 So, the plan of this lecture is to give a macroscopical counting for the entropy of ADS-4 black hole. I have notes. I was planning to put them in tech, but I was too lazy. So, at the moment, there is only an end-written copy that at a certain point, once I have corrected all the mistakes I will put on the web. And maybe later on I will write the same thing in Latin. I hope what you find is readable. And the plan of the lectures is the following. Thanks to Stefan, we already have a lot of background on black holes. But nevertheless, in the first of these lectures, I will discuss the black hole of interest, those that have an embedding, and I will give you some details on the object, and then we will study from the point of view of field theory. So, the plan of the first lecture is to interpret this black hole using holography and to tell you what I have to compute. I will try to be slightly more general than discussing only ADS-4, where the computation has been done. So, in this first lecture, I will discuss essentially ADS-D black holes in the greater of equal than 4. That can be embedded in string theory and theory and so on. So, they have a consistent string embedding. And tomorrow I will discuss how to construct the field theory using localization, the partition function that is supposed to count and the number of ground state of this particular class of black hole in D equal 4. And in the third lecture, I will discuss how to evaluate this partition function in the larger limit. I want to compare with supergravity using holography, so I will need to perform a larger limit of a partition function of a matrix model and I'll tell you how to do that. And in the fourth lecture, but I'm not sure I will have time to go so far, I plan to discuss something, the same picture for ADS-5 black string that can be reduced to ADS-4 black hole. But I cannot promise that I have time to do all of these. Okay, so today we start with the gravitational description, the holographic interpretation and essentially I want to write the object that corresponds to the entropy and for which I need to find a field theory computation. So, as I already said, I'm interested in black holes that are syntotically ADS and can be embedded in M theory or string theory. Actually, there are plenty of these objects that can be embedded already in a maximally supersymmetric background of M theory or string theory. So maximally supersymmetric background relevant for holography are this one. There is an ADS-4 times S7 in M theory. There is an ADS-5 times S5. This is an equal for superior means and there is an ADS-7 times S4. This is M theory. This is type 2B and this is M theory. And all of these are maximally supersymmetric background and for all of these guys we know what is the dual field theory. Well, this is the basic of the holographic correspondence. This is an equal for superior means. We know quite a lot also about ADS-4. This is the model that we'll discuss in detail and this is the Chern-Simons theory in 3D, maximally supersymmetric. This is the ABJM theory. We will come back to this later. And even for ADS-7, we know what is the theory. This is the famous so-called 2.0 theory in 6D. It's not a grandeur, but some sense is the mother or father of all the other theories because by compactification you can get an equal 4 and everything else. So we have a pretty well understanding of holography for this background. And there are plenty of black holes in such backgrounds that are characterized by a set of charges and in general also angular momentum. At the very beginning I will be more general, so I will consider rotating black hole. Then the main example that I will discuss is actually the one that Stefan discussed in the previous lecture, so static black hole in ADS-4. But for the moment, let me be more general because as I will tell you there are still many open problems here and many of these problems involve rotation. Let me general just to give a picture of what you would like to compute. So what kind of charge you can put? So the idea is that you want to embed the black hole in a consistent way, for example, in type 2B. So consider ADS-5 times S5, for example. Here you have an isometry which is SO2,4. S5 as an isometry which is SO6. And here clearly there is a group of rotation which is inside SO2,4. So you can introduce two spins for this guy, J1 and J2. Essentially I'm always looking at the carton of this non-Abelian symmetry. As Stefan discussed, there are also non-Abelian version of black hole but they are much more complicated. Everything from now on whenever you see a non-Abelian symmetry, a non-Abelian symmetry, at a certain point I will take the carton of it. Taking the carton of SO4 means you choose two angular momentum inside ADS-5. You can choose two of them. And here you can select cartons of algebra, which is you want to the cube. And you can put three electric charges for the carton generators of this. You want cube. So in general you would expect to find a black hole that can rotate in ADS-5 with two independent angular momentum and it can rotate inside the sphere with three, if you want, these are angular momentum inside the sphere that appears electric charge for your black hole. Curiously supersymmetry impose a constraint among these guys. So in principle you have five parameters, but the BPS conditions tell you that there are four independent ones because there is a constraint. This has been a long-time puzzle. People ask why there should be a constraint. When I will discuss the interpretation, I will interpret the microstate of this black hole as states in N equal 4 superior mills with particular spin and particular charges. And in N equal 4 superior mills there is no such a constraint. You can construct many states that are supersymmetric with the spin that you choose, the charge that you choose. So this is a long-standing, well, say, puzzle why the black hole exists only for a constraint among the charges. There is a recent paper by Santos and Markiewicz this June where they say that they found a new black hole corresponding to the missing charge. I don't know yet exactly what is the status. This black hole seems to have some mild type of singularity, tidal force, and this kind of thing. So it's not clear. Maybe there is even a full family depending on all parameters. But for the moment let's keep the standard law. There is a constraint among charges. OK. In this game actually ADS-4 is special because here you can have dionic black holes. Let's consider the same set of charges for ADS-4 times S7. S7, sorry. Here the relevant group is SO2, 3 for ADS-4. It's SO8 for S7. So this time you just have one possible spin that you can introduce in ADS-4. So you can have the spin for your black hole. And here you can break to the carton SO8 and you have four charges. So for sure you can put four electric charges. But also since you are in four-dimension as Stefan explained, you can also have magnetic charges. So here the number of charges that you can put is much bigger. Even in this case actually supersimilarly tells you that there are constraints. And essentially as we will see better later in this lecture there are two constraints. One is a sort of linear constraint of the magnetic charges that will be very important. That's related to the topological twist of the dual conformal field theory. And again, BPS tells you that there is an extra constraint among the spin and the electric charges. I will come back to this constraint later. Now, and you can play the same game here. Here you have three rotations and two electric charges. No magnetic charges. Magnetic charges are relevant in four dimensions where you can have the ionic object. Now, why I'm stressing the magnetic charges? Because the magnetic charges makes a big difference in the holographic interpretation of the black hole. Now, given your black hole, let's consider the effective field theory that you have in ADS-D. So you start with ADS-D time sphere. You dimensionally reduce. In many cases all these black holes that have been found actually have been found in the lower dimension of theory and then uplifted. So you may think that this effective theory in ADS-D is some gauge supergravity or unequal to gauge supergravity as discussed by Stefan or something more complicated, some consistent truncation of your string or theory background. But in general for what I'm going to discuss is not even necessary. So suppose that you just truncate an effective theory. The effective theory lives in ADS-D. You want to use holography. So in ADS-D you will have a very complicated Lagrangian. The relevant terms that are of interest are obviously the kinetic term for gravity and kinetic term for a set of gauge fields. As we saw in Stefan lectures, typically in these theories you also have scalar fields and fermions and other stuff. Many other objects in particular scalar fields and typically the kinetic term for the gauge field is a complicated function of the scalar fields that you need to specify. What are the scalar fields of interest? Here these vector fields have new lambda. The ones that are interested for this discussion are those that are related to the isometries of the internal manifold. So suppose that this is ADS-4 times S7. You have for sure a full set of SOA gauge fields in the bulk. If you're just interested in the carton, you take four, a billion vector, that you write here. These new lambda correspond to the massless guide that you find in your Lagrangian. They correspond to the isometry of the internal manifold. So they correspond typically to the r symmetry of the theory. If this reduction is ADS-5, in principle you have an SO6 bunch of gauge field. If the theory is ADS-4 times S7, you have an SO8 and so on. I will always break to the carton. Now I want to use holography to interpret the black holes. According to the rules of holography, a gauge field in the bulk correspond to a global symmetry in the boundary. So gauge fields in the bulk are global symmetry on the boundary. Again, example, in this case, the global symmetry is the r symmetry of the dual field theory. So the black holes that you want to write and Stefan wrote explicit examples are asymptotic to ADS-D. So this means that there is some radial coordinate r such that the metric for large r is of this form. There is a term dr squared over r squared. And in my choice of coordinates, there is an r squared times md minus 1 squared. So this is a standard form for a space time that is asymptotic to ADS. There are corrections here in one over r when r is not strictly infinity. And here this guy here is just the boundary of your ADS space. And so whenever you write black holes, you typically are working in sort of global coordinates for ADS. Stefan wrote a metric for the black hole where this boundary term contains time and contains the metric on a two-sphere, OK? Which means that they are using as coordinates two angles, theta and phi, time, and the radial coordinate. There is a parameterization of ADS, which is called the global coordinates in ADS that precisely write ADS in this form. So in general, in all the example that I'm interested in, obviously there is time. And so one component, one element of this manifold d minus 1 will be time. And in most of the black holes, the other, the remaining coordinates parameterize a sphere. So for example, these are the coordinates theta and phi of the previous lectures. Or in three-dimension, you have coordinates on a three-sphere and so on. So most of the time, the boundary metric is time, time as sphere of dimension d minus 2. And this is the case I will discuss most because it's simple. Looking at sphere is simpler than looking at more general object. But in principle, you can also construct more exotic black hole if you want, where you just replace this with another manifold. Typical example, this I will discuss in detail, is ADS4. In ADS4, you have the radial coordinate, you have time. Here you have a sphere. Well, what you can do very easily is to replace the sphere with an orbital Riemann surface. There is nothing essentially that changed in the black hole. I don't know if you want to still call it a black hole. It has an horizon that is not a sphere. It is a torus or is a Riemann surface. This object can be constructed. And from the point of view of holography and the point of view of localization, there is not much difference between the two classes. So I will discuss the general case. I will talk to you if you like black hole with, say, hyperbolic horizon. Hyperbolic means that you take a Riemann surface or Iger genus totes the hyperbolic plane with a quotient. And in general, if you go to Iger dimension, you can replace these guys with other stuff if you like exotic black hole. So the first message is the following. Whenever you see a metric in the bulk that is asymptotic, that has an asymptotic of this form, well, what you are doing is the following. Since this is an effective theory in anti-deceptive D, you assume that there is a conformal field theory somewhere. Conformal field theory with one dimension less, D minus one, this is holography. In a sense, the conformal field theory is on the boundary. In standard application, in the original application of holography, one was considering the metric of ADS and this is really either flat space in Poincare coordinates or time hemisphere in global coordinates. So the logic is that when you replace this slice here with something more general, you are just taking your conformal field theory in dimension D minus one and you put the conformal field theory on this manifold, which can be a curved manifold. But we learn in the lecture by Francesco that it's not a problem, not even for supersymmetry. There are many manifolds that support supersymmetry. So we know how to do that. So this is the first message. The black hole that I will consider correspond to conformal field theory on manifold of this form, where there is a time direction and typically there is a sphere or some other manifold. But this is not the end of the story. This is the simplest part. The point is that since there are gauge fields and we saw that you need gauge fields to give charges and have regular black hole, in the bulk also the gauge fields have in general a profile. So what you discover is that in the very example the profile is slightly different but there will be a profile of the gauge field that near the boundary is some power of R that you need to specify times in principle a non-zero function of the coordinates on the boundary manifold. Now, so what is the interpretation of this gauge field in holography? Let me remind you another general paradigm of holography. Recall that subscript of the... It's mu, sorry. Well, let's write it as a form. If you want, I put it here. It's a gauge field. Recall that whenever you have a field in ADSD which I write as a function of the coordinates on this manifold and the minus one and the radial coordinate. And this field has an expansion near the boundary in power of R. So here there are some coefficients that you need to compute and they are typically related to the conformal dimension of your field. Since generically all the fields satisfy second-order equation of motion you can always find an asymptotic solution with two independent parts. And typically the point is that one of the two in a generic situation is a non-normalizable deformation of ADS. So the norm of this finite is infinite. If you compute the natural norm in ADS while the second one is normalizable. This is the generic situation and there are caveats for a particular value of the conformal dimension but don't go in this detail. In general, you can split phi into pieces and the interpretation of the two pieces is different. So the interpretation of the non-normalizable mode is non-normalizable so it's doing something severe at the boundary on your conformal field theory. What it's doing is this guy, if you have a profile for a field in the back where this guy is different from zero what you are doing is deforming your conformal field theory. How you deform your conformal field theory? Well, this field in anti-desitter is due to some operator on the boundary and the deformation is simply you take your conformal field theory and you add to your Lagrangian the guy that you see here suppose that this is a constant at the boundary but it's not necessary so you take this guy here you multiply by the conformal operator and you deform in this way the conformal field theory. In general, if you have a non-normalizable deformation you are deforming the conformal field theory so you are using the conformal operator associated with the fields. What's about the normalizable? Well, the other paradigm of holography is that instead this guy is a source for the operator so when you turn it on you turn on the operator the sub-lead term, the normalizable one is not a source but is a vacuum expectation value. So the interpretation of this second object is essentially a VEV for the operator associated to O. So whenever you have a solution that is only asymptotically ADS and you have fields that runs you can have an interpretation of what's going on in the dual conformal field theory in this way. You look at the field which is a non-zero non-normalizable term and then you say, okay, you are deforming your theory in a particular way you are adding the operator that is associated with the guy phi. So in this case you are doing a deformation and in this case if this is different from zero you can even ignore the second one. If there is a second one that will tell you that there is also some VEV if you start deforming your conformal field theory you will get VEV for the operators and everything else. So the other relevant situation is the situation where the guy is zero. If the guy is zero so if there is no lead in term then there is only this one. So what's the interpretation? You are not deforming your conformal field theory. You are just studying your conformal field theory in a different vacuum so you take the conformal field theory there, the operator has a VEV so for example in N equal 4 superior mills obvious VEV that you can give is to go along the Coulomb branch you take the scalar fields in N equal 4 you give a VEV and you may want to study your theory in this new vacuum. So this is the general paradigm. Now typically the VEV is determined by equation of irregularity in the bulk as a function of finite. So you will find typically when you deform a theory if you compute the vacuum expectation value of all is no more zero is something which is determined by the deformation. So essentially there is always some VEV but the main deformation that you are doing is by adding terms in telegram. So this is the general paradigm that you find on first pages of any ADS CFT review. Now I want to interrupt the black hole in this language so there are gauge fields at least there may be other stuff scalars and other stuff but in general there is gauge fields because you need charges and then what is the interpretation? So here is the crucial point and the difference between magnetically charged and electrically charged black hole. So let's distinguish the two cases. So for an electrically charged black hole in all the cases that I discussed ADS 4, ADS 5, ADS 7 typically a mu lambda fall off at the boundary as a normal normalizable solution. So the term phi naught is strictly zero. The leading term is strictly zero. The same is true for spin and rotation. Rotation you cannot interpret as a gauge field. Rotation is sitting here in your metric. Your metric instead of being dt is dt plus something generically. You still are deforming your solution with term that belongs to the metric instead of the gauge field. Same question, you look at the terms in the metric, you see how they goes at the boundary and you still discover that they goes as normalizable modes. So electrically charged and the spin correspond to normalizable deformation of your boundary theory. So they correspond essentially to the deformation value. So if you want to study a black hole which is electrically charged and rotates, what you have to do is just to take your conformal field theory in d minus 1 put it on the curve manifold in the simplest case is just times times the sphere. So you just put the, you do in conformal field theory what's the name the you go on the sphere times times so you do radial quantization so that's a pretty standard way of studying the conformal field theory but you need to study the conformal field theory in a state with a non-zero expectation value for what? Well, the operator dual to the gauge field is essentially the current so essentially you are studying your conformal field theory in a state with non-zero charge. Yes. The chemical potential for the gauge field? So the chemical potential would be the non-normalizable piece but if you look at a black hole when you fix the charges so it depends on the choice of ensemble so the black hole you can fix charges or chemical potential if you fix the charges that's the gauge field is such that it is normalizable at the boundary so what's the interpretation? Well, you are studying your conformal field theory in a sector where the electric charges and the spin let me call it J I hope there is no confusion with J mu which is the current associated to is q as an effect yes so you are just studying the conformal field theory the electric charge is different from zero and the spin is different from zero what's the meaning? Well I want to know what happens to the conformal field theory in a state with given electric charge and given angular momentum this is a black hole should correspond to an ensemble of states and if you are after the microstate you say okay what are the microstates well the microstates of this black hole should be just a set of spin-full and charge states in the conformal field theory what spin and what charge well this is specified by the spin and the charge of the black hole so if you want to study the entropy of the black hole you just need to count the number of states that have a given spin and a given charge okay that's simple enough and pretty natural this is for an electrically charged black hole if you add magnetic charge and to add magnetic charge you need to be in AdS-4 well you discover that the magnetic charge in 4D behaves quite differently so if you look at the profile of the gauge field in the bulk you discover that the profile of the gauge field in the bulk for larger is not a normalizable solution in this case a mu lambda is a non-normalizable solution I can even write for you because Stefan already did the point is the following suppose that you have a black hole in AdS-4 with a spherical horizon so your boundary what I call M3 is just in Lorenzo signature just time times S2 and the point is that as we learn in in Stefan lectures there is a magnetic charge which means that you have a gauge field F such that if you integrate over the sphere you get a number actually because of quantization in suitable units integral and divided by 2 pi you should get some integer this is just the Dirac quantization condition that Stefan mentioned so and this is should be true at all possible value of the radial coordinates in particular at the boundary at r equal to infinity so if you want to write as please the gauge field well you can for a sphere is very simple okay you see that there is no dependence on r here which means that the gauge survives all the way to the boundary is there you can compute the norm in AdS the norm is infinite so this is a non-normalizable deformation of AdS or you can easily generalize this to a Riemann surface cannot write for you I don't know how to write it well you need to choose coordinates on the Riemann surface but it will take long and it's not important you can replace the monopole with the natural object with magnetic charge on the Riemann surface there is the analogs of the monopole I will call it monopole from now on there is a monopole on the Riemann surface and again the charge is quantized so what's happened to the dual field theory as I told you this is not normalized so you are changing the conformant field theory how? the first thing that you do is to add the background field a mu a lambda mu and what is the natural deformation? well obviously from the point of view of holography every gauge field is associated with a current for a global symmetry so what you are doing is turning on a standard gauge coupling between this guy which is a global symmetry what you are doing is turning on a background gauge field for some global symmetry this you can do in field theory which simply means that you deform your Lagrange by adding the current multiplied by the background gauge field so the point that these black holes have a different interpretation they are interpreted as the formation of the conformal field theory where you put background gauge fields for the values global symmetries to understand a better what is going on it is useful to look at how supersymmetry is preserved so remember that there is always this this guy here essentially a monopole that you turn on if you look at how supersymmetry is preserved you need to look at the variation of the Gravitino Stefan wrote it for us and schematically my notation obviously are not precisely the same as those of Stefan but you can easily make a dictionary well obviously there is the spin connection here and in as he may discuss carefully engaged supergravity the guy is charged under a collection of fields so there is an asymmetry of Gravitino that you are engaging and quite schematically let me call it in this way should put an I here the variation of the Gravitino so what happens is also Francesco was discussing this morning if we are in AGS4 so the boundary is time times S2 time is pretty boring let's take everything constant on time is a flat space it's not interesting S2 is curved and you need to have some BPS equation that tells you how supersymmetry is preserved the spin connection on S2 or even on a Riemann surface is just an abelian gauge field here you have an abelian gauge field so if you turn on an abelian gauge field that cancel the spin connection well the your equation of supersymmetry can be solved very prettily easy in this way you choose a constant spin I'm cheating a little bit because I started with rotating black hole it's a mess so for static one this is the relevant equation that you have to write but even for rotating there is a generalization with things but let's focus on the static case so essentially the point is that on the sphere the spin connection you can do your computation by yourself compute the spin connection on the two spheres you will find a very simple expression the simple expression will tell you that the spin connection on the sphere is just a unit monopole it's a sphere it's precisely of the same form if you write it as a form tomorrow we write it better the spin connection is exactly cos theta d phi so it can be cancelled by the background that you turn on and it is indeed cancelled yes think of the CFT on S2 as being almost two dimensions CFT yes but it is on the sphere and with now I'm coming to this point yes but I want to deform it with this gauge field so what you get and what I was saying right now is that what you are doing is topological twist of this theory indeed as you saw and we will come back tomorrow to this point in details as you saw in Francesco Lectra this is one of the way of preserving supersymmetry on the sphere and as he mentioned it's very old correspond to a topological twist tomorrow we will discuss precisely the meaning in all details but if you remember the whole story about the topological twist as introduced by Whitton there was a change in the nature of fermions so it was mixing the Lorentz group with some asymmetry and mixing the Lorentz group with some asymmetry in this more transparent means turn on the background field for some asymmetry in such a way that cancelled the spin connection so that the equation that you typically cannot solve on a sphere in a well you can solve in a sphere but you get complicated objects the covariant derivative in the presence of this guy becomes just a standard derivative and you preserve supersymmetry by taking constant spin you do what you do typically in flat space essentially so this is the topological twist so the meaning of the interpretation of this black hole is that you are doing a topological twist actually this guy here you may have many gauge fields in the bulk as Stefan was discussing this is a particular combination the one that he called zeta lambda e lambda mu before so there is a particular combination of your gauge field that enters here and you need supersymmetry to SUL you should better choose the right charge for this particular combination in such a way that it can solve the spin connection then you may also have other global symmetries once their symmetry is fixed all of them have this profile this monopole profile so coming back to the interpretation of a magnetically charged black hole is essentially now the interpretation is the following you are taking your conformal filter you are deforming in this way this is just another way of saying that you are doing a topological twist in the sense of written so the magnetic charge correspond to a topological twist they are deforming your theory and what's about the rest if you have electric charges rotation well they behave as precisely as before so the magnet the interpretation of a magnetically charged black hole with electric and charge and spin is the following you take your conformal filter you deform it according to magnetic charges and then you start counting you count the number of states that have a given electric charge and a given spin so you always counting problem counting the number of states with given electric charge and given angular momentum but in the case of this black hole you just count on the conformal filter when you have magnetic charges you count the same states the topological twisted version of the theory we will see that topological twist helps with the computation because give you a computation that you can match with with holography let me also mention that again for the difference between this black hole here the spin is constant very simple and here is a mass especially when there is rotation I couldn't even find the famous class of black hole it's very difficult to find explicit expression but it's a mass typically so typically there are some of them that are just partner and supersymmetry of what I'm doing here so typically in the example that I will discuss there are running scalars so they also induce some deformation but the deformation that they induce is just the supersymmetry part of this then you can have more general black hole with hypers that's more complicated things that you have to do the one idea of this idea of localization as Francesco was discussing that you can compute something but integral or correlation function of certain specific operator yes it's two in a sense but this is a full quantum field theory on S2 so there are excitation that are not topological the only point is that I cannot compute those using localization and I will not need to compute those or what I will need is the partition function that I can compute now so let me summarize what you again what you have to do enumerate all the states with electric charge let's call it QI and spin that can be more than one so let me call J.I. the spin in the conformal field theory on R times d-2 for electrically charge and rotating black hole that's what you have to do or in the topologically twisted CFT on R times Sd-2 for magnetically charge black hole so the magnetic charges enter here in the topological twist now enumerating states is equivalent well this is the way we like to package things in statistical mechanics on instant theory is equivalent to knowing some partition function particularly some I will introduce some grand canonical partition function so I will introduce chemical potentials for the electric charges and the spin and I can package the information that I want this is the number of states with charge QI charges QI and spins JK this is what I want to compute I claim that these are the this is the essentially the term is the entropy of the black hole is the number of microstates I can just package this by summing over all the charges and the spins with chemical potentials I call delta the chemical potential for the electric charge and omega the chemical potential for the spin and I can write a generating function for this object this obviously can be also written as a trace if you introduce operators that correspond to these conserved quantities let's call this operator so the charge operator and let's call I a mover with a letter let's call JTIL with some funny notation the operator whose eigenvalue is JK this is simply computing a trace on the Hilbert space of I delta I the corresponding charge plus omega K the spin operator so these are the operators in the your conformal field theory or something that I should mention I probably was not very precise here you can't hold the states but you don't want really hold the states the same supersymmetries of the black hole so I restrict this sum here to the supersymmetric states those that preserve the same supersymmetry of the black hole now what? or the other states probably will contribute to the entropy of some non supersymmetric black hole extreme non supersymmetric black hole ok the macro state of this one should satisfy the same symmetry constrained as the black hole so this is what you would like to know and then the entropy of the black hole is just hidden in this coefficient here so if you know the partition function the coefficients are easily found by Fourier transform Legendre transform whatever where you integrate where you put your contour of the pencil but essentially what you can do is integral to extract the coefficient if I put an i here I put a minus i here delta i q i plus omega k j k and so that's a if you really know this function for all value of delta and omega you can compute the number of states number of states is the exponential of the entropy of your system so this guy here you want to identify with the black hole entropy so this is what you have to do in all cases even a problem in some lecture always the same structure the goal is to compute this one the generating function of all entropy for all black hole now I will be discussing holography where there is a large parameter as large n so essentially these guys here will be very large so in that case I can evaluate the integral by saddle point approximation okay so I will be always in the case where you can use the saddle point saddle point you know how it works you just punish at this and you take the mean the extremum of the exponent and the extremum of the exponent give you the value in this large limit of this quantity so in the limit where the charges are large or n is large when you can use gravity essentially what you have to compute is the following delta i qi omega k jk evaluated on the extremum of this quantity so you determine the quantities such that the derivative with respect to delta and omega of this quantity here log of z minus i delta i qi plus omega k jk is 0 okay this as you know is also called a Legend transform is something that you do every day with statistical mechanics where indeed you take the thermodynamical limit which is the analogous of the saddle point here and this is telling you that in this limit the entropy is the Legend transform of your partition function quite natural the statistical mechanics so the point is how can I compute this object here can I do that well but okay it's hard because not always you can enumerate all the states in a conformal field theory with a given charge and a given supersymmetry you can do some times for example in n equal 4 superior if I ask what can I enumerate all the one half bps states with a given electric charge yes it's a pretty simple problem I could solve for you in 10 minutes for example complex geometry and nothing more one half bps but if you remember the black hole that I'm interested in are one quarter bps the problem of enumerating all the one quarter bps states in n equal 4 superior is open people tried 10 years ago very hard they couldn't find any solution for the problem there are partial results so this is too hard to compute so well I can hope to be lucky and compute something else hoping that it gives the same result the something else that I will compute for you in this series of lecture is the supersymmetric partition function what's the difference when it's written in terms of a trace of a real space the main difference is that instead of being a trace there is a minus one to the f that counts in a different way bosons and ferns clearly it's not the same object this is an index because you can for free introduce your Hamiltonian here with a parameter which is a fictitious temperature there is no temperature in this black hole there are bps and indeed there is no dependence on this beta and the reason is the old with an index idea so whenever you have supersymmetry the states that have no zero value of the energy are paired for every boson there is a fermion you apply supersymmetry they have the same energy they have the same charge because charge commutes with Hamiltonian all the massive states cancel and you are left with just the states with zero energy which is precisely this sum here and as you know from test book whenever you have a trace over e to the minus beta h you can rewrite as a pat integral pat integral on what well the pat integral is the pat integral on the manifold that I discussed before actually in order to interpret this you need to go to the Euclidean so what you do when you have finite temperature is to take your time direction compactify on a circle of radius beta so you make a circle of radius beta you integrate on the manifold so in all the cases that I I already had a compact manifold I had a circle the radius of the circle will be relevant because the partition function supersymmetry is not actually depending on this beta but you can formally introduce it and this is the partition function with some chemical potential turn on of your conformal filter so this is the the object that I can compute and in general this is different from this one because one counts all the states the other counts all the states with a sign all the ground state but with a sign if they are boson and minus if they are fermions so if you are lucky and many times this happens in theory you are lucky there are no cancellations in the sense that the majority of your states in the larger limit are either bosonic or fermionic if this is the case the physical content of the two partition function in the larger limit is the same and I will show you that for radius 4 indeed there is such large cancellation of between there is no such large there is no large cancellation between boson and fermions so the majority of states are all bosonic or fermionic and this partition function correctly reproduced the entropy of the black hole now this will work for radius 4 as I will mention at a certain point if you try to do the same trick for radius 5 this was done more than 10 years ago you fail miserably the same partition function for the same partition function there is a large cancellation between states this guy is different from this you have to explain an entropy that goes like n squared in field theory you find a number that goes like 1 cannot explain the entropy for some reason with the topological twist remember that the main difference is the magnetic charge that it's over okay no no I can stop here getting there or over it's time it's time, okay good okay so this is the object that I will start evaluating for you tomorrow morning actually I will spend the first 10 minutes telling you precisely the form of the black hole I'm interested in exactly I already had Stefan lecture most of the topics so that part will be quick and after that we will start computing for radius 4 this supersymmetric partition function so the point is that I will take a conformal field theory I'll tell you how to implement the topological twist and how to evaluate explicitly the partition function