 I'd like to start with thanking the organizers and ICTP for giving me the opportunity to talk about some attempts or some studies that we have recently carried out in understanding the microstates of, or we are hopeful that it will lead to understanding of the microstates of extremal black. So this is a paper in collaboration with two of my students and Jeff will compare. So that's the outline. I'll first try to motivate why this question of studying black holes and it's trying to understand the source of the thermodynamical behavior that the black hole shows is important. Then I'll go through this specific example that I'm going to discuss about. That's basically a geometry which is related to extremal black holes. Then I present you the three laws of energy, the mechanics which basically parallels those of black holes. Then I move from this classical purely GR classical picture to one step ahead. They're basically trying to construct what I call the phase space of this geometry. Then I'll tell you about the conserved charges and associated algebra and try to eventually argue that why it should be all relevant to the microstates of black holes. So there are more than 40 years of work basically trying to establish a connection. The fact that semi-classically black holes are behaving like thermodynamical systems, you can associate the temperature to the black hole and other chemical potentials to the horizon of the black hole. Then you can associate charges. Basically these are like no third charges that you can associate with the black hole. They are defined at infinity or asymptotics of the space, whatever it is. And eventually you can associate an entropy again as a conserved charge to the horizon. These thermodynamical concepts and quantities of course satisfy the laws of thermodynamics. A black hole could be hence viewed as some system at a fixed temperature. It has a black body radiation. That's the Hawking radiation. And we know that within Einstein GR plus semi-classical quantum field theory coupled to Einstein GR, that any matter, almost any matter would generically undergo the gravitational collapse. And then after black holes formed basically the Hawking radiate and completely evaporate. But the problem, which was also alluded to yesterday, is that this process of formation and evaporation of black hole is not unitary. But then in the standard physics we have another part of the physics which basically, although the underlying physics is unitary at the surface in thermodynamical limit, we don't see unitarity. And the non-unitarity arises because of as an artifact of taking the thermodynamic limit. So let's entertain the same idea. Could be that in black holes we have a similar feature that there is some underlying physics which is unitary. But in the black hole description we are dealing with some thermodynamic limit and we don't see unitarity manifestly. Then there is also a law which has several arguments behind it. And that's basically the micro states or that underlying system of the black hole are basically residing on the horizon. So if you focus on the horizon you should be in principle able to see where the micro states are. So that was just for the general picture. Usually there's a laboratory in the set of black hole solutions, the extremal black holes, which are used to study or address these kind of questions. The reason why they are interesting is that they are at zero temperature. They do not have to radiate. And hence there is no evaporation. So they are much easier to study if you want to address the question of the micro states. Then these black holes are essentially what we are focusing on. Although they are at zero temperature, they do have non-zero entropy generically. And they could also be related to candidates of black holes that we see in the sky. So it's not just pure theoretical studies. Then within the set of extremal black holes there are supersymmetric black holes, which even gives you more control over the system. And you can study the micro states even better. Actually, for a certain set of supersymmetric black holes, the project of micro state counting has been carried out successfully first in 1995 by the seminal paper of Strom and Jernel Waffel. And since then many people, including Oshuk Sen, have contributed to this. Just to give you an idea of the general picture of the micro state counting attempts that has been carried out, there are two different approaches, which I've called top-down and bottom-up. In the top-down approach, basically they try to embed or model the black hole in some theory of quantum gravity. For example, string theory. And this approach only today works for the BPS. But there are many attempts to basically go beyond the BPS black holes. Then there is the bottom-up approach. And basically the idea here is that let's try to start from the gravity and try to add semi-classical features and see how far we can get. So the examples of top-down is, of course, this Strom and Jernel Waffel, the D1, D5 system of what Oshuk Sen has been doing and collaborators, including Otish Double Kerp, and for some certain BPS ADS5 black holes. Then examples of the bottom-up approach have given two examples, but there are more. The standard Kerr CFD and the ideas by Cardi, basically. These are relying more on the geometric gravitational picture. But generically, in either of these two approaches, the microstate counting is relying on using Cardi formula, finding some 2D CFD or identifying the central charge of it would be 2D CFD and basically trying to reproduce the entropy. Here I'll be just giving another bottom-up example. So that's what we are doing here. OK, the geometries that I'm going to focus on are not exactly black holes. They are related to these black holes. They are near-horizon geometries associated with extremal black holes for short NHEG. These are a class of solutions to GR with SL2R across some U1 isometries. And they could be BPS or not. But what I'm focusing on here is not the supersymmetry. So supersymmetry will not be so relevant to my discussions here. And basically, the idea is that, as I mentioned, we believe that the microstates of the black hole should be residing on the horizon. So if you focus on the horizon and consider the NHEG, it should be enough for understanding the microstates of black holes, or at least counting them if you don't specify explicitly. But then there have been many attempts within this set of geometries relying on the fact that you have SL2R. And geometrically, that corresponds to some ADS2. I'll show you the metric in a second. And then try to use ADS2 CFT1 for doing this counting. But there are various issues associated with this duality. And that project has not been really fruitful today. So basically, I'll try to take in completely different routes, not using ADS2 CFT1. Actually, I'll try to argue that ADS2 CFT1 would be misleading for this specific class of geometries. OK, so these NHEGs are not black holes, meaning that they do not have even horizon. But actually, we showed that there are infinitely many bifurcate killing horizons in this geometry. So although they are not black holes, they show some properties that are useful for basically carrying out the standard GR analysis, as we'll see. Then there are uniqueness theorems as we have for some black holes. And more importantly, actually, there are perturbation uniqueness theorems, or oftentimes called no-dynamic theorems. And the essence of this no-dynamic theorems is that on these NHEG geometries, you have an ADS2 part. And you cannot really turn on fluctuations on the ADS2 part, which do not destroy the asymptotes of the ADS2. Reason is very simple that ADS2 is just two-dimensional. And basically, it does not afford putting any fluctuation on it. So there is no normalizable fluctuation on the ADS2, which satisfies the standard boundary condition. So that's the statement of this no-dynamics. OK, so these are the facts that we know about the NHEG. So let me just focus on a specific class of NHEGs, which are the vacuum solutions in generic D dimensions. So this is the generic form of this geometry. The blue parts are associated with the ADS2. And this red part is associated with the U ones. And these gammas are some functions which you can determine if you impose the equations of motion. And these are the SL2R U cross U1 killing vectors. OK. These geometries, so let me go back here, are specified by this set of vectors. So these are D minus three-dimensional vectors, K. So remember that. We'll come back to this. OK. Then there are D minus 3 number of angular momentum associated with either of these U ones. And actually, I told you that there are infinitely many killing horizons. And they are generated by this killing vector field for each and every given point on the ADS2. So this Th and Rh are arbitrary. And so basically, for any arbitrary Th and Rh, we have a killing horizon. And interestingly, all of these different killing horizons have the same surface gravity. So irrespective of where you are on the ADS2 sitting, you see the same thing. And actually, this is the key point that I'll be using in my construction. So just to give you how this spacetime, the causal structure of the spacetime, so that's basically the R and T part of the spacetime. I've suppressed the D minus two-dimensional H surface. And there are also some discrete symmetries. And basically, all different points on this space, as far as the micro states are concerned, should be equivalent. So that's basically the basic idea. This is why you should not be able to use ADS2 CFT1. ADS2 CFT1 is basically trying to put the dynamics or whatever it is in this ADS2 part. But that's not possible. So this is just the flow of this vector, killing vectors, later H. So here we have bifurcate killing horizons, no matter where you are sitting. OK, so let me just briefly mention the laws of energy-geomechanics. So the surface gravity for all these H surfaces and the constant T and R surfaces are equal. You can associate the conserved charge to this. I mean, the conserved charge associated with the killing vectors, later H, which is the entropy. And it satisfies this relation. So that's the first law of energy-geomechanics. Then we have the entropy perturbation law, which states that if you have any perturbation, like delta phi, phi is a generic field, which satisfies these two conditions. It's invariant under two of these SO2R generators. The associated entropy perturbation and the angular momentum perturbation is related in this way. So for the background, now I want to tell you how we can think of these microstates or where they can come from. That basically means that we want to construct the phase space of fluctuations around the energy-geomechanics. So basically the idea is that this phase space, which could be relevant to the problem of microstate counting, should be viewed as some fluctuations on this given metric. And these fluctuations, as I mentioned, they could not be physical fluctuations, which carries in non-zero energy and non-zero angular momentum perturbation. And hence, we are confined to consider defiomorphisms. But in the standard GR, we are told that all the geometries which are defiomorphic are physically equivalent, unless you can associate non-trivial charges to these fluctuations. And that's exactly what I'm going to tell you and show you basically that there are some set of defiomorphic metrics which are not physically equivalent. And that's basically how I construct the phase space. So basically, I construct my phase space through some defiomorphisms. These are the infinitesimal form of the defiomorphisms. And I show you the finite form of these defiomorphisms. And I show you the explicit form of the metrics, of course. So just for the notation, I'll be using the vector notation to denote this K i and J i. These are the d minus 3-dimensional vectors. The dot is basically the dot on these vectors. And then this round, or partial, is basically the directional derivative along these five i's. And the background metric that I'll choose is basically in this coordinate system that I showed already the metric in that coordinate. So this is the form of the defiomorphisms. This is a vector field which generates my phase space. There is a function epsilon of phi, which is a function of all these d minus 3 values. And of course, my perturbations are generated by standard lead derivative. This phi bar, in my case, is just the metric, the background metric. So you may ask, this is a very specific defiomorphism. And the way I fixed it is basically trying to impose these conditions. The first condition means that I'm imposing the fact that all different points on the ADS2 are physically equivalent. That's what it amounts to. They are volume preserving, meaning that the divergence should be 0. They are preserving the form of the ansatz that I showed. Basically, the variation of the larger top form is 0. They also keep the horizon structure. That's what I need to require. And I want all these metrics or all these geometries that are produced through these defiomorphisms are smooth. So that's also a strong condition. And more importantly, I want there should be integrable and well-defined concept charges associated with these defiomorphisms. So actually, through solving this equation, which is basically requiring that I want the form of the defiomorphisms to be embedded all over the phase space, fixes the finite transformations. So I showed you the infinitesimal transformation around some given point in the phase space. Now I'm basically giving it through this generic transformation. So this transformation is specified by a single function, which is a periodic function, and a psi function, which is defined through the derivative of this part. And just to show you explicitly, one can really readily see from this that the null direction, which is given by this combination, is remaining embedded. So these are basically the set of metrics. This is one parameter family of metrics, which are specified by this function f, and there are elements in my phase space. And let me speed through this. So schematically, basically, these are the set of all these geometries are defiomorphic to this specific point. This is the metric that I started with. And this plane is basically constructed through the defiomorphisms that I just mentioned. All the points on this plane have the same angular momentum, J i, because they are built through the defiomorphisms, and J i is a physical charge. So that's basically where my phase space is. And importantly, one can readily, given these chi's, the generators of the phase space, you can find the lead bracket of these two chi's. And this is basically the algebra of the generators. And actually, there is a strong theorem that the algebra of the charges, if you can have a well-defined charges associated with this, is going to be the same thing up to some central extension. So I'll be going through the details of this. So to make this set of metrics to a phase space, I need to give you the simplistic structure. Simplific structure is a two-form, which should be finitely closed and non-degenerate over the spacetime. And it's a d minus 1 form in the spacetime. So it's a two-form on the phase space, on the tangent space of the phase space to be more precise. So these are tangent vectors in the phase space, variations of the fields. Delta is an exterior derivative on the phase space. D is a standard exterior derivative on the spacetime. And there is this very elaborate machinery of covariant phase space method developed in these papers that basically allows us to construct for any given morphism invariant theory to construct the simplistic structure. And so let me just give you very briefly how this works. One can define a pre-simplific potential whose delta derivative makes the omega. So that's the simplistic structure. And basically, for a given Lagrangian, you can define this theta, the pre-simplific potential, in this way, just given the Lagrangian. But this only defines it up to some boundary terms, which are basically defined in this way. So basically what we have shown is that there exists some well-defined y's, which makes this set of metrics into a well-defined phase space. So the consistency conditions on this omega is that they should be closed on shell and actually all over. Not only d of that should be 0 on shell, omega itself should be 0 because we want the flux of this to be vanishing. So that's the conservation condition. The integrability condition amounts to having this. And there is a very standard but very straightforward, tedious, long algebra to show that there are some y terms which does this job for us. So you can find the details in this paper. OK, then there is the fundamental theorem of covariant phase space method, which basically tells us that omega on shell is d of sum 2 form. And through this k, you can define the variation of charges. And if the charge is integrable over the phase space, through this delta h, you can define or construct h itself. So in principle, I have given you how to construct omega, how to construct k, how to construct h. And now I give you the result. And basically, if these h's are the charges associated with these transformations that I introduced, this is the algebra of this, which is exactly the algebra of the vectors y, the generating vectors, up to the central term. And there are ways to compute the central term in this covariant phase space method. It's basically standard. And we computed that. And actually, the central charge appears to be the entropy of this object. So that's how it is. So basically, so I gave you the algebra of charges, but you may also ask whether for any given element in this set of metrics, I can specify what's the charge associated with that. And this is basically to give you the answer. Because yes, we can specify explicitly what are the charges in terms of this function, psi, which appeared in the metric. So these charges are Fourier transforms of this, you may call it a stress tensor, t, which has this form. So for those of you who are familiar with the Liouville theory, this is basically very similar to what you would call a Liouville stress tensor. But here, the fields are d minus 3 dimensional, unlike the standard Liouville. And this is the directional derivative. So this psi in this setting resembles a weight 1 field. OK. So far, it was just classical phase space, classical simplicity structure. But you can use standard quantization this way and basically promote these charges to algebra of generators. And of course, the algebra that you would find is basically this one. And the energy phase space that I constructed just by construction is invariant under this symmetry group. So let me just give you some more words on this energy j algebra. So it resembles a virus or algebra, but it's not a virus or of course. Because here, the indices on the generators are vectors. They are not just numbers, they are basically points on the lattice, on a d minus 3 lattice. These vectors, k that I showed you, they appear in the geometry appears as the structure constants in this algebra. And importantly, the entropy appears as the central charge. So let me just give you some more words on this algebra. So this is a new kind of algebra that we have found through this study. It has not been identified in physics or math literature before. If we are in four dimensions, k is just a number, is not a vector, and when k is just a number, these are numbers, and this reduces to standard virus or. But for higher dimensional cases, this is a different object. And actually, this algebra has infinitely many virus or subalgebras. And I've given the construction how it goes. So for any given direction on the lattice by this vector e, you'll have a virus or a subalgebra. And there are infinitely many such directions. So it's a real extension of the virus. OK, so let me try to summarize and try to give an outlook where we are heading. So I showed you how one can construct the phases space of geometries, which are all diffeomorphic to this given energy g. These are one family set of geometries specified by a periodic function, f, that we call the Vigil function. And schematically, basically, given a family of the energy g's, which are specified by the value of angular momentum, that's basically the geometry which I denoted by g bar. You can construct the phases space by applying the diffeomorphisms that I just gave you and construct the whole phases space through that. But the point is that for any given point in the phases space, you can associate non-trivial charges. These are basically charged under the algebra that I showed. But these charges are well-defined because this set of geometries provide us with the phases space because there is a well-defined symplectic structure. And the algebra of the charges associated with these geometries is an extension of the virus around. And the entropy is the central charge. Note that there is no usage of cardiform. Entropy directly appears as a central charge. And this is how it is different than all the previous attempts, conceptually different. So perhaps for those of you who are familiar with the CARE CFD analysis, this is very close to that. But let me try to compare that with the CARE CFD. First of all, we are giving the phases space. It's not just a set of perturbations with a given fall of behavior. We've completely specified the phases space. Our symmetries, hence, are symplectic symmetries. They are not asymptotic symmetries. Of course, any asymptotic symmetry is asymptotic, but not the other way around. Second, the entropy appears as a central charge. And our central charge is not, or the algebra is not an extension of the isometry of the background as it is in the CARE CFD. Actually, our phases space is completely invariant under these isometries of the background, because these are a set of geometries which are constructed through the few morphisms. And we are treating all these U1 directions in a democratic way. That's not what appears in the CARE CFD. So then I introduced the representation of the charges in terms of this Leoville field. And you may ask, what's the role of this Leoville type field here in this setup? The interesting thing is that in this set of HNs, the H0 is positive definite. And it could be viewed as the Hamiltonian over this phase space. And that's basically the idea, which could be useful. So that's basically one of the directions that we are exploring. Then there could be many interesting extensions of this idea, for example, extensions to other extremal geometries. And finally, whether it's related to microstates or black holes, and whether this Leoville theory has anything to do with the microstates, that remains to be seen. Probably that's at the moment the only comment that I can make on this issue is that this Leoville type theory would be the long-stream sector of whatever that quantum theory is. Thank you very much. Questions? A comment? So normally, if you study Lirasoro representations, then you come up with the formula by looking at the dimension of the module. But in your case, the entropy appeared. I mean, so how do you study the representation of this generalization of Lirasoro? And do you get this as really the dimension of creating the states, or this is? Actually, that's an open question. We are studying this now. So we expect exactly what you just said, that if you have this so-called coadjoint orbits of this new algebra, then you find the volume of that coadjoint orbits you would expect to get the entropy. So that's the idea. That's how we think that we can carry out the microstate counting. We are not identifying the microstates, but you can do the counting over the counting of the representations, the volume of the representation. Any other? I could ask or say something. In a dynamically evolving black hole, because the time scale is associated, so you have to cause great energy to get to entropy. Namely, that entropy is entropy density. For extreme on black hole, it could be like a designer's in ground state. It could be really like a real designer. So it's very different. You cannot see the dynamical aspect in this way. But the idea why extreme on black holes could be relevant to a generic black hole microstates is exactly that if you identified the theory here, that would be the ground state of that would be theory. And then you have the theory you can presumably go to non-extremal. Well, some purpose is useful, yes. So you need to have some control. For generic black hole, you have little control, so it's very hard problem. OK, thanks, speaker again.