 Yeah, okay, so yesterday we started talking about black hole thermodynamics and then we saw how for supersymmetric black holes yesterday we just use the extreme limit one has a definition I mean Sen has proposed a definition of quantum entropy and I don't know if this is really visible I can enlarge a little more I don't know how to do this here I guess that's the best one so there were these so it's all defined in the near horizon areas to region there are these falloff conditions that one puts on the metric and the gauge field and the scalar field and then you want to integrate over the fluctuations these these pieces here that's what you want to integrate over I want to emphasize this because there was some confusion yesterday I got a few questions about this so what I developed in the main part of the story yesterday was the classical entropy so that's a an extremization problem and there everything is constant and the on-shell value of the action is the entropy the Legendre transform of the action but what we really after is this we actually want to do the functional integral okay so I just want to emphasize this and then there was second part of this that what we want to do is whatever your bulk Lagrangian is you want to add so whatever your Lagrangian is you write the bulk action in ideas to add some boundary term now this Wilson line and then such that you choose the boundary term such that this S3 normalized is finite and then you want to do this path integral for the fluctuating fields see here five graph means all those o of one and o of one the sub leading pieces that you want that are fluctuating that's what me five graph means okay so that was yesterday's discussion I'm going to switch this off now start now so today I'm going to do yesterday we gave a very general context we started with four dimensions with a simple example of rational nostrum and then I said that actually this kind of formalism is more general applies to any supersymmetric black hole and in order to do that more general thing you have to go to a two-dimensional way of thinking today we'll give ourselves a very precise context and the context is black holes in n equal to supergravity and this is exactly the thing that Stefan discussed in his first two lectures so I'll be talking about asymptotically flat black holes so ungaged supergravity so there's a very precise context and what we want to do is to apply this formalism to these kind of black holes and these black holes are all spherically symmetric so I'm just making a choice so as to make the discussion clean and therefore we'll always work with four dimension four dimensional actions and Lagrangian again I'm saying this because I got some questions which showed some confusion so whatever system it is it tells you what Lagrangian you have okay so this was sort of a more general discussion but we don't need to do that okay so so why are so it's a class of black holes we're considering but it's a very general class of black holes and in fact these black holes appear in string theory in type two string theory on Calabi how compactifications and what you get is an n equal to graviton plus some nv vector multiplate graviton multiplate vector multiplates plus and age hypermultiply okay so I think this everybody knows so here in the graviton multiplate there is this gravity photon there'll be some nv gauge fields AI and here there will be some number of hypermultiplets which I'm not going to discuss very much today and in these vector multiplates there are also let me say a1 to nv and x1 to xnv those are the complex scalars of the n equal to vector multiple okay and together our notation will be AI okay I goes from 0 to nv okay then as I said I'll only be discussing un gate supergravity here in these lectures and the formalism that I'm going to use is the super conformal formalism that Stefan very nicely discussed and I'll review it because not everybody may be familiar with it I'll I'll take whatever I need out of that but let me just tell you why this formalism is nice to attack such problems firstly there's off-shell supersymmetry which means that off-shell means that you don't the supersymmetry transformations close without using equations of motion their auxiliary fields and they just close algebraically and this will be crucial later when we start to use localization there's also fairly easily allows for higher derivative interactions so this I'll talk about tomorrow today I'll stick to derivatives but the formalism as Stefan he didn't quite talk about this but he hinted how higher derivative interactions can be incorporated it's it's fairly easy and this third thing is a consequence of the first two the Susie transformations are I'm just repeating independent of the action okay and this is nice both technically and conceptually so technically Stefan already talked about it a little bit but as we'll see even conceptually this this fact is going to be very useful in our calculations okay and the theory is based on gauging the n equal to super conformal group algebra let me even say this so in this super conformal formalism I'm going to take this theory is rewritten as so the graviton multiplet is often called the vile multiplet you have nv plus 1 vector multiplets and h plus 1 hypermultiply okay this Stefan already talked about the plus 1 is reflects the extra gauging variance of the symmetry of the theory so the super conformal supergravity has many more symmetries compared to the Poincare supergravity and this plus 1 for instance is a compensator field for the dilatations that you have and this plus 1 is a compensator field for the SU2 gauge transformations you can think about it like that but in any case once you fix all the gauge transformations and so on of this theory you get back Poincare supergravity with nv multiplets and nv vectors and nh hypers alright okay so let me just briefly remind you of the theory it's based on gauging this algebra and of course it's not just a regular gauge theory you have to put some constraints I won't discuss this in detail but if you really want to understand all the calculations you know in all its depth you really have to understand these and encourage you to look at the original papers and so let me just tell you what the symmetries are so the local gauge symmetries are so there's general coordinate transformation okay which is the same as the themorphism although I want to range this this local Lorentz transformations then so these are just rotations local rotations there is the rotation which certainly is not present in Poincare supergravity and then there are special conformal transformations which also are not present in Poincare supergravity there's an SU2 times u1 r symmetry and this q and s supersymmetry okay all of these are local transformations okay and so these are you can think of this as based on some some gauge principle okay and a word about the notation I'm going to use and this is important today as well as the next two lectures so my notation is a mu is a spacetime index okay a which is 0 1 2 3 is the tetradindex or the tangent space index I is 1 comma 2 is an SU2 r doublet okay and these sort of make so if I have a field with some indices then these symmetries are manifest that the representations are manifest if I have some field okay so the reps are manifest and the other symmetries that you want are so Lorentz is as Stefan said is automatically taken care of by this so even say this so diff I just remind you these are all basic things but let me just remind you x mu those two x mu plus c mu right so the mu index tells you how the local translations act a index tells you how the local Lorentz transformations rotate the field and then you're left with so this one gets gauge fixed you're left with what you also have to say how this is this acts so this this one and the rotation and you and R and Q and s are not manifest from the from just the letters here and so you actually have to tell say what those things are so there are tables if you go look at the papers that the theories always presented like this there are some fields which carry indices here and then each field you have to say what are the charges how they transform under all these transformations so for example take so the basic field is the wheelbind emu a so you can see that this has a mu index so how it translates it has it rotates as a vector and but then I have to say what are the charges so under dilatation so under dilatation a mu goes to lambda inverse the emu okay so it has charged minus one lambda affects some local scale transformation and another example is the x i still complex scalars in the in the vector multiplied they have charged plus one under this you and R and under dilatation also they have charged plus one okay so like this so you can just write the whole table I just give you examples okay so that's my notation okay any questions so far either about yesterday or about this much okay let me to remind you what the field content is the field content this is a as I said there's a while multiplied which has the following fields so there's the wheelbind there's the Gravitino rather to Gravitino there are two gauge fields a mu and b mu then there's an s u 2 gauge field called v mu i j so there are three of them and then there are auxiliary fields t i j mu nu and another fermion chi i and the field called D which is a scalar so this is a boson is a fermion boson boson boson boson fermion boson and overall there are 24 plus 24 of shell degrees of field okay this field is is anti-symmetric anti-surf duo okay and we'll be using combinations of the following type t minus t minus a b is defined to be epsilon i j t i j a b or mu nu okay so I'm going to I might go between between the two I mean just a second yes which multi-plate how many delete on why so how many sorry what how many degrees of freedom this year it's another it's another formulation of this okay yeah so I'm going to stick to this formulation what was the question okay then there are vector multiplets so that was t minus and then there was t plus which is t minus mu nu star yeah maybe in the fourth lecture I'll be a slightly more general and my answer right now is yes please do go ahead and pursue pursue that so a vector multiplied has a mu x x is a complex killer then there are two k genie omega I and then there's an auxiliary field why I IG that's a symmetric so there are three auxiliary fields I know this is a reminder for most of you but I don't think that's too bad and there are eight plus eight degrees of freedom okay and I was from zero to NV as we said okay then there's also hyper multiplied which I'm not gonna discuss and you need at least one of them in order to gauge fix the theory but a full-off shell formulation is not known but it's fine so that's my theory and what I want to do now is is to talk about black hole solutions so the black hole solutions of the type of state fund discuss in the first two lectures the participating fields are these so participating fields are mu a T mu nu I j in the vial multiplet vector of course and also the scalars okay and the main difference from this simple right known as term solution and Maxwell Einstein that I presented yesterday is really the fact that the scalars play a very crucial role in this theory okay so you'll see in a second and it's really it's it's that that's the technical difference and once we just know how to deal with that the philosophy is exactly what we did yesterday okay so before telling you what the solution is let me write down what the action is so the action in this case is very exciting unlike in some previous lectures and because of that I will write it down so here is the Lagrangian is good to write it down once it's actually quite exciting so so first let me say that let me say somewhere here in the action is the action that I'll write here at two derivative level is completely governed by what is called the holomorphic pre-potential by one function of Xi and this is a homogeneous function of degree 2 okay what that means is that it should scale if all the Xi scale if you take Xi goes to lambda Xi the function should go to lambda square times f okay that's all it means so it could be x1 x2 or it could be x1 x2 x3 over x0 anything like this okay and my notation is be here is f i is ddxi of x and similarly f ij is 2 derivatives g mu of i xj sorry xi plus i f ij f ab minus i minus plus and minus here means self-tool and anti-self-tool minus quarter x bar i t minus ab times f minus ab j minus a quarter x bar j t minus ab okay here is this function f ij they are all functions of x minus and eighth times i fi times f plus ab i minus a quarter x i t plus ab times t minus ab minus i over 8 f ij yi ij yj ij minus i over 32 f times t minus t plus square plus so these are all written in some kind of holomorphic notation they are always minuses here okay and so that's the chiral part of it and you have to add the Hermitian conjugate okay and that's the full action okay at two derivative level that's the full action well not quite you have to add the gauge fixing terms which I won't write down which you won't see written down in a book immediately because that depends on your context okay and if you want you can add higher derivative pieces okay so let's look at this action it looks essentially a generalization of Maxwell Einstein what are the differences one difference is here right in the fact that the it doesn't start with Einstein Hilbert but there is some I didn't tell you what k was e to the minus k is defined to be minus i times x i fi bar minus x i bar okay so that's a combination which will keep coming up it's very important in fact let me make one more comment here that there is a gauge freedom in the theory which is dilatation and if you remember I've written somewhere x scales with charge plus one and the wheelbind scales with charge minus one and therefore the metric which is the square of the wheelbind scales with minus two and therefore this combination g mu nu capital which is e to the minus k times g mu nu so let's go through slowly x scales with plus one f is a homogeneous function of order two so it's like x square so one derivative also scales with plus one so this is plus two so e to the minus k scales is plus two and g mu scales with minus two so this is actually gauge invariant okay and so this is the let me call this the physical metric okay and if you make this change of variables this e to the minus maybe I should write it here that just becomes square root of g that plus this of course square root of g times r of g okay so that's this is a transformation many of you might have seen it's like so essentially this e to the minus k of x acts like a dilaton if you've seen it either in string theory or in some other context okay just make this transformation and in terms of the physical metric you have just Einstein Maxwell as you show okay so yeah so maybe I've already solved for something it's possible just maybe I have cheated and solve for just for D which is algebraic maybe yeah so I said I cheated a little bit so there should be there are a few more terms here which I got rid of because for what we're going to say next it doesn't matter but you're actually right there are boring terms of the type D and you can get rid of the boring terms and write it in this clean way thank you very much it's a good point there are other terms if you want the full if you just check off shell transformations of this it doesn't quite close so there is some term here which is there's a coefficient here r and minus minus D and you have to solve it and so on now so what I'm saying is that you might as well think of it as the full off shell theory I didn't in all my calculations I really take the action that you give so I just didn't want to clutter the board so the cheating is only for presentation it's not for calculations all right yeah here no so that's the one I think I get after solving the the constraint there's some 1 6th usually and then you solve it and sorry here I'm pretty sure of this but I just check this these are all good questions unless someone knows already either Stefan or Val or one of the experts the Dutch school Flemish school no I think it's okay I think it's okay okay in any case so if there are errors please go and correct them I just wanted to show you the form of the action so that's one difference between the usual Maxwell Einstein okay the teachers in in when you begin teaching they say that you should you should always make small mistakes so that people get engaged and so on so that's not why I made it but it seems to know but here there's no mistake I think okay let's go on the other difference which I want to point out is that the kinetic terms so this the fact that the kinetic term is not canonical it's just a gauge artifact okay so you get really the canonical kinetic term for the gravity for the vectors and the scalars it's not an artifact really the kinetic term is a scalar dependent it's a field dependent kinetic term and that's the main difference in these n equal to supergravity is compared to simple systems okay okay so now I'm going to talk about the in this with this action there is a bps dionic solution okay and we're just going to go directly near horizon like according to yesterday's philosophy so I'll just go near horizon so the bps is actually one half pps so it preserves four of the eight superchargers and the near horizon geometry actually is a full bps solution in its own right it's a you can just decouple it and the metric I'm just going to remind you of this metric of yesterday so I square minus one d theta square I'll be quick now d r square plus 2 square so I'm going to use star for this near horizon system of this black hole that's going to be my notation for this it will be called the attractive black hole so this particular system of black holes that I'm studying I'll call it star so fi is minus i e i star over 4 pi d r wedge d theta plus pi over 4 pi sin psi d psi pi we've seen all this together and x i is some constant x i star and there's another field which I said is t minus r t is also v star and the other t's are determined by the anti-symmetry and anti-self duality of this okay and these are the only non-zero fields these and everything related to this by symmetry the only non-zero fields in the solution okay now we can now ask what is this classical entropy and the classical entropy you can just use this extremization prescriptions that I procedure that I described yesterday here is the action here is the non-zero horizon solution all these are constants this plug it in to a Lajano transform minimize it so this is an exercise for all the students slightly involved but still eminently doable you'll spend 20 minutes half an hour and so what did the classical entropy function did do it did two things first it told you the values of all these constants in terms of the charges and secondly told you what the entropy is okay and the answer is the following the answer is i x i so the imaginary part of x is p and the real part of x such star the real part of x equals this electric field okay so the imaginary part of x x is complex is equal to magnetic field and real part of x is the electric field so another way to write that is x i star is e i plus i p i over 2 so that still hasn't completely solved the problem because I need to now say what is e i e i should be a function of the electric fields and that's essentially the difference as I said if I have one just one Maxwell field and the canonical kinetic term e i e equal to q the electric field is equal to just charge but now because of this mixing because this field dependent netic term essentially it's a linear algebra problem we have what is called kinetic mixing right so you have to then diagonalize the thing and this is a solution for that problem and that the solution is written like this I fi minus sorry fi bar minus fi so that's the imaginary part of f i is given by q i okay so it's a system of equations so this is some function of x i you have to solve and of e i and p i have to solve this to get what e i is so that's the solution of this problem and v star equals one so you speak up please that's great so I'm just going to make a comment about it in a second yeah let me just make a comment about this and then if you have a question you can ask it so first let me just explain the yeah this this set of equations oops are also called the attractive equations okay sometimes they call the stabilization equations depending on which language you speak and I think Stefan also discussed this and essentially this goes back a long way to Ferrara Kalosh and Strominger and then many other people also Ferrara Gibbons and Kalosh yeah and there's an alter so the historically the there's an alternative route which was followed to get this equation so this here we really haven't used super symmetry we just use extremality and we just use that there's an ad s historically what was done was just use full bps solutions and and just that turns out to completely constrain the problem so you get first order equations and then you get this as a consequence so the reason is called an attractor is as I said if you think of the full geometry this is a row star which is the horizon and think of Xi so this is some Xi star if you look at the full geometry this scalar fields can take any value at infinity and if you follow the bps equation the bps equations give you first order equations which are like flow equations and those flows essentially just always bring you at least when you're within a certain range to the same value x star okay that's why it's called attractor it attracts it's an attractive point it's just a first order system okay now about Jean's question so note that here these equations are not gauge invariant right x has a charge as I've been saying so the way I wrote it it's not gauge invariant and that was common then indeed I have I have chosen a certain gauge in which I write these equations and there's a way to write this in a gauge invariant way which I'm not going to do but it's it's fairly standard and the gauge I've chosen can be just thought of to be the fact that V star equal to 1 remember V star was essentially so square root of oops if you look at the metric the square root of G is V star square and if I because the metric itself is charged under the notation like we had this discussion here I could just use that to to gauge fix and that's what I've done and it's in those coordinates okay so it might if you've never seen this before it looks it might look a little funny that it looks like the area of the horizon is one but that's a gauge artifact because the real area of the horizon in the physical metric is that time some function of the scalars like we just discussed okay so it's in this capital G and indeed you can ask what is the what is the entropy the classical entropy continuing this this program is minus pi q i e i star plus 4 pi times the imaginary part of f of x i star okay so that's part of the exercise so that's exercises to show this and that okay and again just to connect to other treatments sometimes you might see at two derivative level which which is what we have been talking about so far you can this you can show that this expression is also equal to pi times e to the minus k and that's in some of the original papers that's how the entropy was first presented I think it's a nice paper by I'll have me Gibbons and kalosh and two or three is some subset of Gibbons kalosh Ferrara stronger okay okay so so that's the end of one topic let me pause for questions I'm not making any claim about that but I think the answer is yes if you give the black hole charges such that the area is positive then there is a bps black hole solution right so I don't need to the charges actually can be fixed at infinity so the answer is yes but I'm not using using that in this in this system I have ungain supergravity so in this in all the as I said in the beginning I even wrote it down today and tomorrow and the day after I'll only work with ungain supergravity given some charges of the black hole this you can measure just at infinity just by one over our square fall of rates I did because so that's what I'm saying so sorry so what I said what I didn't write but I said was that there is a half bps dionic black hole this is the solution that Stefan wrote and I wanted to avoid repeating yesterday's discussion all over again so I directly went to near-horizon okay because all my entropy business is anyway we saw yesterday is sort of only defined near the horizon okay other questions very good so you see two things there are two remarks to be made one is that you see that the classical entropy has a very nice form for these black holes so the action the whole action was at two derivative level is completely governed by this one function f it's called a pre-potential it's a very important function that completely determines the theory at two derivative level that's the thing that tells you how the kinetic mixing takes place and the entropy should also be only a function of that and the charges and it's a very beautiful function of that it's just a Legendre transform of the imaginary part of the pre-potential evaluated at these attraction another small comment is values later if you work this out if we so everything of course is a function of the charges so if you scale qi pi to lambda times qi pi you can see that s black hole is classical entropy scales as lambda square okay and the reason is related to the so of course now this is this is not a gauge invariance anymore I'm just saying if you really change the charges the scales but of course technically it's related to the fact this as that this has way to under scale okay and that's going to be important later very good so any other questions okay very good so let's move on to the quantum entropy now okay so what do you want to do we want to now you want to fix fall off conditions sorry I mean fix boundary conditions fix boundary conditions these are v star e i star x i star etc t star etc okay and then there are these fall off conditions that I gave you yesterday that we saw in poor handwriting over there at the beginning and then you want to integrate over these fluctuations so you want z radius 2 is the exponential of minus i qi integral e i so ad is to here means precisely that there are fall off conditions and finite means that you have to take the action there will be divergences and you have to renormalize it in the sense that I talked about yesterday before I actually so the goal in the rest of today's talk and and the next two lectures is to actually calculate this function functional integral for the system that I just showed you so it's a very concrete context and everything is so there's something there's a black hole in n equal to supergravity with some charges and you want this function and this as we saw yesterday is only a function of the charges okay and I want to actually compute this function okay that's my goal and before I start doing that I want to make a few comments but let me also again pause for questions if there's any confusion about what I'm doing so yesterday I had some question about near horizon versus maybe I'll make the comments and become clearer okay so one comment the first comment is that if you take maybe I should repeat the philosophy once more because there was confusion so what we've done is yesterday we said that the classical entropy can be understood purely near horizon that we proved and then there's a proposal that the quantum entropy can also be understood in the near horizon you can take that as a definition of the quantum entropy and then in string theory you'll have to match it with some counting and you have to make sure that that counting agrees with that near horizon entropy okay so that's the philosophy so one comment one technical comment is so everything reduces to the calculation is some path integral on ad s2 so if you think of Euclidean ad s2 there's always this cutoff so that's r and just take a free Maxwell field on this ad s2 you'll find that so just Maxwell field and solve the equations of motion near the boundary you'll find that there are two solutions it's a quadratic thing and you'll find that two solutions go like this so a is let's say a theta is a times r plus a1 times r plus a2 so there are two modes which solve the equations of motion one linearly diverges with r the other one is a constant okay now this is the gauge field and therefore this this component is the first derivative with respect to r and therefore this component is the field strength e okay or equivalently the charges okay those are just related by some algebraic equation this component is the potential now typically if you think of ad s spaces in in d dimensions for d greater than two what happens is that lots of have two two modes but the potential modes always dominates at infinity over the the charge mode this is in flat space it's almost trivial just take a charge it goes to one over r square potential is constant therefore potential wins at infinity but this is also true at in ad s space for d greater than two well d greater than three strictly for d equal to three there is some there are two possible choices of boundary conditions which I won't discuss but in d equal to two you see that gets flipped so the charge mode actually dominates over the potential mode okay and therefore if I want to do path integrals functional integrals on ad s too it's that that I have to keep fixed okay so that means that I am oops that I am in the micro canonical ensemble okay that's very important and that's the fact that this path integral is only depends is only dependent on the charges it's a function only of the charges yes oh there no so just a second so let me first make a comment and then I'll ask the expert to to answer so I don't think in ad s you can make a choice in fact it's so okay so the common sense choice of the canonical choice is always that the one that dominates you keep fixed right because okay now you can ask what happens when that's not the case that's not being studied very carefully there will be the some papers by I think Don Marov and maybe maybe other people here I don't know and that's a very strange quantization you're asking the sub leading term is fixed and the leading term fluctuates so this is in general is it's not a good theory maybe there are some situations where you can do this it doesn't make sense but maybe sometimes you can do it but okay but what I wanted to say something else no not at all the standard one is that the dominant one is fixed no but it's not in the in the window of double that's some very subtle thing you're talking about so sorry yeah that's that's that's that's one case yeah that's right that's right that's right in the thing yeah yeah yeah yeah but generically in see if I just say if I have if I give you some ads d say one two three four five six whatever right the easy one to do the canonical one to do is keep the dominant one more fixed that's the one which has that's that's that's what my answer was but there might be cases where where both are allowed but okay I'll be very curious to know how to do the quantization in ads two with the other choice because that's interesting for a certain reason let's talk about this after the after the thing okay but generically you always keep only the dominant mode in ads cft so that there might be situations where you have a certain field of a certain charge and a certain mass such that you land in a window where you can do both and maybe in low dimensions that's that becomes that window becomes bigger but as far as generally ads cft or ads path integrals are concerned the most straightforward thing to do is just keep the dominant one fixed this and and integrate the subleading okay no but you mean the the finite piece coming from the boundary yeah so that's the boundary term right the renormalization is an addition of a boundary term right the bulk Lagrangian is given to you I wrote it down there so it is the final result when defined only up to no no so let me answer that so first let me clarify the question the bulk Lagrangian is given to you by the theory okay that has some divergences and you want to renormalize it renormalization means that you add a boundary Lagrangian to to cancel the divergent pieces okay so yesterday made this comment but I'll repeat it in theories where you don't have non-canon where the kinetic terms are all canonical or derivatively coupled you can easily prove in this case and I sort of went through it yesterday that there is no ambiguity it's a very let me just tell you again take a gauge field it goes like that and there are maybe one over r corrections okay so if you look at the gauge invariant variable which is f mu nu so this constant that's zero and that's one over r square one over r square integrated over the boundaries one over r so that dies constant integrated on boundary is linearly divergence and that is just killed by a cosmological constant there is no constant term right and that's true for all gauge invariant variables in the theory or local gauge invariant variables okay the type of variable is not true for is just a itself so Wilson line itself does contribute a constant term and if I have something which is just x it will and I'm going to I want to discuss this today but maybe a little bit tomorrow but otherwise it's yeah and in our case what makes it unambiguous is supersymmetry supersymmetry okay so let me just continue on my comments and we can take up the discussion later so it's micro canonical ensemble and in the so in the definition so this is implemented by the insertion of a Wilson line right what I mean is that if you just do the classical let's do classical dynamics in ideas to suppose you have a Maxwell field f mu nu f mu you do a variational problem you'll get a mu box a mu plus a boundary term right now that boundary term usually becomes zero because you freeze the you keep the potential to be the boundary term looks like something like delta of a d a or something like that so if you say that that zero then this boundary term is zero but here it's not you want to keep this fluctuating so and this is some q so this is in our case this just becomes delta of a times q and so you want to cancel that and that's exactly what the Wilson line does for me okay so the Wilson line insertion just implements this this ensemble for you in the gravitation here all right just ask me if this was not clear later it's a very simple thing I'm saying okay now the next comment is you see you have some functional integral on ideas to it's natural to think of some CFT living on this boundary all this are not and the partition function of the CFT is this trace of e to the minus beta times the Hamiltonian of the CFT where beta is just 2 pi r not that's the the length and you see that so suppose h is let's take a CFT where h is positive definite states which have non-zero energy so this will become there's some ground state which starts like that okay times you'll have a trace of one okay so this comes from states which have zero energy and plus there are subleading contributions which don't contribute when r not goes to infinity okay so that's just 2 pi r not is the argument clear okay so it's minus e to the minus 2 pi r not e times trace of one now if I do a renormal if I do a change of the zero point energy of the boundary theory I can just make this to be zero that shift is exactly the same as a as the fact that I killed the linear linearly divergent piece in the boundary theory okay so that's so that's one thing so after renormalization so suppose I just say that e not equal to zero it means that z CFT one is just trace of one over states with h equal to zero and that's just just the number of states okay and this is what is called the d micro whatever the theory is I'm saying that so according to ADS CFT logic if there is some ADS 2 functional integral quantum gravity function integral it should be due to some CFT one I'm saying that's CFT one you just count the number what happened the partition function of that CFT one it's just the number of states of that CFT yeah so you're given some so you you you're saying I need to specify what this is yes indeed in in some string theory this is possible so I won't discuss this so Joao might discuss some part of this so yeah so typically we'll have so what I'm saying is that the e to the so z ADS 2 is equal to z CFT one so that's supposed to be ADS CFT is the same as e to the s s black hole quantum equals d micro okay so this what Baconstein Hawking had said at at leading order in El Plunk gets promoted at least in the supersymmetric black hole context to an exact statement it's just that the quantum entropy equals d micro that's what that was one of our goals we wanted something like this and that statement is just exactly the statement of ADS to one one consequence of ADS CFT okay can I just finish because otherwise and then I'll take both questions just just one second I have one more comment and then so the third last comment for today is that sorry there are two quick comments one is that if the maybe I'll postpone maybe I'll postpone this comment and instead make the fourth comment which now becomes three so what is S bulk okay that's another clarification I want to give I gave you a Lagrangian here so S bulk is just the integral of the Lagrangian but what I really want to do is full quantum gravity right so what I showed you this action this exciting action that I showed you here was a two derivative action okay but as we already saw yesterday in general there could be higher derivative terms in the action local higher derivative terms and I should keep all of them if I really want to do the problem well I should keep all of them okay so S bulk contains and you know all possible terms so including higher derivative and then you might think so you might think then this is some infinite action and so it's not even well defined as far as physics is concerned and in some sense that's that's true and that's the problem of the the fact that gravity is not UV finite it turns out that for this particular problem of quantum entropy because you have super symmetry we're going to use the method of localization and although our priority looks ill-defined because the original functional integrates not well defined because of this we're just going to follow our nose and just use the same rules of localization and you get some very nice well-defined answer and then you can turn it around and take that as a definition of the quantum entropy okay so that's the but I want to stress this that what I told you today is just two derivative tomorrow I'll do higher derivative but that's where we are heading we want to keep all possible terms and then it's a technical problem how to deal with it okay so I'll stop here thank you so so there were two questions