cross H2. But what is the asymptotic behavior for large r? So you might guess that it is AdS4, but this is not true. So for r goes to infinity, the metric goes to this metric here. And you see that in the bracket, this is, I changed the signature, sorry, this is AdS2 and this is H2. But you have a conformal factor in front of the bracket. So the UV geometry is conformal due to this conformal factor to AdS2 cross H2. And this is actually very similar to what is called a hyperscaling violating geometry. So let me briefly explain what this is. So in d dimensions, a hyperscaling violating geometry has this form. You have, OK, the i goes from 1 to d minus 2. And you have here these two exponents, the theta and the set. And the set is what is called dynamically critical exponent and theta, the hyperscaling violation exponent. And if you scale your coordinates in this way, you divide r by lambda, multiply the axis by lambda, and t lambda to the set times t, then the metric is not invariant, but it transforms covariantly. So it takes this overall factor of lambda. Let me just mention that geometries of this form have been instrumental in applications of the ADS-CFT correspondence to condensed matter physics. The systems that you consider in condensed matter physics are typically non-relativistic, so a space and time they scale differently. And well, actually, our UV geometry exhibits a similar scaling behavior. And in order to see this, let us introduce new coordinates x and y on the hyperbolic space that are related to phi and theta by this equation here. And then you can write the asymptotic geometry in this way, so now the h2 is written in this form, which is Poincare-Half-Plane. And then if you scale in this way, then you see that this metric transforms as dS goes to dS over lambda. So this is very similar. And one interesting point is, as I showed you, that the scalar potential has an ADS vacuum, but the solution does not go to this vacuum asymptotically. Instead, it goes to a hyposcaling-violating geometry. Let me come to the summary. So in the first part, I gave you an overview on known black hole solutions in various models of Abelian FI-gaged supergravity and showed you for which class microstate computation is available. And then I introduced this solution-generating technique still in the FI-gaged case. And this involved the stabilizer of the vector of FI parameters under the u-duality of the un-gaged theory. And as an application, I showed you how to add rotation to a BPS black string in the five-dimensional FI-gaged SCU model. And the last part, we considered coupling two hypermultiplets. And we saw that one can construct a BPS black hole that interpolates between ADS2 cross H2 in the IR and a sort of hyposcaling-violating geometry in the UV. And it would be interesting to extend this to include also magnetic aging to see if one can construct analytically the solution for which a microstate computation was done. Thanks.