 Svaj, je stejn. Ti stejn. I'm going to scoop your next paper. OK, we can start the afternoon session. We have Sameer Monti from KINX, who will join all what we have seen so far, localization, supergravity in so on. We talk about localization in supergravity. Wait. Thanks. So, let me begin by thanking the organizers for this invitation or maybe I should say push to give these lectures. This is not here yet. So there is. So my title is Quantum Black Hole Entropy and Localization in Super Gravity. And what I would like to discuss is a set of ideas about the quantum entropy of black holes. I'll explain what I mean by that as we go along. That have been developed in the last 10 or 15 years. And in parallel there's a set of techniques which have been developed to address such problems. And the main technique is localization about which we heard a lot applied to field theory. And here what we want to do is to apply this technique to supersymmetric theories of gravity. So that comes with its own set of challenges and problems. So I'll try to explain that as we go along. So my handwriting is not very good. So at some point of time I'll try to type up what I write and put it on the website. But for now at least I'll scan in my notes and do it after every lecture. So for now I think this should be already on the website. There's just a plan of the lectures that I have. So I have four lectures and I've divided it into four parts roughly corresponding to one part per lecture. I don't know why it came out like this. I had single pages but now I'm sure it cannot read it with my handwriting. So I'll just try to do it like this. Maybe this is better. So the first part is, which is today, which is, I'll just tell you about the idea of quantum entropy of black holes. So I'll just review some basic black hole thermodynamics, which I'm sure all of you know, just to put all of us on the same page. Then I'll talk about supersymmetric black holes and their classical entropy and then start to define what quantum entropy means. The second part, which is tomorrow's lecture will be giving a precise context to it and the context will be what Stefan discussed very nicely in his first two lectures, namely supersymmetric black holes in four-dimensional n equal to supergravity. This is in asymptotically flat space. So that's tomorrow. And then part three is the discussion of localization applied to this problem and the computation of exact quantum entropy in this class of black holes. And part four, so this has a whole bunch of stuff. So maybe a bit of this will already be tomorrow and some of this will be on Friday. And then part four is, so loose ends, but actually I replaced this. I call this advanced topics because this looks very casual. But based on a little bit of the kind of questions that have arisen here about one loop determinants and so on. So I just want to summarize our status of knowledge about that. I'll just try to go through this in some detail. But applying these two supergravities is actually, there are some serious issues there about which there's been some progress. Some of it you'll hear in the workshop by Intox of Bernaduid and also of Imtak Jeon, who's here, who's not here, I's here. And hopefully I'll come and at least give you an introduction to the topic so that you can then really take advantage of those talks. Okay, so today will be just essentially an introduction. Okay, very good. So, of course, please feel free to interrupt with questions. And also if this is not visible, just pick up. I just want to make sure that, so is this sort of readable in the last row? Okay, yeah. No, when I was sitting there, not all the lectures were readable, so just, and no one was picking up. So please tell me if I write something illegible or if I say something, which you don't understand. Okay, so let's start with some very basic notions. So start with black hole, so ideally I should be drawing some kind of space-time diagram, pendulum diagram, but it actually helps a lot to keep this very simple minded picture that there's a black hole with some even horizon, okay, with some radius r. Maybe I'll sometimes call this h. Okay, and the story starts with the fact that black holes have thermodynamics. So the first statement is that of Bekenstein, so I'm not being completely chronological, but there are lots of works of Bekenstein and Hawking and collaborators, but I think logically this is a good way to say it. So the statement of Bekenstein was that black holes carry thermodynamic entropy, and the argument for this is very simple. I'm sure all of you know it, but just to put everybody on the same page. In fact, here is the paper of Bekenstein and he says, let's start by considering this very simple thought experiment. Suppose you have a black hole and suppose a body carrying entropy s goes down a black hole. Just throw something which carries energy and entropy and throw it into the black hole. So something like this, okay, you have a body which carries a lot of entropy. And then what you want to do is that it goes and falls inside the black hole. So actually Bekenstein had Mexican roots, so I'm sure he will be happy with this. Sorry for the resilience, I know there are a few of you. Okay, sorry. So I hope Bekenstein's spirit forgives me for this joke. But anyway, based on this, so you argue the following. You argue that there is a generalized second law, and the argument is very simple. It says that consider the second law of thermodynamics, okay, and you know that the total change in entropy of the world of the universe should increase. So if you think of this inside and outside of a black hole. So you've lost some entropy from the outside world. So this is entropy of out, okay, and this is negative, right? But this should be positive, and the only way in which this is even possible is if you assign an intrinsic entropy to the black hole itself, okay. So this should be positive, okay, and that's the argument that the black hole must have entropy. So very simple minded argument, but a very profound one, and in fact you can sort of carry this further to say that a black hole must also be the thing that packs entropy in the most efficient way, because whatever you throw in, the total entropy has to increase, okay. So now the second statement is that, well, if black holes have entropy, then they must also, sorry, if the black hole has entropy, they must also have thermodynamics and temperature, and this was a hard calculation, a beautiful one by Hawking. And so let's say, let's take a spherically symmetric black hole, okay, spherically black hole, okay. Then, so the horizon radius here is 2 gm, this R, and Hawking's beautiful calculation showed that the temperature of the black hole is h bar over 2 pi times the Boltzmann constant times the surface gravity. So this is gm over R square, and if you plug that in, pi and this is equal to h bar over 4 pi kb times 1 over R, okay. So that's the temperature. So now use the first law of thermodynamics, and it says that the infinitesimal change in the entropy is dm over t. Just plug that in, and you'll get kb over h bar times g times 2 pi R dr. So you can integrate this equation now to get famous formula, which is kb over g h bar c cubed. I've reinserted a c times area of the horizon divided by 4, and this is what we call area of the horizon divided by 4 in planke units, okay. So that's the famous area law of bacon, stun, and Hawking, all right. So it's a very beautiful equation with all the constants, most of all the fundamental constants of nature appearing. Now in any theory of quantum gravity we expect that we are the Boltzmann equation. This thermodynamic entropy should be equal to Boltzmann constant times the logarithm of the number of microstates of the black hole, okay. And this should be true in the thermodynamic limit when things simplify. So these dots means that things depend on the details of the system, which just gets washed away in the thermodynamic limit. So when the black hole becomes very large, we're just left with this, okay. And for reasons that will become clear soon, I'm gonna call this the classical entropy. I should really be saying semi-class, but it's easier to write class, okay. Now, these little, yeah, S is what? Well, I've made a KB in both cases, so it's, sorry, KB. Thank you. There's a KB here. Any other questions? Yeah, so L-plank square is this, of course. Now, this law of bacon, stun, and Hawking is universal in that it applies to any black hole in generativity, and so that's a very nice thing. Universal laws are good, but it also means that it's, that's a limitation of, it's a self-limiting thing. If you want to understand what is the fine structure of gravity, what's the ultraviolet properties of gravity, all those things get washed away in this thermodynamic limit, okay. So if you want to understand the quantum structure of gravity, you actually need to understand these corrections, and that's what the notion of quantum entropy is about. So these corrections, you can think of either as finite size effects, okay, and by size, I mean, size is the area, the horizon in plankerings, okay. So that dimensionless object, I'm going to call size, when the black hole is large, this is correct, and now there are going to be corrections when the black hole, when the curvature is not so small, okay. And because there's an h bar there, this is also a quantum effects, or in fact, because it's actually a plank, these are quantum gravity effects, very, very broadly speaking, all these things are true, okay. So that's what we're interested in. So this, you want to ask, what are these effects? So this whole thing, okay, I'm going to call. So this right-hand side, I'm going to call s black hole quantum, okay. And from now on, I'm going to put kb to 1, so you don't see this issue. So what I'm asking is really some macroscopic question. So in Boltzmann's equation, this is micro, and this is macro, okay. And I'm just saying that in a theory of quantum gravity, there must be an equation like this, and the question I'm asking actually is just about the gravitational theory, okay. So let's be a little bit more precise. So let's ask ourselves some questions. So one question is, what is the definition of this thing? So Baconstein and Hawking gave us the definition of the classical entropy. What is the definition of this quantum entropy? Okay, so it must be something defined in the gravitational theory, which reduces to Baconstein Hawking when the area is very large, and otherwise agrees with the logarithm of the number of microstates of the system, okay. Yeah, so once you have the, yeah, so once, they have entropy because they have area. That's not how I set up the argument, so you should think of extremal black holes as zero temperature limits of non-extremal black holes. And in the limiting procedure, if you ask what is, so think of the entropy of the black hole as limit t goes to zero of entropy of the black hole of this, you get an area, yeah. So, yeah. Okay, other questions? Yeah, yes. If it's a quantum effect, so shouldn't one anticipate so UV sensitivity at some higher order? Absolutely, so that's actually, that was my point, so you said it in a better way than I did. Thanks. What I'm saying is that if you, these things are not universal, they depend on the ultraviolet completion of gravity. In that sense, if you manage to compute this, actually get some information about the UV theory, okay. What I'm saying is you can actually try to, so there are just looking ahead a little bit in string theory, we have sort of two notions of what a black hole is. One is that of some effective theory and a solution and so on with higher derivative and quantum interactions and the other notion is some microscopic interactions. So, the idea is that, sorry, microscopic excitations of some ensemble. The idea is you want to use the first effective type description and see how far you can push that. And in the supersymmetric case, it turns out that you can go very far with these ideas. Okay, thank you. Other questions, yes. Yeah, so in fact, you'll see that they can be positive or negative. They're small compared to this one, but you'll see actually that in the one example where I know how to actually compute this all the way to the end, sometimes they're negative. It's because they're fermions. Yes. Yeah. Yes. Vox. Yes. Gravity in 10. Indeed. Or just the ones from the black hole. I understand. So, the answer is the latter and I'll answer this in five minutes. It's a very good question. I'm going to come to that very soon. Other questions. Okay, so one question is what's the definition? The question is can we compute in any sensible model? And another question which I'm not going to discuss today is compare to microscopics. So, in string theory, as we all know, to the work of Sen and Stromingenwaffe, we know that a black hole is really some, at least in the supersymmetric situation, is ensemble of some kind of microscopic states which you can count and you get some integer here. So, the question is, is the exponentials, if I can calculate this, is the exponential of this equal to that integer? How far can you go? What I'll be talking about in this set of lectures is just these two topics. Joan Gomesh, who is studying tomorrow, will talk about the microscopic side of the story. And in my third lecture, and maybe in his third lecture, we'll try to somehow, at least in the future. So, just keep an eye out for his lecture and try to connect with this, with this, what I'm saying. Okay, now, whatever this, let me make one more comment. Is that this kind of, whatever this quantum m2p is, just based on mathematical analysis, it starts like this. It just starts with this and then there'll be some logarithmic and power law suppressed, right? And maybe some exponential suppressed corrections. Again, I'm being schematic. Here, I'm sort of roughly assuming that a h is the only small, sorry, large, one over h is the only small parameter in the problem. Sometimes you have more parameters, so it's a multi-dimensional expansion, but roughly speaking, something like this is true. So, I'm just putting this out to sort of, so that you know what to expect towards the end. Okay, so these are the questions I want to answer. So, let me first begin by asking why at all are there these corrections, the origins of, what is the origin of quantum corrections? Well, one obvious answer is the following. You have, so write down the gravitational action. So, it starts like this, okay? At low energies is this coupled to other fields, this Einstein-Hilbert coupled to other fields, okay? But at higher energies, so whatever the theory of quantum gravity is, I think there's a universal agreement that it's not just this, okay? Whether, whatever philosophical bias you might have from an effective field theory point of view, you know that there are operators of this kind that are going to correct the two derivative low energy action, okay? L-plank square, R-square here, I'm schematic, this could be any four derivative term and so on, okay? Now, in the, in such, with such an effective action, and it's not just GR, the Bacon-Stein-Hulking law does not obey the laws of thermodynamics. I forgot to say this, but I think all of you know that this work of Bacon-Stein-Hulking and Carter and I'm missing one more name, Bardin, all of, they showed that in fact this entropy and that temperature obey the laws of thermodynamics. So, the first law I already used, but also that in any physical process, the second law is true and so on. Okay, so the first, even the first law is no longer true for such an action. Now, this problem has been solved with some assumptions and oops, I shouldn't erase that. Yeah, and there's something called the Wald formula developed by Wald and Ier and their collaborators, which says that given some, suppose this is a local effective action, okay? Then, there's a formula, which says that the black hole entropy is, so let's say this is spherically symmetric again, it's something like this. So, given some black hole solution, you take the action, oops, and you take the action, the action can be some complicated things. There's a set of rules which goes here that you might have some functions of the Riemann tensor or there can be other covariant derivatives and so on. There's a set of rules that, when you have anti-symmetrized combinations, you have to make it symmetric and all the anti-symmetric guys, you have to write in terms of the Riemann, then you think of Riemann as an independent object and you just vary the action with respect to this and compute this formula. I'm not going to go in this direction, but this is an extremely well understood formula and it's completely explicit and this obeys the first law of thermodynamics. It was rigged up, so as to obey the first law of thermodynamics, okay, in given some local effective action. Now, this is not the direction I want to take because there is another problem which is in a sense as interesting and more interesting, namely that in the quantum theory, the correct action of the theory, the correct means one PI action which respects all the symmetries of the theory, need not always be local effective action because massless particles run in loops, okay. So suppose you have some loop of this kind, I'm not telling you what observable this is, but something like this, this is going to give you some contribution of this type where epsilon is some IR cutoff, right, and this one versus log epsilon, okay, and that's certainly non-local. Think of this as some box, we'll have log box or something like this. It's a very non-local effect and once formalism is not enough to take care of that, so you want some formulation in which you want to think of being able to compute such effects, okay. So that's what I'm going to turn to next. So this, so the breakthrough came to work by Ashok Sen in 2008 in the context of supersymmetric black holes. Before I go on one technical comment, through all these lectures sometimes I'll talk about classical and quantum entropy. So far I said classical is area over four. Sometimes people also call this wall entropy the classical entropy just because if there is a local effective Lagrangian which you can minimize, that's like a classical problem. It might have higher derivatives, okay, and you talk about these loop effects as the quantum effects. So it's a question of semantics, but if I might switch between the two, if it's unclear, please ask me. So look at supersymmetric black holes. So these are extrema zero temperature black holes and as was already anticipated, the way we should think about it is to think of a very low temperature black hole, compute all physical quantities and then take the zero temperature limit. That's how we'll think of extrema black holes and this avoids all kinds of problems which people have invented when you directly take extrema limits, not invented, found, but I think this is the right way to do it. Okay, now the fact is in the limit these black holes are isolated quantum systems and that's a very important thing philosophically and this comes to your question. Sorry, I don't know your name, but the question. So usually you have a black hole which is interacting, which is environment and to ask what is the number of states of the black hole? Suppose I want to be ultra powerful and say I want to compute the full integer number of states associated with black hole. For a real non-zero temperature black hole, this question is not even well defined because there's Hawking radiation which goes in and out. So you can't say that this quantum belongs into the black hole or outside. It's only the full ensemble, the box as you said, which has a well defined entropy, but there you might have a canonical ensemble. So those are the kind of things which one has to compute. But for extrema super symmetric black holes they have zero temperature, they're completely quiet objects and they're isolated quantum systems. So there is a sense in asking what is the number of quantum states associated to that black hole and that's the question I want to answer, okay? So that's an important point. So let's then do this in order to do this. So the first idea is that the entropy of a black hole is somehow stored near or behind the horizon of the black hole. The horizon is the reason why there is an entropy. So the first step is to sort of go near horizon and before going to the quantum entropy I want to do an exercise which was done by various people, some of whom may be here, namely to express this classical entropy, this wall entropy for the case of super symmetric black holes as a quantity which you can purely compute in the near horizon region, okay? So let me do that and in fact, so there are two advantages. Firstly you'll see very soon, it's a very elegant formulation, it's a very simple formulation and secondly it allows us to go beyond and define quantum entropy, okay? So I'm going to take some kind of near horizon limit and in order to do this I'll just be very specific just for now. Let me talk about this Einstein Maxwell theory in four dimensions, 16 pi. Okay, so there is, in fact, Stefan even wrote the Reissner-Aston solution, I'm going to repeat it in slightly different variables. So here is the solution, it's 1 minus rho plus over rho, 1 minus rho minus over rho, d tau square, tau is time, plus d rho square over 1, this thing, plus rho square d omega to square and it's going to be more precise, d psi square plus sin square psi d phi square, okay? So that's your Reissner-Aston solution and if you take the extremal limit what you'll find is that, first I should tell you more about this, finish the solution, so there are two parameters, rho plus and rho minus, this is not how Stefan presented it, I think, but it's completely equal and I should also tell you what the fields are, electric fields, gauge fields are, so there's an f rho tau, so these are charged black holes and there's an electric charge q here and in fact you can also take a black hole which is dionic, so let's take that, that's the most general thing, p sin psi, okay? And rho plus and rho minus are the positions of the two event horizons which are defined like so, rho plus plus rho minus is 2 gm and rho plus times rho minus is g over 4 pi times q square plus p square, okay? So that's the Reissner-Aston solution in all detail and the extremal limit is a limit in which the two horizons coales, rho plus, both of them go to a common value which I'm going to call rho horizon, rho h and that turns out to be equal to square root of g times q square plus p square over 4 pi and m equals 1 over 4 pi g times q square plus p square, okay? So that's the extremal limit so the mass is completely determined by the charges and the two horizons coales, yes? Yes, so I haven't yet done that, that's exactly what I'm going to do next, let me just do it and then you can ask a question if it remains, so now what I want to do is, so far I haven't talked about so I want to take a near horizon limit but I want to take a simultaneous limit so this is the comment I made earlier so what I want to do is to go to some take a scaling limit where I go near the horizon but always keep the horizon at a finite distance so if I do this naively then the horizon goes up to infinity but I want to do it in a way that I keep it at a finite distance so you do, take a simultaneous extremal plus near horizon limit as follows so you do so plus minus rho minus is 2 lambda and I'm going to take lambda to 0 at the same time it take rho minus rho horizon equals lambda times r I'm going to define a new variable called r okay and tau, so if I do this and look at the metric but tau also has to be rescaled in the other way otherwise there's no sensible metric tau over rho h square t over lambda okay and then I take lambda goes to 0 okay in doing so what I've effectively done let me finish the analysis so what I get is that the metric is v times r square minus 1 dt square plus dr square over r square minus 1 plus d omega 2 square where v is rho h square okay so this side just freezes to the horizon here so this is an s2 and this is two-dimensional anti-decider space okay but just to answer your question I think there was another question about this it's not your someone else's that I've done this in a way so that effectively this also looks like there are two horizons to it although it's all in the near horizon limit and that's important okay so what you get is some near horizon limit to be ads2 times s2 now this fact ah sorry and what happens to the gauge field so f equals let me write it as e over 4 pi times dr vh dt plus p over 4 pi sin psi d psi vh d phi okay so what I get is s2 times s2 and I get constant electric and magnetic field okay so for those of you who have never seen it I recommend this exercise I've spelled out everything it'll take you less than 10 minutes but it's nice to do it where e equals where e equals q so it's just q over 4 pi and p over 4 pi so that's what you get in this resume question tau has disappeared from the story yeah thanks okay now this fact so it's very easy to do in this metric for this theory in fact this is much more general so let's stick to 4 dimensions and give ourselves an arbitrary theory of the metric coupled to some bunch of h fields and a bunch of scalar fields and maybe fermions this is the kind of theory we've been seeing in the last few days okay so suppose you have so this is very general so this is general suppose you give yourself s which is before x the square root of minus g times some Lagrangian of metric some bunch of scalar fields a mu i and some bunch of sorry bunch of gauge fields bunch of scalar fields phi a and maybe fermions this is still true you always get in the near horizon limit in ads 2 times s2 in this theory in asymptotically flat spaces in a second the changes are that let me take the question because I am going to take 2 minutes to explain the Reissner Nostrom this is just ads 2 times s2 this is the near horizon geometry of the black hole you've really gone to the horizon and taken that patch near the horizon and that's what this is I mean in the pendulum starting the horizon looks like that you really have some very small patch next to it that's this okay so what happens is that this is much more general so given any such Lagrangian you write a Reissner Nostrom type solution so there's some electric and magnetic charges now the near horizon region of the supersymmetric or extremal black hole always has ads 2 times s2 symmetry so this means there's an sl2 associated with this and su2 associated with this and if I can just write down directly what the near horizon geometry should be so this always looks like ads 2 times s2 with some value of v the field strengths become constant because those are the only things allowed by the symmetries and now where should I write it so comma I'll go on I wanted to fit it here comma so phi a is also constant and all the fermions are 0 okay so these are the this is the most general configuration consistent with the symmetries and this relation that the electric field was equal to the charge for one gauge field now gets changed that e ai is a function of all the qis okay yes so the near horizon is a very thin shell that's right so this is still true this is true now v is some v constant f i becomes e i plus p i e i is some function of the electric fields scalars are constant fermions are 0 at the same time yes yes that's correct that's correct okay and now in this limit sorry in this near horizon configuration this Lagrangian effectively becomes just a function in these fields now it just becomes a function of of course q1p qi p i semicolon now it's just the parameters v e i phi a I hope I didn't miss anything sorry is there a question yes that's the solution that's the solution in its own right so maybe I said this too you said it's the near horizon limit of a solution so yeah that's the first statement but then it's a solution in its own right meaning that that is a solution now so I showed you this sorry so I should have added one sentence so in the case of just Maxwell Einstein you start with Reissner-Norsturm take the limit you get this thing I should have added a sentence saying that itself solves the equation of motion and this fact is much more general I'm saying that there's a very easy way to do it you just assume the symmetries of the near horizon sl2 times su2 and that's the most general configuration I can write down consistent with the symmetries and the Lagrange I still haven't solved equation of motion just give me a second but there is you'll see that this is a solution the Lagrangian now becomes a function of these fields of these numbers of these real variables so what was a complicated what was gr now it just becomes multivariable calculus ok and now in fact this is very general it's not just general but very general in the sense that it's also true if you don't have four dimensions so what then changes is this part of course if you don't have four dimensions if you have five dimensions there will be something else so this correct statement is now you always have ads2 times a compact manifold and it's so you can be any dimension any d and any theory so including the type of theory that Stefan talked about in his second, sorry, in his third and fourth lectures need not have asymptotically flat space kind of thing that Alberto was talking about you can have ads2 times some Riemann surface you can have s3 whatever so the statement is that as long as there is a consistent limit to gr the near horizon region always has an ads2 factor ads2 times a compact factor constant electric fields and these magnetic fields now become some scalar fields think of it as a two-dimensional theory and constant scalar fields so what we're going to do is to think of this as some two-dimensional theory on ads2 just compactify the compact part think about it as a two-dimensional theory so yes so this is thus far a very classical statement about equation of model I'll finish the classical thing here quantum so it could be what, quantum what at the quantum point of view this could be a troncation that is not so I'll discuss that once I start here just give me 5 minutes yes so the charges are just given to you the parameters that vary are the size v yeah I'll come to that so I'm saying that the most general so maybe I should have done the whole thing very quick the most general most general configuration consistent with the near horizon symmetries is this where v is arbitrary e i are arbitrary and in this case p is and phi i are arbitrary but now you have to impose equations of motion and here's the statement so firstly I wanted to say that it's not just four-dimensional with spherical symmetry so it's even more general than that so in that case this L you should think of some two-dimensional Lagrangian which comes from compactifying this higher-dimensional Lagrangian in our case is just equal to v sin theta sorry sin psi d phi times L of the four dimensions and this is just 4 pi v L and S is just another integration of that and you get a v of this two-dimensional thing just look at the metric there's a v in front of the radius 2 so the square root of g of this two-dimensional part is also v and this part is also v and that explains all these factors of v that I've been pulling out I just do the integration over the two-dimensions I do further integration over the radius 2 I get this and now so is that clear was that too quick there ok so now comes the statement the statement is that statement is that the there's something called a classical entropy function which are defined as follows e is function of q i pi v so those are fixed v e i phi a which is defined to be 2 pi times e i times q i minus v times l2 it's just a Legendre transform of the action of the theory with respect to the electric fields ok and the two statements are then one implies it's just exactly the same as a variational principle delta e is equal to 0 and I said this is just multivariable calculus so that means what I mean is d e d v is 0 d e d e phi a is equal to 0 and d e d e i is equal to 0 so the statement is that if you just take your Lagrangian think of Einstein Hilbert plug in this ansatz what I'm saying is that the near horizon symmetries have completely fixed the tensor structures completely so the only things that are left to vary are the sizes so the shapes have been fixed by either supersymmetry or extremality as you want to think of the only things that are left are the sizes but if you take the motions of motion then just become this extremization problem ok so this is just extremization ok and notice in particular that this is the same oops sorry this one is the same as saying that q i is d v l 2 d e i which is the same as Gauss now and part 2 of the statement is that the classical entropy of the black hole is the value of e at this extreme ok so I'm not going to derive this for you you can think of so please take this as an exercise they're both sort of medium level exercises if you want to do a nice review paper by Ashok Sen in 2007 you can do this so Jaum Reis sitting there he's also worked on it you can bug him but anyway this is not a very difficult problem in my opinion ok so that's the classical entropy statement ok any other questions about this did I answer all questions or did I postpone something which I still haven't answered Leopoldo did you it's clear now quantum quantum but I'm going to come to that in a second so for the classical it's just you can prove it as I said all tensor structures are fixed and there's nothing else you can do it as a 2 dimensional problem or a 4 dimensional problem it doesn't matter because of the symmetry I mean here I'm assuming there's a lot of symmetry if not suppose this thing is some Riemann surface like Albert then you'll have more complicated problem ok so in this case let me be careful and say this ok the main point is this equation implies that that's always true and the entropy is equal to that under a large more or less the other way is also true but let me not commit right here we can discuss this later so before writing here before general and very general I said let's stick to 4 dimensional or simplatically flat space this statement is true even for when I break such assumptions so the kind of theories that Alberto was discussing Stefan was discussing all of that is true now of course I'm assuming supersymmetry so what I've assumed here is extremality actually so that was my assumption in looking at rational nostrum but that was because I'm interested in supersymmetric black holes which then become extremal so the technical assumption so far is extremality yes no so that's what I said so the first example I did was spherically symmetric then it remains spherically symmetric but the Lagrangian changes now I'm saying even that's not true any theory arbitrary alright of course the details will change if you have some other symmetry this is not a sphere it will be something ok now let's come to the quantum entropy no so far it's completely general what I said was completely general sorry I want to stress this try to say this I really mean any sphere so the only assumption I've made is extremality or soon I'll say supersymmetry but so far it's just extremality but now I'm going to go towards something more specific and then maybe at the end tell you about the generalities sorry who's the chair can I take five more minutes because I have lots of questions already I mean I have five minutes and that's great please continue like this but it's just that I want to do one little thing before I finish so I've restated so if you want that you can think of that as a definition of an extremal daco that in its near horizon that fact is true for you can prove that it's true for again an arbitrary theory of gravity coupled to matter any extremal black hole as long as it has a limit to Einstein's theory in the sense that the corrections that I put to Einstein's theory must be small if I within that set of assumptions you can prove this I forgot who did it it was kunduri, luciati, bunch of these people but otherwise for our purposes you can just take that to be the definition that's correct what if I add a total derivative total derivative no I think I've not made any assumptions of that sort so I think it's because these are constants I think any Lagrangian is okay can we just postpone this for a bit I think that's correct but let me just think it through okay so what I want to do now in the last x minutes is define the notion of quantum entropy so the classical entropy problem has been recast as an extremization problem in the near horizon geometry so since idea was that the quantum entropy for super symmetric black holes should be a functional integral so the classical problem is an extremization as was said there's an on shell problem so now you make it off integrate over all the fields so that's the sort of moral idea so extremization now is promoted to a path integral so what I want to tell you next is how to do it in the next 50 minutes so the first thing I want to do is to go Euclidean because otherwise the circle is not well defined so in order to do that I'm going to take t to i theta so what I'll have here is now the classical problem will become plus become theta square and here I'll get a minus i times EI so that's the only effect on these fields alright and now what is the idea so that's Euclidean ideas 2 so that looks like a disk it's a hyperbolic disk with some asymptotic boundary the idea is that so in the classical problem everything is constant as we just saw in the quantum problem I want to fix the boundary conditions according to the classical problem and then in the interior I want to let all the fields fluctuate integrate over these fields so I need some falloff conditions so as r goes to infinity I want to say that g mu nu is that classical one which is on the other blackboard times 1 plus o of 1 over r square a mu is a mu i classical which is there times 1 plus o of 1 over r phi a is phi a classical which is constant times 1 plus o of 1 over r now people have analyzed this that this is a sort of consistent boundary conditions so another way to say this is that the ds square in the ads 2 part will now become like this v times r square plus some o of 1 times d theta square plus dr square times 1 over r square plus o of 1 over r to the fourth that's the kind of fluctuation I am going to take and a i is minus i e i times r plus o of 1 o of 1 and phi a is constant phi a plus 1 over r constant plus 1 over r and now given this conditions I am going to define quantum entropy so now I think I can erase this more time define the quantum entropy so e to the s quantum entropy for black hole is the path integral z over ads 2 is a function of q i and p i which is defined as the expectation value in the quantum theory of the exponential of minus i q i times wilson line of all the gauge fields under which is charged and this wilson line is placed at the boundary of ads 2 as r goes to infinity and this is with asymptotic ads 2 boundary conditions of the type that I just said and there is a certain renormalization that you have to do which is what I call finite so that's the definition and I want to spend at least explaining how one should think of this path integral then next time I will pick up with that so this is euclidean ads 2 I am going to put a cutoff and r0 so there are many many issues with this with this path so the way what is the definition you think of the two dimensional theory this includes gravity and all the fields that you have all the whatever fields are in the theory and then there is this wilson line according to all the electric charges of the black hole the magnetic charges in two dimensional description becomes some scalar fields I give myself these boundary conditions and integrate over all fluctuations what are the issues the issues is well firstly how should I think of this integral I really think of it as an integral d over all the fields e to the minus exponential times this many issues with this one issue is that this action itself is divergent because the volume of ads 2 is divergent so this is an infrared problem from the point of view of gravity so let's first tackle that so think of some bulk action which is what our Lagrangian was so you have dr0 from some interior to sorry dr from something to r0 and d theta square root of g times l bulk now remember that the Lagrangian evaluated on the classical field started with a constant we already saw that and therefore this thing will start as as something like that so constant times the volume the volume is r0 it diverges as r0 so it's going to start like this then there's going to be some finite piece and then there's going to be some 1 over r piece ok this is going to be my definition of the finite part so in general in ads cft or in ads when you think of ads quantum gravity as we heard in Janis' lectures there's a whole procedure of holographic renormalization and that's what you should really apply here it actually just becomes very very simple and I'll make some comments about this because the only divergence is this linear divergence which you can essentially subtract I'll make some more comments about this but for now I can just take this as the definition just take this c1 to be the finite piece just one second now similarly for the Wilson line so think of now the Wilson line lives on the boundary the proper length of the boundary if you just integrate the length you'll get v square root of v times 2pi times r0 and therefore the Wilson line eI qI times integral ai also goes similar behavior there is some r0 piece plus some c1 prime piece plus order of 1 over r and again I'm going to take this and a better way to say this so I'm going to define something called renormalized action which is s bulk plus sorry minus i qI integral ai plus some boundary term boundary counter term that I have to add and you choose it so as to cancel this leading order divergence and I'll make comments about this after I finish and now with this definition z ads 2 I should really think of as really a path integral d of all the fields phi gravitational which this metric et cetera et cetera is s renormalized of these phi vector and now I'm going to say one more thing and then stop so suppose you want to take the classical limit that's the first thing to check so when the black hole is large that implies v goes to infinity remember v was the size of the horizon and if v goes to infinity remember l the Lagrangian started its life by with a term which is proportional to v times l2 in what I wrote therefore the action grows therefore you can do a saddle point approximation okay that's just saying that in the classical limit you have a saddle point because v there's a 1 over h bar in front of v and in that case the s bulk let's be more careful so r remember sorry r went from 1 to r0 d theta went from 0 to pi v times l2 so this is just 2 pi this is a constant in the classical limit r0 minus 1 times v l2 and minus i qi integral ai is just minus 2 pi times qi ei times r0 minus 1 and that implies that normalized is just 2 pi times this is the finite part so it's this one and that is this one I just drop the r just get qi ei minus v l2 and that's precisely what the classical entropy was so this is not a surprise of course I've started with the classical thing and went up to the fluctuation problem just to make sure that everything is consistent so if you do the saddle point evaluation of that so in this limit you get s e to the s quantum just reduces to e to the s classical so this is just s classical so I think I'll stop there next time what I'm going to do is to apply this to this n equal to 2 black holes and we'll take it from there thank you very much sorry I already postponed some questions