 Okay, the mic is on. So let me start by thanking the organizers of this wonderful workshop. It's been extremely interesting so far, as well as the school last week, which was also very nice. So in fact, it was maybe a little bit too interesting because there's a lot of talks that already happened where some of the material that I present were already there. But I guess repetition is not necessarily a bad thing. So apologies if you're already familiar with parts of the story, which concerns supergravity localization and ADS black holes. So this is based on this recent paper that we wrote together with Kirill Ristov and Ivano Lodato. So I'll start with some very basic introduction and motivation. Our starting point are black holes. And this is because essentially black holes are a theorist laboratory to try and understand gravity due to this key property that they have an entropy. And this entropy, as we all know very well, semi-classical physics shown by Beckenstein and Hawking gives you the leading term and some large area expansion that's the A over four. And here I reinstated all of the fundamental constants so you could see this H bar. But then you also expect that there's gonna be corrections to this Beckenstein Hawking entropy, which will take the form of an expansion in a large area. So you're gonna have some log of the area term and then a bunch of other stuff and even possibly even some exponentially suppressed terms that are coming from some non-perturbative effects. So when I refer to black hole entropy, I really wanna talk about the full exact black hole entropy and not just the Beckenstein Hawking entropy. And these corrections here, they're actually interesting because they start probing the quantum gravity regime. And in fact, while Beckenstein Hawking is completely universal, all these coefficients and these expansions, this alpha, beta, et cetera, might depend on the details of the theory and become sensitive to all of this. And one way to see this, and this was a natural question that arose immediately after Beckenstein Hawking started associating entropy to black hole, is can you give a Boltzmann interpretation of this exact entropy here in terms of some microscopic degeneracies making up the black hole? So the question to ask is, is there such a quantity, the micro, for which the exact entropy can be written via the Boltzmann law? Right, so here I'm just gonna very briefly review some of the progress that has been made in trying to understand and answer this question. And much of the progress has been made for supersymmetric models, so that will be the focus of my talk. I'll talk about supersymmetric black holes only. So for asymptotically flat black holes in the seminal calculation of Schrodinger and Waffa, they showed that string theory can successfully account for the Beckenstein Hawking entropy of the black hole by realizing it as some kind of brain system with some strings. And we had a very beautiful talk by Coulomb Waffa for the inauguration of the new center here that sort of reminded us of this very fundamental calculation. I also wanna remind you that in fact, this string theoretic or brain picture is actually very powerful because not only can you recover Beckenstein Hawking, but you can also compute the subleading corrections to the entropy of the black hole in some particular cases. And here I'm just quoting some of the important papers in this direction. And in fact, for certain supersymmetric black holes, the microscopic degeneracies are fully known as functions of the charges. And that essentially was achieved because the generating functions for these microscopic degeneracies can be obtained from some topological invariance in the string theory, such as the elliptic genus and generalizations. So that's a very beautiful story. There's still much to be said as also Coulomb Waffa this morning was explaining. But really a lot of progress has been made beyond Beckenstein Hawking for these black holes. So now I just wanna turn to asymptotically ADS black holes, which are another class of black holes. And here, much more recently, some progress was made specifically at first for ADS-4 spherically symmetric BPS black holes. And the microscopic counting for their microstates was achieved in the dual field theory, which lives at the boundary of this ADS-4 space. And this was in this beautiful paper by Benini Iristof and Zaffaroni. So this is much more recent. This was in 2016. And here the idea was to compute a quantity in the CFT-3, which is known as the topologically twisted index. And you are able to do this calculation exactly using supersymmetric localization in the CFT. Now, what you get is a matrix model out of this. In principle, this is an answer which is valid for all n, but it's quite excruciatingly difficult to evaluate exactly. However, Benini Iristof and Zaffaroni were able to show that at large n, you can take some approximation on this matrix model and it reproduces the Beckenstein Hawking entropy of the black hole. The sublating contributions or corrections to this Beckenstein Hawking entropy are still encoded in the matrix model, but it's much harder to sort of extract them and try to get some log n term or one of our n corrections. And here I just want to mention that after this followed a lot of generalizations to other models, other dimensions. I'm giving a list of names here. I really apologize if I missed yours. Please come at lunch and insult me. But like there's been a lot of recent work in the last two years, essentially on these ADS black holes. So that's the story about the microscopic counting using the dual CFT. But in this talk, I want to ask kind of a, well related but somewhat different question. So given the degeneracies that are computed in the microscopic picture, can we define and hopefully compute the corrections to Beckenstein Hawking entropy and get this complete quantum entropy directly in the microscopic picture, i.e. the dual gravity theory. And if so, of course, if you can do a calculation on the gravity side and on the CFT side, do the two calculations and descriptions of the black hole agree. And if so, at which order. So I want to examine these questions using the notion of quantum entropy for these asymptotically ADS black holes and supersymmetric localization, but directly in the super gravity or gauge super gravity theory where the black hole lives. And here I just want to mention that these ideas of applying localization in gauge super gravity were already put forward in these papers where no black hole really was considered. It was more about just ADS space. And these two notions here, the quantum entropy for asymptotically ADS black holes are more generically black holes and applying localization directly in the super gravity theory were already presented and discussed in the two very nice talks that we had yesterday by Bernard David and Imtac Gian. So some of what I'm going to say is going to be familiar from these last two talks. Right, so that was the intro and motivation and kind of like what we're going to try to do. So I'll go forth and set up the problem and then show you what one can achieve with localization in gauge super gravity and the quantum black hole entropy. And I'll end up with some conclusions. If there are any questions at any point just feel free to interrupt. Right, so setting up the problem. In general, we want to examine solutions of four dimensional N equal to two gauge super gravity and these black hole solutions of electric and magnetic charges and ADS asymptotics. The previous talk by Dietmar gave us a very nice summary of various solutions that exist for these 40 N equal to two gauge super gravity. So typically the full black hole solution will interpolate between some ADS four vacuum at infinity and there's a near horizon region which has the form ADS two times S two or more generally as Dietmar reminded us some more exotic type of near horizon topologies where you can have a Riemann surface or more general things than the two sphere. In this talk I will just for simplicity focus on ADS two times S two and I'll come back to a more general case perhaps at the very end. So the full solution preserves two supercharges it's quarter BPS, but in the near horizon region there's an enhancement and the region itself preserves four supercharges and it's half BPS in the language of 40 N equal to two. This enhancement is in fact a general feature of the attractor mechanism. This might be familiar to you from un-gauge super gravity. This also happens in gauge super gravity as was already explored by these papers. So what I wanna do for the rest of the talk is to focus on the contribution to the quantum entropy coming from the near horizon region itself. And the reason I wanna do this is that this will allow me to consider a large class of asymptotically ADS black holes all at once namely all of those who have a near horizon attractor geometry ADS two times S two. So I'm not gonna be worried too much about where the black hole is coming from or what kind of asymptotics it has. I just want for it to have this ADS two times S two near horizon geometry. Hopefully in the second part of this talk after lunch Kiril Laristof will tell you about more of the asymptotics, but here I just wanna focus on the near horizon region. And the other reason why the near horizon is interesting is that there's this ADS two factor for these BPS black holes. And that's a key ingredient for us to explore the quantum entropy of black holes. The reason is this proposal that was put forward by Ashok Sen in 2008 that stated that if you want to compute the quantum entropy of the black hole you should compute the following expectation value. And this should also be familiar to you from the talk by IMTAC yesterday. This was the same quantity that he mentioned. So I'll denote this by W in the rest of the talk. This is essentially the expectation value of a Wilson line that you're inserting at the boundary of your ADS two space. And this Wilson line here is here to make the variational problem well defined. And in fact it enforces the micro canonical ensemble where I'm gonna look at black holes with fixed electric and magnetic charges. And that's kind of like the boundary condition that I wanna impose when I'm computing this expectation value. Now on top of this there's also some regularization scheme that one needs to put forward because of the essentially because of the infinite volume of ADS two. So that's what this superscript finite here refers to. And it was already shown by Sen that in a suitable large charge limit this expectation value you can do essentially a saddle point approximation. And this will reproduce for you the Beckenstein-Hawking entropy of the black hole or more generically if you want to consider theories with higher derivative terms this will be even the full Beckenstein-Hawking walled entropy for the black hole. But here you're really doing an expectation value so you're hoping to really capture all of the quantum corrections to Beckenstein-Hawking entropy. And just for some final motivation for why this proposal is interesting this was argued in the original paper on the basis of the ADS two CFT-1 correspondence where holographically you expect this to correspond to some Witten index in a dual CFT-1 counting the number of ground states which you interpret as the microstates of the black hole. So that will be the main player of the game and I just want to basically compute this quantity for the black holes engaged supergravity. So yeah, as I said, this defines the non-perturbative entropy of extremovps black holes. I want to stress as well that this is a Euclidean path integral. And we all know that supersymmetric localization very generically gives you exact one loop evaluation for path integrals. So what I want to do is apply supersymmetric localization in four-dimensional line equal to two gauge supergravity. Now to apply localization it's very convenient to have an off-shell formalism. And as Bernard was explaining yesterday, this is in fact crucial if you want to go ahead and make sense of the localizing supercharges that you want to use in supergravity. And to ensure off-shell closure of the underlying gauge algebra, I'll make use of the superconformal formulation of gauge supergravity which was put forward a long time ago already. And so, right, that's essentially the theory. I have some superconformal off-shell Euclidean gauge supergravity in four dimensions and equal to two. So the field content consists of a via-multiplet, which essentially is the multiple that contains the graviton as well as a bunch of other fields. And I'll be considering NV physical vector multiplets. But because I'm using the superconformal formulation, I need to introduce some compensating multiplets in order for the theory to be gauge equivalent to Poincare's supergravity. And this is why I have this extra vector multiplet here and one hyper-multiplet. So these will be my gauge compensators to sort of compensate for the extra conformal gauge symmetries that I introduced for the off-shell formalism. But physically I have some gravity multiplet and NV vector multiplets, sorry. So the steps will be as follows. First I want to go through and specify this near-horizon half-BPS field configuration for my vial vector and compensating hyper-multiplet. And then we'll look for off-shell fluctuations, BPS fluctuations with respect to a localizing supercharge that I'll show you. And so these fluctuations are around the half-BPS attractor background. And I also want these fluctuations to satisfy the boundary conditions for the quantum entropy, meaning that these fluctuations should go to zero at infinity and I should recover the attractor geometry. So that should also be familiar from the boundary conditions that Bernard was mentioning yesterday in a much more general setting where you had some background field split and then the quantum fluctuations should die off at the boundary. Right, so let's get into a little bit more of the details. So this is the Vy-multiplet. As I mentioned, it contains the graviton or the field bind. There's obviously the gravitini. Here, I is an SU2R index because I have n equal to two. And there are a bunch of gauge connections for the extra conformal gauge symmetries. So this is a dilatation gauge field. This is the U1R gauge field or more appropriately in Euclidean, the SO11R gauge field. And this Vmu ij here is the SU2R gauge field. And then I have some auxiliary fields that are required for off-shell closure of the algebra. So there's a rank two tensor. There's some extra fermion here, chi, and there is a scalar field, d. That's my Vy-multiplet. Moving on to the vector multiplets, as I mentioned, will be in Euclidean. And in fact, the vector multiplets you might be familiar with in Lorentzian typically contain a complex scalar. Here in Euclidean, there are actually two real scalars that I'll denote by x plus and x minus. And in fact, with Bernard DeVitt, we worked out the full off-shell formulation of these Euclidean supergravity theories. And that's where you see these two real scalars appear. Morally, you can think of them as the real and imaginary part of your complex Lorentz scalar. There are also vector fields, obviously. I'll denote them by W mu, just so that it doesn't clash with the A mu up here, this S-O-1-1 R gauge field. And then there's some Gagini, and of course, some auxiliary fields, which again are required for off-shell closure. So these guys are just triplets of the SU2R. And I have a collection of these vector multiplets with I running from zero to Nv. And finally, there's this compensating hypermultiplet. For the moment, I'll just discuss its field content on-shell and then we'll come back to putting an off-shell later during the localization procedure. But on-shell, this just consists of some scalars, and they're from Yannick's super partners. And this is a good way of illustrating where the gauging comes in, because we're gonna talk about a gauging, which is specified by, so typically also as Dietmar reminded us, we have these symplectic vectors of gaugings. Here I wanna consider only electrical gauging. So I'll set the magnetic counterparts here to zero. And so the gauging is specified by these Feyyay-Liopoulos parameters, XII, as well as some generators, T-alpha-beta. And you can see this very clearly. For instance, if you look at the covariant derivative of the scalars and the hypermultiplet, where again, you have all these connections from the y-multiplet, and the gauging entering here, which charges this hypermultiplet section under some linear combination of the gauge fields in the vector multiplet. Now, this is in the conformal theory, so that's in the conformal compensator, but then if you gauge fix all of this to Poincare, you'll see that this induces a charge for the gravitini. So here I'm really talking about gauge supergravity with these electric gaugings. Right, so let's talk about the half-BPS near-horizon configuration first. So here in trying to keeping in touch with Bernard's notation from the talk yesterday, I'll denote the on-shell attractor configuration by a ring here or a dot above the fields, and this will really be my background fields, and then I'll think of fluctuations around this background. But for the background, so we have this metric, that's the near-horizon region. It consists of this ADS2 times S2 factor. Notice that there are two parameters here, V1 and V2, or sizes. In gauge supergravity, they're not equal, which is also a difference with the ungaged supergravity case. And then we have some non-zero configuration for this auxiliary T tensor, as well as for the scalar fields that sits in the vi-multiplet, which are all given by these V1 and V2 constants. Now there's, as I said, this hyper-multiplet compensator. Now in the background, I mean, this guy is just here to fix the SU2R gauge when you go to Poincare. So in the background, you can really set it to a constant. And essentially here, this will identify this alpha index, which was an SP1 index with the SU2R in the theory. And finally, for the vector multiplets, again, the attractor configuration. So you're gonna have some linear combination of the charges or the field strength specified by this V2. And the scalars are also, here are their attractor values in terms of V1. And then because you're on shell, you can also solve for the auxiliary fields in the vector-multiplet. And these will be given by the following expression, Nij, by Nij here, I mean the same period matrix that Dietmar was using earlier, which is given in terms of some pre-potential for the theory. So this is my attractor configuration, almost complete. There is one more non-zero field. And this is the SU2R gauge field that sits in the V-multiplet. And this guy here on the background is related precisely to this combination of gauge fields in the vector-multiplet. So these are all the non-zero fields, all the other ones are zero, and of course I'm just looking at a bosonic configuration. So now that we have our background, we're gonna look for off-shell BPS fluctuations or like the quantum fields that Bernard was talking about. And for this, the very first step that one should do is solve the gravitini supersymmetric variations for arbitrary metric and arbitrary killing spinners that respect the boundary conditions for my quantum entropy. Now of course in general, this is a very hard problem. And here I wanna remind you in un-gauge supergravity, this was solved for fluctuations around the full BPS attractor geometry. And in particular in this full BPS attractor configuration for un-gauge supergravity, this SU2R gauge field vanishes. Recall that otherwise it's precisely given, it's non-zero for us because of the gauging. And so this problem here was solved in a great paper by Gupta and Murthy. And what they showed is that the only BPS configuration which is allowed is the attractor geometry itself. So essentially there are no off-shell fluctuations for the vi-multiplet fields and they're all frozen or pinned down to their attractor full BPS configuration. Now what we tried to do was essentially to go through this analysis again, but because of the gauging, because of the non-trivial backgrounds that we have for the SU2R field, et cetera, we rather quickly hit a wall. And so here will come the central assumption for the rest of the calculation. Here in gauge supergravity, we will assume that this is also the case and that the vi-multiplet is frozen to its attractor geometry. So this basically the sort of slogan is this w equals w naught. And this also means that I can use the attractor or background killing spinners to parameterize my superchargers with which I'm gonna affect localization. Right and so now I just wanna look for off-shell BPS fluctuations but in the gauge and matter fields only, so in the vector multiplets and the hypermultiplet. Right so we have, as I said, the background is half BPS. We have four superchargers and out of these four superchargers, we can pick one which we're gonna call Q lock, that's the charge with which we're gonna do localization and it has the following algebra, the one we picked. So this should look very familiar to most of you but this is essentially a lead derivative along the tau direction which was this Euclidean time at the boundary of ADS2. And then we have some SU2R transformation as well as some gauge transformations which are parameterized by the scalars in the vector multiplets. This type of gauge transformation is really inherent to the supergravity formulation. It also played a role yesterday in the talk by IMTAC. But essentially this algebra should look rather similar. This is just some lead derivative and some R transformation. Right so that's our setup. We have the localizing supercharge. We've made our assumption that the via-multiplet is frozen to its background value and now we can just go ahead and analyze the localizing equations in the vector and the hypermultiplet sector. So in the vector-multiplet sector, we'll just examine the consequences of imposing the vanishing of the Gagini variations. Now here comes another specificity of gauge supergravity and one of the important differences with similar calculations that have been done in ungauge supergravity. And the point is that the supercharge here Q lock is parameterized, as I said, by an attractor or background killing spinner. But because of the non-trivial SU2R gauge field in this background, the gauging actually affects a twist on the two-sphere of the ADS2 times to near horizon geometry. And this means that if you solve for the killing spinner in this background, you'll find that the killing spinner is actually constant on the two-sphere. There's no angular dependence at all because you've canceled essentially the spin connection with the SU2R gauge field, which is non-trivial. So in a way, the localizing equations that I wanna solve here, they're non-trivial in the radial direction of ADS2, but they're much less constraining in the angular directions because I just don't have anything non-trivial happening along the sphere. And this is a crucial difference with the ungauge supergravity or flat black holes. Nevertheless, let's not worry too much and press forward. So we want to look at a new find solution to these kind of equations and we'll parameterize the off-shelf fluctuations. Again, here I'm just trying to use a notation which should echo what Bernard was mentioning yesterday where we have a background and some fluctuations for our scalar fields, X plus and X minus, the field strength and the auxiliary fields. Remember, I'm looking for off-shelf fluctuations so the auxiliary fields might also have some non-trivial profile. I'm not requiring that these fluctuations solve the equation of motion, but simply that they solve the localizing equations for that one, particularly supercharge. So these equations ultimately will lead to some constraints on these fluctuations, X plus FAB and YIJ. Right, so you write down the equations, you go and try to solve and the end result that we reached is that also taking into account the boundary conditions, as I said, these fluctuations should die off at the asymptotic of the near horizon configuration and it should really recover the attractor geometry there. What we found is that the scalar fields here, they can be excited away from their on-shell value and the excitations, you can parameterize them, so each of the scalar fields, X plus and X minus, has a parameter C plus and C minus, but in fact, there's a whole tower of parameters which are indexed by this index K and these fluctuations die off as one over R to the K. So that's already quite a lot of parameters and I wanna just, I haven't written down the expressions but essentially the gauge fields and the YIJs, they're off-shell fluctuations are also gonna be given in terms of these C plus and C minuses. But not only do I have a bunch of those for all values of K from one to infinity, but these parameters are actually dependent on the two-sphere coordinates because as I said, I don't have any equation and any localizing equation to constrain their dependence so these guys are really just functions on the two-sphere. So I have two real functional parameters or rather infinite families of functional parameters for each vector multiple. Right, that doesn't look too good and there seems to be a lot of off-shell fluctuations allowed, however, if you analyze the localizing equations, you'll find an additional constraint and that is that there's a very specific linear combination of these fluctuations parameters C plus and C minus which turns out to be independent of the angular coordinates and that will be very important later on. So just to summarize, essentially this localizing manifold, the set of BPS solutions to the localizing equations in the vector-multiplet sector is given by these two types of CK plus and CK minus. There's one for each vector-multiplet, and V plus one and for every K but you can sort of repackage them into a combination that I'll denote by Phi plus which is really independent of the two-sphere that's really a constant parameter and well, all the other ones that are orthogonal to this particular linear combination that I'll denote by Phi perp and these guys are really angular dependent. Yes, yeah, especially at the horizon that R equals one, you want these sums to be finite. Yeah, it doesn't, I mean, yeah, there's some additional things here that I'm not, we can run through the constraints but essentially what you find is that it's not enough to completely kill off the angular dependence or the fact that you have this tower of excitations. One more caveat in this solution is that we've assumed smoothness of the BPS solutions. Of course, there could be additional singular configurations that might contribute. Here, I just want to briefly remind you that this was also the case in engaged supergravity with these kinds of geometries but for the moment I just want to set that aside and keep on with the smoothness assumption. Yes, they solved that one BPS equation for that one supercharge Q lock. Yes, that's my localizing supercharge with which I'm, right, right, no but then of course if you take your four supercharges then you'll find that these guys have to be zero because I need to recover the attractor geometry. That's the half BPS one that, but here I'm only really asking for one supercharge which is the localizing supercharge that I'm using. For the compensating hyper-multiplet you can also analyze the variation of the hyperini with respect to the localizing supercharge. Now here, I briefly mentioned earlier I need to put these guys off shell and for this we used a technique that was introduced by Berkovich and also beautifully illustrated by Hamanozomishi and it consists in the following. Basically you want to modify the standard variation of the hyperino by introducing some auxiliary scalar fields H and some constrained parameters side check. So these side checks have to verify some non-trivial equations in terms of Xi which is the parameter of my localizing supercharge but if you introduce these auxiliary fields and you give them an appropriate transformation under Q lock you'll find that you can close the algebra without using the fermionic equations of motion. So again we impose the vanishing of the hyperini variations and of course you see here that in the variation itself I have the gauging which is entering once more and the gauging here is selecting this linear combination of scalar fields. The scalar fields have off-shell BPS fluctuations so these are gonna get related to the off-shell BPS fluctuations allowed for the hyper-multiplets. So essentially what you'll find is that the fluctuations of these scalars are controlled by the parameters that I introduced earlier CK plus and CK minus. But in addition there is one more constraint that you can get out of this equation and that's a constraint on the linear combination that was independent of the angular coordinates and essentially you should satisfy this constraint that is specified by your choice of gauging. So that constraint comes out of the hyper-multiplet compensator. So in total we have characterized this localizing manifold in the gauging matter sector and what you get is phi plus phi perp which is angular dependent and this constraint for phi plus. Now what you wanna do is evaluate the action of gauge supergravity on this field configuration just as a brief aside in the Euclidean Formalism the action is actually specified by two pre-potentials which are functions of these X pluses and X minuses they're not related by complex conjugation but now comes an important observation when you plug in this field configuration you realize that the bulk action of gauge supergravity specified by a generic pre-potential here actually only depends on this constant parameter phi plus and it doesn't depend on phi perp. And in fact you can go further if you add the Wilson line contribution and the appropriate boundary terms to renormalize according to this finite prescription you reach the final form for the action on the localizing manifold which is only a function of phi plus and it takes a very simple and surely to most of you familiar form very schematically this is coming from the action this is coming from the Wilson line but everything only depends on phi plus which are these constant parameters. So the phi perps are really zero modes of the bulk action and the Wilson line. So because there are zero modes they can only enter my localized quantum entropy function not via the action itself but via the one loop determinants that I still have to compute and put into my formula. Now we haven't computed or these one loop determinants precisely but we wanted to argue the following that we can actually split these one loop determinants into a contribution from these phi plus modes and the ones from the phi perp modes and this one seems rather dangerous because it still really contains a functional integral. I have functions on the two sphere and so I've localized but I still have some functions that I need to integrate over and the reason for this as I mentioned many times is the gauging and the resulting twist that we had on the two sphere. But putting everything together and allowing for a bit of a formal notation what you'll find is the following. So here this was my WPQ from the beginning. I put a hat on top of it just to remind you that these are for smooth BPS off shell configurations. So I have an integration over these phi plus. This is a standard finite dimensional integral one for each vector multiplet. I have the constraint which was coming from the hyper-multiplet compensator which I'm putting in as a delta function. Then there's the exponential of the renormalized action so that's our QIFI minus PIFI. I should just mention but this is rather standard notation like this FI here is the derivative of the pre-potential with respect to one of the scalars. And then I have some one loop factors one of them which will depend on phi plus and some possibly horribly complicated messy perp factor which is coming from these angular dependent integrations on over phi perp. So as I mentioned for the moment we have what we did was characterizing the localizing manifold in the paper evaluating the action on this but we're still trying to compute these one loop determinants and I'll come back to this towards the end. Right but let's forge ahead a little bit and just see what we can get out of this formula. So setting aside these unknown factors we can analyze the saddle point of this of this integral. And for this you can define what are perhaps to some of you familiar combinations which I'll denote by Z and L. This L also made an appearance in Dick Mars claim talk sorry earlier. And so if you compute the saddle point you'll see that there is a constraint that L must be equal to two pi that was our constraint coming from the delta function. And the saddle point equations coincide with the standard attractor equations engage super gravity where you're supposed to extremize this quantity Z over L. And furthermore the value of Z over L at the extremum is the Beckenstein Hawking entropy of the black hole. So even though we have still some unknown and possibly very complicated factors in this formula we're kind of on the right track because if you start looking at some saddle point approximation you find the standard gauge super gravity attractor equations and the value of the Beckenstein Hawking entropy. So this is a strong hint that these one loop and perp factors will not contribute at this leading order level. Otherwise that would spoil the agreement that you find with the gauge super gravity attractor equations. And at least for the one loop factor this is reminiscent again of what was happening in engage super gravity or for the flat black holes where the Z one loop that IMTAC give us a very nice review of yesterday and the complete form. This contributes at order log of the area and beyond. So it does not spoil Beckenstein Hawking. Okay, there's some other checks that we can do on this formula that we obtained. So now you can sort of try and specifically think about, okay, I have this black hole in engage super gravity with some ADS asymptotics. For instance, with some ADS four times a seven. So as I mentioned in the beginning and we've heard now many times this has some dual holographic description as some ABGM three dimensional field theory. And there is this topologically twisted index that's supposed to capture the microstates of the black hole. So the index itself was computed using localization but rigid localization. So here really in the field theory. And the degeneracies for the supersymmetric ground states are given as a function of the charges after euphoria transform the index and here you get DCFT. And the interesting thing is that it takes the following form. So the Z here is the topologically twisted index. It's a very complicated expression at finite N but at large N you can actually evaluate it in a variety of examples including a BGM but some other examples as well. And what we showed in the paper in the number of examples is that if once again you sort of forget and set aside the one loop and per factors, the integrand of our W hat here matches the one of these degeneracies at large N. Once you identify the off-shell fluctuations phi i with the chemical potentials delta in the computation of the topologically twisted index and you also have to use the old graphic dictionary to relate your N to your Newton's constant and your charges. So it takes an eerily similar form by just doing a calculation directly in the near horizon of the black hole in the super gravity theory. Right and now I just want to make also an observation. So this is the result that we got in gauge super gravity. Here I'm just trying to be schematic so I'm dropping a bunch of factors and things like this but essentially what you get is some finite dimensional integral over a bunch of variables and V plus one of those and then you have some exponents which is specified by the pre-potential of your theory. And some one loop determinant factors. Now, thank you. You should remember or I'll remind you of the un-gauge super gravity result which was pioneered by double cargo mesh and murty where they showed that for the flat black holes you can also write a very similar formula again with Nv plus one integration with a finite dimensional integral and some exponent which again depends on the pre-potential of course not the exact same combination but you can see the similarities here and some one loop determinant factor which again IMTAC nicely reminded us yesterday of its full form which is given in terms of the killer potential which once again is specified by the pre-potential. So what I want to emphasize here is just that these two formulas both for ADS and flat black holes they're extremely plug and play. You can just pick whatever pre-potential you want to specify your theory because you're coming from some string compactification because you just want to do it as an academic exercise and you can just plug this into these formulas and start seeing if you can even evaluate these integrals exactly. And in fact just as a remark for very specific models here I have in mind some type two B compactification on T6, this W flat here can be computed exactly. You can even take these integrals and actually get an exact result purely in terms of the charges P and Q and this led to a very precise non-perturbative match with the exact string theory results that you can derive in these compactifications. And here also comes a, if you'll allow me to be subjective, some very beautiful connection to number theory via the Rademacher expansion of which Komlund Vafa also talked about in his inaugural lecture. So this was for flat but of course for ADS we're far from there. We still have these one loop determinants to compute, et cetera but I just want to sort of end with this remark on the similarities that you can get engaged and engage supergravities. Right, so let me move on to my conclusion. I believe in the paper that I mentioned at the beginning with Kirill and Ivano, what we did was really presenting the first step towards the computation of the quantum entropy for BPS black holes engaged supergravity with a near horizon ADS two times S2. And again, I want to emphasize that this is a contribution from the near horizon only and not from outside of the horizon. That's important to keep in mind if you really want to do some precision matching with the calculation in the asymptotic CFTs. So perhaps unsurprisingly, although it's always kind of reassuring the saddle point evaluation of the quantum entropy that we have reproduces the semi-classical Beckenstein Hawking entropy and you see here the usual attractor mechanism engaged supergravity. And the answer takes a very similar form to the computation of the topologically twisted index in the dual field theory. This is at this stage just a formal observation but I would say that this is rather encouraging. So for some future directions, obviously one of the main priorities is to complete the one loop determinant calculation engaged supergravity. And this should really lead to a better understanding of the role of these mysterious functional zero modes that are not entering the renormalized action but we need to see what happens in the one loop determinant for these guys. And if we are able to do this calculation this is also something very concrete we can do to try and reach perhaps some predictions for the dual matrix model at finite end because here at large end you recover Beckenstein Hawking, you have these n to the three half scaling in the specific ABGM case but in the gravity side, again, modulo the caveat that this is all near horizon, if you know the one loop determinants you can start picking up some finite end corrections and then you can go back to the matrix model and try to ask how these things talk to each other. So that's our main priority for priority for the one loop determinants. Of course, as I mentioned, there's a central assumption in this work that the via-multiplet is frozen and pinned to its attractor value. So that should really be examinated thoroughly and that's quite high on our to-do list. And eventually later, I did mention this at the very beginning, you can think of trying to generalize these calculations to perhaps a more trivial, more non-trivial topologies for the near horizon, namely with some generic Riemann surface of some arbitrary genus and of course, also more general gaugings. Why not go to dionic gaugings and really turn on the full electric and magnetic F5 parameters. So I think these would be interesting generalizations of the calculation. Of course, these are two much more important points that we hope to be able to report on in the near future. And, but as I said, you can already say a lot about the formulas that we had and hopefully after lunch, Kiril will keep saying a lot about the formulas that we have. So thank you for your attention.