 OK, so I'm going to do a bit of a recap, because we went quite fast last time. But pretty much we've done all the, at least very quickly, we went over an important calculation that will, if you understand this calculation, help us to state, finally, how the ADS-CFT correspondence works, and actually makes it rather, the way it's phrased, rather natural. So to recap, what we did was we calculated the dynamics of a scalar field in ADS in the Poincaré chart. And the natural dynamics, if you remember, the way we'd like to think about it is we start with some boundary condition on this characteristic surface here, this Cauchy, or null initial data surface, the past Cauchy horizon, or rather, Poincaré horizon. I guess I'm not quite sure I should call it a Cauchy horizon. It's null, but anyway, it's initial characteristic data surface. And then we have the conformal boundary, which isn't a real boundary, remember? But I'll just refer to it as a boundary with that understanding. And we'd like to impose something like a Dirichlet boundary condition to say what our field is doing on this boundary. And then there'll be some dynamics. And because we've chosen things to be trivial down here, the dynamics will be induced by what we make the field do later times on the boundary. And then as we discussed, we looked at the scalar field equation. And rather schematically, last time I showed you, I argued that it will have a form like this, an asymptotic form like this near the conformal boundary, where there are essentially, in terms of this, what I might call a radial coordinate, z, which is some normal to the boundary in these coordinates, there are two power series, one controlled by this power d minus delta, one controlled by delta. This is, if you recall, I've restricted myself for simplicity to think about delta greater than d over 2. In fact, that's not necessary. One can consider a slightly wider range of delta, but for simplicity, I've also restricted to cases where I don't get log terms. You can handle those as well, but it just makes everything more complicated. And then the two bits of data, if you like Dirichlet and Neumann data that this field have, are really this function. This is like the Dirichlet data. This is the leading fall off with this. And this is the sub leading fall off. So this function and this function, this function determines these terms up to the power where this kicks in, then you have this undetermined data, and then the subsequent terms, which depend on both this and this. So this probably is rather familiar in different contexts to all of you. And delta is the larger root of this quadratic, which is determined by the mass of the field. OK, so then we imposed this vacuum condition that we don't have any radiation coming in through this parsnall surface. And that relates this to this in the way I described last time. So this can be written as an integral of this against some kernel. Now then we took that solution and we tried to plug it into the classical action for reasons that will become apparent later. So here's our classical action. And if we really try and integrate this over this whole domain here and put in our solution, we found, well, when you integrate this by parts, you get an equation of motion and then you get a surface term. And the surface term on a solution, therefore, determines the action. And that surface term diverges just essentially because there is a finite number of terms here until you reach this order. And these naughty finite number of terms have a divergent behavior towards the boundary, which means this action is just infinite. Unless you restrict yourself to trivial data, phi 0 is 0, in which case these are all 0, this is the leading behavior. And then things would be finite, except that if you haven't got anything coming in, your whole solution is then 0. So it's not terribly interesting. However, the key observation is this divergence is rather trivial in the sense that all of these, the nasty finite set of terms that cause the divergences are all determined by this. And this is your data anyway, so you know it. There's nothing interesting or dynamical about it. You just fix it at the start. And so there is a procedure to remove these divergences from the action to get what we call the renormalized on-shell action. On-shell physics speak for evaluated on a solution. How do we do that? Well, there's nicer ways to think about this. But pragmatically, one way to think about it is we don't take all of the ADS, we cut it off at a finite value of z, let's call it epsilon. So now we're going to think of our action just integrated over this region, which I call m epsilon. And that has a boundary here. It's also got these boundaries, but we argued that actually they don't contribute in the action, if you recall. So the contribution is only going to come from the surface term evaluated on this boundary. And what we then do is we construct another boundary term, which we call counter terms in physics speak, but just some other action. And this is constructed from purely intrinsic, well, purely intrinsic as an intrinsic functional of the scalar field and induced metric on this surface. Nothing extrinsic. The reason being that, of course, once this epsilon boundary is taken to zero, because of this expansion, all of these divergent bits, the extrinsic, any normal derivatives in the z direction, are just related already to this data. So you don't need anything extrinsic. So we've just got an intrinsic functional of phi and the metric induced on this surface here. So this, in particular, if we need this term, this derivative is the induced metric on that surface, is the connection of the induced metric on that surface. The number of terms here, remember, when we plug this into the surface term you get when you integrate by parts, there's only a finite number of divergences. This will correspond, you know, something of this form will correspond to the leading one. Something with two derivatives in will be the next subleading one and so on. And so you only include as many terms as you need, as many terms as there were divergences. In physics speak, this is the number of counter terms is determined through power counting. It's an analogous statement. And then what we do is we add this to this and we evaluate it. And we think of it as a functional of what we want to fix. What we really want to fix is this phi zero, but we don't want to say it in terms of this expansion. We want to say it in terms of the field itself. So we do that just by saying we'll evaluate this with this epsilon cutoff taking the field at position epsilon x and epsilon to be J times the appropriate power of epsilon so that when epsilon goes to zero, J will just be phi zero. So think in the limit of taking epsilon to zero, we're fixing phi zero, and then we take this limit. And the bottom line is what that basically does, and it's slightly more subtle than this, but not really, is it picks out, it kills off when we substituted this into here, it kills off the divergences and just gives you the finite part. This is called holographic renormalization. But it's not rocket science, obviously. It's not as hard as it's, it's really fairly straightforward. And if you know geometry and you know hyperbolic spaces when you think about things like the renormalized volume of a hyperbolic space, this is exactly the same thing that you're doing. And when one does that, one finds that the renormalized action on shell action then takes this form, it's determined by this data, your Dirichlet, I think this should be a capital J, sorry. It's determined by your Dirichlet band recognition, think of that as phi naught, but I'll call it J. Okay. And this, the second bit of data here, which is determined by solving the equation. And as we said last time, could be written in terms of an integral of J against some kernel and explicitly took this form. So I haven't given you any references for further reading at this point. So before I go on, let me just give you some references as I wrote them up. It's a very nice short review of ADS CFT, I think written for physicists, but from the horse's mouth, and I don't mean to say that Juan Maldesena is a horse in any way, but I just mean he's the person who started all of this in the first place. So it's I think nice to see his way of presenting it, but it's a very digestible short review. There's a very detailed and standard reference on this issue of holographic renormalization, presented slightly differently to what I've, the way I've presented it here, slightly less geometrically in terms of these aren't necessarily thought of as living on the boundary of this surface, but it's equivalent. And then there's a quite up-to-date review on ADS CFT here, which you may find interesting. And there's also, well, I might mention this one later. This is something else. Okay, but there's three sort of reviews on ADS CFT. Now let me state what the conjecture is. That was the whole point of all of this. What we've seen is that ADS acts like a box. It needs some boundary condition. What is that boundary condition? In this calculation, it's this J. We specify J. Think of it as specifying Dirichlet data for your field at a real boundary, but it's a conformal boundary and so on, but roughly it's the same. Once you specified that, there's some dynamics. You can compute some action. But also remember, when we talk about what we want to do is say some gravitational theory is equivalent to some conformal field theory. How do we think of conformal field theories or any quantum field theories? We think of them, or one way to think them is through the path integral. And what do we need to specify in the path integral to sort of determine the behavior and dynamics of a theory? The path integral has a load of fields in it, and there are things we might want to compute. And the way we get, for example, at correlation functions, one of the most basic objects, or if you like, the basic object in a quantum field theory, one way to do it through the path integral is to add sources for all of the operators that we might want to calculate correlation functions for. And then as I was explaining in the first lectures, you can functionally differentiate this generating functional to get any correlation function you like. And the key observation is that in ADSEFT, these sources from the point of view of the quantum field theory are just related to the boundary conditions J over there for fields. Every field in the gravity corresponds to some operator in the conformal field or in the field theory. And the boundary condition for that field in the gravity corresponds to a source for the operator in the field theory. So it's a rather sort of simple construction, in fact. So the CFT lives, or is defined, on the conformal boundary of this asymptotic ADS space. And as we've talked about it, that will just be Minkowski at the moment, but we'll say something a bit more general later. Let's assume it's Minkowski. So the CFT is living on Minkowski, but we should think of that probably naturally as the boundary of this space, the conformal boundary of this space. And roughly speaking, by bulk I mean on the gravity side, boundary is CFT, bulk is the interior, the gravity. Every bulk field, so that might be a scalar, it might be the metric. If you've got other fields, whatever you have in your theory, has a dual. We call it operator. Let me call it an operator or field in the CFT with some scaling dimension delta. So for example, if we talk about a scalar field, as we just have been doing, how does that work? If we had gravity in a scalar field and we thought it was some CFT dual, so a massive scalar with a mass M, or maybe I should say mass squared, M squared, is then dual. So this is in the gravity. Over here is the gravity. Over here is the CFT. It's dual to some scalar operator. By scalar operator, I mean it's something in the field theory, which from a Lorentz point of view transforms as a scalar. So actually it's some scalar field in the, or it's some scalar, it may be, it itself is some sort of scalar field, but it may in fact not be, well, anyway, let me not, sorry, I'm digressing. And this has some dimension. Let's call it O. This has some dimension. For example, we've seen examples of these scalar operators when we talked about free fields, the field itself in a CFT is a scalar operator. But it's also true, for example, that Lagrangian is a scalar operator. The Lagrangian of a field theory itself is a scalar. It's like a scalar field. It's not fundamental. It's composed of its composite of other things, but nonetheless it's a scalar. It's got some definite dimension and so on. So, and in particular, the relation between delta and m is exactly as we've seen. That is the relation. Another example is that the metric is due to the stress tensor, which is, again, some operator. The metric is obviously some sort of field in the gravity. It's a dynamical field in the gravity. It should therefore be due to something, as we've said. What is it due to? It's the stress tensor. Note, that's very deep, because all field theories, at least of the sort of normal variety that I want to talk about, have a stress tensor. It just tells you about energy in the theory. So if you don't have a stress tensor, it's a very peculiar theory. So the fact that every CFT that we would want to consider has a stress tensor tells you that the theory that's due to it is always going to have a dynamical metric, may not, as we've discussed, it may not look like an Einstein gravity theory. It may be something more complicated, maybe highly quantum, but it will have a dynamical metric. So as far as I understand it, personally at least, you won't write down ADS CFT where the ADS theory is non-gravitational. It doesn't have a dynamical spacetime. That's sort of a basic. That seems to be a basic built-in feature, which makes it interesting, but also tricky, because of course, precisely what we don't understand well, quantum mechanic clears, there is with dynamical spacetime. OK, so these are some examples. And then how does it work? As we've discussed, let's think of our theory, or let me go and draw it here. We've got this gravity theory here. We've got our CFT, which we should think of as living on the boundary. And the gravity theory needs some data on this boundary. So the gravity theory needs a boundary condition for its field. And the CFT, this field over here, let's think of a scalar, phi. There's some dual operator, let's call it O in the CFT, and that needs a source, J. And this boundary condition for phi and the source, J, are just the same thing. So let me call this boundary condition J. And you have to do this matching for every single field in your theory and principle if you're interested in writing down the complete correspondence. And then the statement is that the partition function of the conformal field theory, which is a functional of all of these sources that you've added for all of the operators that you want to think about in your theory, is equal to the partition function of this quantum gravitational theory where, in this quantum gravitational theory, you have to impose boundary conditions for all of your fields on the boundary. And this is a source for an operator. These are, well, that there are many of them, but let me just call them J generically. These are the sources. And these are the boundary conditions for the corresponding field to the operator that that's a source for. That's the ADSEFT correspondence. By the way, you may also say, in some sense, this is actually simpler to think about. A little bit simpler to think about in this global chart. You may think, oh, hang on, but what about this? Isn't there some data here as well? Well, there is some data there. There's some in a Lorentzian setting. You also have to say, what's the initial state? If I'm interested in some dynamics, what's the initial state of my theory? And both here and here, you would have to say, what's the initial state of my theory? In the CFT, you would put the theory in some state and then think about time evolution. And in the gravity, you would again have to do the same thing. So the data you put here is just associated to choosing the initial state here. But for example, if we put this in vacuum here, so this would just be ADSEFT until you deform it with the source, with the boundary condition J to get some interesting dynamics, that would be like putting this in its vacuum state. So it's trivial until you turn on. It's trivial for all times until you make the source do something interesting. That's the ADSEFT correspondence. There really isn't, I mean, note, by the way, as I am, note, by the way, there was this important point that phi d, if you recall, under this transformation, remember the ADSEFT is part of the max is a scale transformation from the perspective of the boundary, the conformal boundary. And phi d, in that sense, transformed as a field would in a CFT with scale dimension delta. And in particular, this phi naught scaled as a corresponding source would scale. So J does indeed have the J over here, does indeed have the correct scale dimension, or properties under scaling, to be a source for an operator with dimension delta in the scalar case, and more generally. Now, where does it look like there's any gravity in there, classical gravity, the classical gravity comes in because in an appropriate limit, this gravity theory, whatever it is, reduces to something that looks like semi-classical gravity. Let's say semi-classical Einstein-like gravity, if we were considering just this universal sector I discussed before, it would just be a d plus 1 dimensional gravity plus a cosmological constant and nothing else. We could truncate it to that. But maybe there's some other fields like a scalar, or maybe you're looking at some 10 dimensional gravity theory, which is rather complicated. But as we discussed, when you go to a large number of degrees of freedom in these theories and strong coupling, you may get this semi-classical behavior. And the point about semi-classical behavior is that this path integral, whatever it is, and we can't write it down for a quantum gravity theory. That's the whole point. If we could write it down, we'd be done. And this would be entirely less interesting, I think. Sorry, what am I saying? This is the gravity theory. It's a functional of j by which I really mean these are boundary conditions, let me just say, of j. So again, even at the quantum level, if I'm looking at some quantum gravity theory which has some asymptotics, I'll still need to impose these boundary conditions. But in a semi-classical limit, whatever this horrible object is, we believe it will reduce, in just some very general grounds, to being dominated by the classical saddle points and small fluctuations about it. That's what semi-classical physics means. So whilst we don't know what this is, what we do know is that when this happens, this should just be given by this on a solution, satisfying the boundary conditions given j. So I don't know if people over here can see, but what we're saying is this is just e to the i times the action evaluated, the classical actually evaluated on the classical solution. And you can go further and calculate corrections to this. So this is just a saddle point approximation to some integral. And you could then go and correct it. But the important point of the leading behavior is just governed by the action of the classical solution with the appropriate boundary conditions. And note, really, we should have been careful to add these counter terms. So all of this had to be done carefully because otherwise we get divergent actions, as we've just discussed in the scalar case. So this action here should be understood as this renormalized on-shell action that I discussed. Is that surprising? No, it isn't surprising, because if we look at a conformal field theory, it's defined by this path integral over its fields. It's a functional of sources, so we've added in sources for fields. But it is also divergent, horribly divergent, and we need to renormalize it. We know quantum field theory is horribly divergent, and there's a systematic way to renormalize theories, if they can be renormalized. And so this procedure of renormalizing the CFT is thought to be exactly analogous to this procedure of renormalizing the gravitational action, hence the name renormalization. And so this is then very profound, because if you understand, even in this classical limit, if you understand this classical action, renormalized on-classical solutions, you know this, but this is just equal to this. So calculating this for some sources, some value of source, is the same as calculating classical solutions, plugging them into the action, with the appropriate boundary condition given by that source. And remember from this object, by functionally differentiating it, or by varying the sources, and then setting them to 0, you can calculate all the correlation functions in vacuum that you want. So to calculate all the correlation functions in your CFT that you want to do, all you have to do is know this object, which is just classical, classical GR. So classical GR, understanding this modulized space of solutions and how to move around in it in the different directions, is tantamount to understanding all the interesting physical data in this strongly coupled CFT, which is rather remarkable. So it's much simpler than maybe, I mean it's actually not very complicated I think. I don't know, maybe we should have a show of hands. He thinks this is horrifically complicated. OK, well that's great. I won't ask the converse in case no one puts up their hand. I think it's really rather elegant. And a lot of it's down to the fact that there's this boundary. And the boundary of ADS, the conformal boundary, has the geometry that the field theory lives on. So this identification is very natural. But it's a full quantum equivalence that's the claim. So it goes in principle beyond this classical limit. It's just we don't, you know, only in very limited circumstances can we make sense of this. Usually we're reduced to just thinking about this. But there are cases where you can say something about the quantum theory. Let's just have an example. Supposing we have a scalar operator in the CFT. Let's call it O delta. It's got dimension delta. Let's calculate its vacuum correlation function. Well, how do we do that? From the rules I gave you before, you add the source in for this. You functionally vary twice with respect to that source. And then you set the source to 0. And that generates for you this object here. But we've just seen that this is just given by e to the i renormalized action with the appropriate boundary condition. So I can now replace, well, this using that. So in this, if you're in the semi-classical regime, zcft of j is just proportional to this renormalized on-shell action, which is a functional of j. And we calculated that the renormalized on-shell action as a function of j was this integral of two sources, well, a source against a source multiplied by an appropriate kernel. So now if we do this, if we put this into here, differentiate, I mean, it's a very simple exercise to see that you will recover. So when you just put that in, I mean, it's a trivial calculation. Let's say some constant, which I've times this. So this factor here, you're just basically when you functionally differentiate twice because of the exponential. And because you then finally put the source to 0, you basically just bring down this factor of 1 over x minus y to the 2 delta. And one of the things I said at the very beginning when we were talking about CFTs is that scale symmetry constrains correlation functions of operators like this to have precisely this form. And you see it very explicitly now. This is something we can calculate. It's important to understand you could go away and calculate any correlation function you like once you know the solution. Once you know this, and we know it for this scalar field now, but you could calculate it for whatever fields you wanted, maybe with nonlinear interactions and so on and so forth, you could then function differentiated many times you like to calculate arbitrary correlation functions. It may be rather technically tricky, but in principle, one can do it. It's just one other thing to note, something nice to note is that if we think about the variation of this action, this renormalize on shell action, it's given by this. One of the sources now gets varied. And so in particular, I can think of this as this variation, the integral of the variation of a source against this subleading bit of data in our expansion that we had before. And what that means is the one point function, if we don't set a source to 0, but remember, if we set sources to 0, as we said before, what you generate is vacuum correlation functions. But we can also think of these sources as actually external fields that deform our theory, maybe like an analogy turning onto a magnetic field that deforms the physics of some theory with charges. So we don't have to set j to 0. But then, of course, we have to understand we're deforming the theory away from its vacuum state. We're doing something to putting it in some other state. So if we don't set j to 0, and if we were to set j to 0 and calculate the one point function, because this action is quadratic in j, you'll just find 0, which as I told you, one point functions in vacuum of a conformal field theory have to vanish. But if we don't set it to 0, and we think we've deformed the theory by whatever this operator is, we will find, actually, this is just so you function in order to compute this, you functionally differentiate once through the respect to j and then don't set j to 0. That's the one point function of this operator, but in the theory deformed by the source. And it's just given because of this by phi d. And therefore, when we had this expansion near the boundary for our scalar, there was this leading part, which we thought of as Dirichlet data, and then there was this sub-leading part, which we is determined by the dynamics and the boundary conditions. But the nice thing is this is the source. You can think of this as a source, and this has the interpretation of the response to that source in the sense that it would be the one point. It gives you this one point function. This is what determines the one point function of whatever the dual operator is to this field, given that you turned on this source. So actually, this data itself is rather physical. It's not something abstract. It's rather physical. Just a comment. If we put, when you do things to the theory on one side, you land up doing them to the other. This is just an identification of their partition function. For example, if we put the theory of finite temperature over in the CFT, we have to put this at the same finite temperature. So in that sense, the physics of black holes, which have Hawking temperature, finite temperature, is what lands up describing the physics of hot, conformal field theories, which is rather profound. I'll say more about that in a minute. What I just wanted to do now was tell you because, obviously, there are some, well, with geometry in mind, I wanted to say something about the metric. So the metric, we said the metric is due to the stress tensor. The metric's rather special, of course. Now, we can choose, let's have a think about how what we did for the scalar field would translate for the metric. In some sense, it's almost a little bit easier, I think. Well, in some sense, it's more complicated, so you cannot say that. But we can choose coordinates near the conformal boundary, which are almost, I mean, they're not normal coordinates, but they're sort of an appropriate modification. So we've got a z-coordinate and these x-coordinates. And what we've done is we've fixed the z-z component of the metric to be just one of z. And we've chosen there to be no off diagonal terms. And then, at least for, this gets slightly modified if you've got matter fields and you do complicated things with the matter fields. But if we've just got gravity and a cosmological constant, we get, again, a power series expansion for our field, now the metric, when we solve the Einstein equation. And the power series expansion looks like this. There's a leading term. This is the analog of this phi 0. This is the Dirichlet data. And then there's some sub-leading terms which are just determined by this. How? Well, in fact, they're just local curvature. They're determined as local expressions that are local in terms of curvatures of this. Derivatives and curvatures of this. And then we have, at some point in the expansion, this actually goes up in powers of z squared. And then at some point, we get z to the d times some new bit of undetermined data. And in fact, this has a couple of constraints on it. It's traceless. Well, roughly speaking, its trace is constrained, let me say. And it's also, its divergence is constrained. And then there's all the rest of it, which is determined by both this and this. One can, again, write down the classical action in order to think about this duality. Remember, we need to understand the classical action. So what is the classical action for gravity? We, again, define this little cutoff in z, because otherwise we're going to run into divergences exactly as we did for the scalar. And up to signs and coefficients, the Einstein-Hilbert action with cosmological constant looks like this. However, we've got to be slightly careful. And this was a subtlety that I didn't mention in the scalar case. You might have said, in the scalar case, we started with the usual scalar action. And then you integrate by parts. And this just becomes a boundary term. So it just becomes something like this. But in order to have the scalar equation, I mean, I could add any boundary terms I like to this to start off with. So I could have added or rather subtracted this from this. I could have started with this minus that. And then just got zero. And then I would have got a completely different answer. So why didn't I do that? And the reason is, in the end, as I was saying, we need to think about evaluating this scalar field with this cut-off on a domain with a boundary. And we should have a well-posed variational problem from the action. So the action doesn't just give you the equations of motion. It also, if you wanted to be stationary to linear perturbations, you also need appropriate boundary terms if you've got boundaries so that the action isn't just stationary in the interior of the problem. It's also stationary on the boundary. And in fact, this action is the appropriate one. If you think about varying it, and you then have a boundary and you want its variation to vanish to linear order, including at the boundary, you'll find that it does. But if you added any amount of this, it wouldn't anymore. And for gravity, it turns out, when you have gravity with the boundary and you want a well-posed variational problem, you have to add what's called the Gibbons Hawking term, which is an integral of what I would call the trace of the extrinsic curvature with an appropriate coefficient on the boundary. And what I guess other people would call the trace of the second, well, is it the second fundamental form? The trace of the second? It's something with fundamental forms. OK, and you would probably call it H rather than K. Anyway, being absolutely pragmatic and it's sort of emotionally nice to do this, what this does when you vary this, or in fact, even if you just, sorry, without even varying, when you evaluate this action actually on a general metric in some coordinate basis, you get second derivatives. And you know that when you vary an action, you don't really want second derivatives. You don't want to start off with an action with second derivatives. And in fact, what this term does is it exactly gets rid of second derivatives in this action, at least in the directions normal to the boundary. In a very concrete level, that's why it makes this the right variational problem when you have a boundary. And you want to fix the metric on the boundary, as we want to do here. However, of course, what we actually also have to do is add some counter terms, appropriate counter terms. And what do they look like? Well, the first one is just a constant on the boundary. You think about that as a cosmological constant on the induced geometry of the boundary, attention, if you like, if you want to think of it sort of very physically. Then depending on what dimension you're in, you'll have different numbers of divergences. So the divergences will be associated to these terms. And this, once you've plugged it into the action, will give you the finite part and everything else after. And so the number of terms you have there, this is going up in powers of z, until you get to d, the number of terms you have depends on the dimension you're in. So in four bulk dimensions, you would have two divergences, and you would need just two terms here. The next term will be involving the riche scalar of the induced metric on the boundary, called induced. And then you would have higher curvatures if you needed them, the right number to subtract the divergences. So this would then be the appropriate action. You could then go through exactly the same procedure as I did for the scalar. Yes, yes, yes, exactly. Sorry, yeah, OK, good, yes, they are all fixed. There's one subtlety. There is one subtlety which I've tried to avoid because it's just unnecessarily complicated in the setting. But please, if you want, look at this for all of the details. In fact, in odd bulk dimensions, there is also a log term, one log term here, whose coefficient is determined by this data as well. And that also has to be removed. And there's a well-defined way to remove that as well, but there's actually an ambiguity left when that happens. And this is associated to something called a vial anomaly, but it's not terribly relevant. Now, once we have all of that, we can compute. We have our action. We can take this limit of epsilon goes to 0 of this plus this. We're fixing, as we did for the scalar, we're fixing, in this case, we're just basically fixing h to be whatever we want it to be to some metric on this surface epsilon. And as we take epsilon to 0, we'll be fixing basically h naught to be whatever we want it to be. So we're fixing the conformal class of the boundary. So h naught then fixes, we fix it, that's our Dirichlet data, fixes the conformal class, well, let me say the boundary metric, but it's really the conformal class of the conform boundary. And we should think of this as the metric that the field theory lives on. So this, by choosing different h naughts, we get to put our CFT on different spaces. And the stress tensor, the vev of the dual operator, the vacuum expectation value of the dual operator, well, just this for the scalar, it turns out basically to be determined by this, this data here. So concretely, putting in some numbers now, it's actually determined in this way. But there's basically some proportionality. And this, by the way, looks very much like the Brown York stress tensor. So for the GR people, there is something called a Brown York stress tensor when you have a boundary of a space in GR, and you sort of think about it, it comes about from a very, in a very similar way. The only difference here is it's basically the Brown York stress tensor on this surface epsilon. But when you take epsilon to zero, the Brown York stress tensor diverges, and therefore you need these extra counter terms to renormalize it. So you can think of this as the renormalized Brown York stress tensor. So it's a very natural stress tensor. And this is the stress tensor of the field theory. So as your matter does stuff in your spacetime, because you turn on sources for it, it does whatever it does, the metric will respond to the mass and energy in your spacetime, this will respond, and this will measure it just as in a field theory, if you start pumping in energy by turning on sources and so on, the stress tensor will respond. There will be energy in your system. By the way, so you can, of course, then compute the two point function of the stress tensor by varying the action twice, not once. And note, I should have said, maybe I should have observed over here, the stress, I mean the operator dual to this, in analogy with the scalar field, this would be the case delta is D. And so the dual operator, which is the stress sensor, should have scaling dimension D, which indeed, as we discussed when we talked about CFDs, the stress center does have dimension D. The two point function of the stress tensor, when you compute it, has exactly the right form, it should have. But in particular, you can compute the coefficient now from the gravitational action. And this is this quantity, this effective central charge, which in certain cases, like n equals 4 super Yang-Mills, you can also compute. So even though n equals 4 super Yang-Mills is strongly coupled because it's got some very nice properties due to its supersymmetry and so on, actually there are various things you can compute at weak coupling using perturbation theory methods that still hold at strong coupling. So there are certain quantities we have control over, even though it's strongly coupled. One of them is this object. And so this here, we're computing it in the semi-classical gravity regime. But in fact, we also know it in the field theory. And what that tells us, I think I wrote this down before, this was what this number would be for n equals 4 super Yang-Mills. This is what it is in the gravity. And so this is a map between how n in the field theory is related to this ADS length scale over G, the Newton constant. And remember, I'm in natural units. Well, I'm not in natural units. I'm in h bar equals c equals 1 units. I haven't set G to 1. So in particular, G we can think of as being the Planck length to the d minus 1. And so this condition that we have a large number of degrees of freedom, n is large, is very obvious here. It's just saying we want the ADS length scale to be much larger than the Planck length. If you do GR with some system that's got a characteristic size L, L had better be much greater than the Planck length, which is the quantum scale of gravity. Otherwise, you've no right to think about classical gravity. So the statement that n, we're seeing very explicitly now, n has to be large in order to recover weak curvatures compared to the quantum gravity scale. OK. Everything else, I have to say, is sort of optional. So let me just rant for six minutes to end. Let me just say a couple of things. Oh, yes. Oh, do I? OK. Well, actually, I can. It wasn't really going to be just six minutes, as I think you must know by now. OK, good. Fantastic. Well, OK. So I mean, what I wanted to say is that's the correspondence. There isn't anything more to it than that. OK, you know as much as I do now. Possibly more, as I forget stuff quite quickly. Let's look at some examples of things that are interesting. And I then want to finish with some things, just some comments about what might be interesting in terms of PDE or geometry as well. But let me say a couple of things just using bits and pieces that we've talked about. We talked about the planar ADS Schwarzschild solution. If you recall, it takes this very simple form. f is just some function of z. z naught is the position of the horizon. It's a very simple calculation to calculate the Hawking temperature of this black hole. As we said, the Hawking, let me call it th. The Hawking temperature, or let me call it t Hawking, was that if you've never calculated the Hawking temperature of anything and you're a geometer, there's a beautiful way to do it by basically just taking t to imaginary time and thinking of imaginary time as a circle, which the circle action is an isometry on this with a fixed point at the horizon where f vanishes and ensuring that that is regular there determines the Hawking temperature of black hole. It turns out very elegant. The stress tensor of this solution we can now compute. So our theory is we should now think of our theory living at finite temperature. So we're not really in vacuum anymore. We've deformed on the gravity side. We're not in the vacuum. We've got a black hole. What have we done? We've deformed the theory to put it at finite temperature. And if we're thinking in equilibrium, the temperature must be the Hawking temperature. And so on the CFT side, we must have deformed our theory to now be at that same finite temperature. If you want to think from a Euclidean point of view, we've just changed the boundary metric to have to be Euclidean with a circle for time. And by choosing the size of that circle, we're just changing the temperature. And it's the boundary metric for the gravity, but it's also the metric that the CFT lives on and thermal field theory is just one way to think about it is just a Euclidean signature field theory on a space with a round circle for Euclidean time, the size of which inversely determines the temperature. Anyway, we can now calculate the stress tensor. What do we need to do? The stress tensor is basically just determined by the asymptotics of the metric. And it's a very simple exercise to calculate the stress tensor from this. It's a nice exercise. It's not entirely trivial because this z-coordinate that I've written down isn't the z-coordinate I had before. By the way, the expansion I had before, which I realize I've now rubbed off, this g equals 1 over z squared dz squared plus h. I should have said this is a terrible thing not to have said with geometers in the audience, where this has some expansion in powers of z. This is called the Phefemon-Graham expansion by physicists, at least. Phefemon and Graham, my understanding, did this in hyperbolic spaces, which is just the Romanian version, but it's exactly the same in Lorentzian signature. In fact, this expansion was discovered, I think even earlier, in a two-page paper by a Russian cosmologist who was thinking about the asymptotics of DeSitter space because they were thinking about how inflation worked. So in fact, it's also the same expansion. But it was Staravinsky. I mean, it's literally a two-page paper in Russian. I think it's in a Russian journal. Anyway, it's irrelevant. But this expansion, the Phefemon-Graham expansion, now this coordinate z, maybe I should call it z bar here, is not quite the same as this because note this factor of f. But it's a very simple exercise to coordinate transform this z to the appropriate one. You don't need the exact form. You just need to do it in an asymptotic expansion. And then you read off the, if we're in d dimensions, the d-th term in this, when you expand everything out, and you'll find that the stress tensor looks something like, let me remove my indices. Why should I call it ij? It's me trying to be mathsy, by the way. Where rho is equal to some constant, which isn't terribly important, times an appropriate power of temperature, well, I mean this constant, I say it isn't important. I mean it's actually just some numerical constant. You can work it out exactly what it is. So it's just some number, I haven't got it here, where t here is the Hawking temperature. And p, in fact, this thing is traceless. So p is equal to rho over something, d minus 1, whatever. It's traceless. So that determines p. The important point is just this. The fact that the energy density, this is the energy density, the time, time component of the stress tensor, the fact the energy density in the field theory scales like t to the d is trivial on scaling grounds. Energy density is an energy density. And if you're in a d-dimensional space time, it must go like t to the d. There's nothing interesting about that. Anyone could have told you that for a conformal theory at finite temperature. However, this constant is just some number derived from, after doing this coordinate transformation, it's some explicit number. If you took n equals 4 super Yang-Mills and said to someone, your favorite field theorist, the best field theorist in the world, calculate in n equals 4 super Yang-Mills the energy density of the theory at finite temperature, they can't do it. But here you've just, I mean this is a trivial. This is like a trivial calculation just fiddling around with black holes. I mean it's very remarkable. And these two-point correlation functions are determined by this scale symmetry. So in some sense you may say, well, obviously we got things that looked like what they would do in CFTs. That wouldn't be true if you went to higher-point functions. They would then be non-trivial. But this is an example of something that's entirely simple to calculate on the gravitational side of the correspondence, but is currently absolutely impossible to do analytically, completely impossible to do analytically on the field theory side. So it's a really rather remarkable and powerful correspondence. Now of course, unfortunately it's n equals 4 super Yang-Mills and all the examples we have are not quite the field theorists we want to do calculations with. For example, that there really were field theorists like QCD that were hot in the early universe would like to understand their thermodynamics and we don't. These theories aren't quite them, but at least their theory is that one can calculate it. This is just one example, but more generally, maybe I start off in some vacuum state here. I put my theory on some curved spacetime H0, not Minkowski. I get to choose it. I've got some scalar field maybe with data phi 0 that I specify for my scalar field. I make them do something. Maybe I keep them simple here. So this is just ADS and then I make them do something complicated and then I get some horrific dynamics. Maybe I form some black holes, whatever. OK, to someone in GR, this is obviously what happens. But to a field theorist, if you said takes n equals 4 super Yang-Mills and throw together some energy density and see what happens, completely impossible. Dynamics is completely and totally impossible in field theory, anything with interactions is extremely hard with strong interaction particularly. So as simple as it seems from a gravitational point of view, the physics this is then describing, for example, what physics might you extract, just looking at the asymptotic behavior, you find your solution, look at the asymptotic behavior near this boundary, what are you reading off? For example, you're reading off, look at the asymptotic behavior of the metric, you're directly reading off the one-point function, the stress tensor as a function of time and space, or the one-point function of the scalar as it responds to the way you've excited it. Completely impossible to do this in field theory, particularly dynamics. It's extremely, I don't think anyone really has any good idea how to do dynamics in a strongly coupled quantum field theory ab initio. OK, so some problems then for, I mean with all of that said, I hope anything you can then say about gravity and asymptotically ADS spaces therefore potentially is very interesting in terms of field theory because essentially in these sort of field theories you can't compute anything on the field theory side. So you shouldn't underestimate the interest, even if it's something utterly uninteresting to you from a GR point of view, it may be really rather remarkable as a statement about dynamics of these quantum field theories. And I think in that sense, ADS CFT is a very exciting area for mathematical aspects of PDs, gravity, and even Romanian geometry just because I think there are all sorts of problems that you probably have almost simple. Well, it may or may not be simple, I don't know, but there may be simple problems out there. They're actually really quite interesting from a physical perspective. So let me just try and outline some of the sort of things one might try to think about. These are just random things that came to mind. Dynamics in what I might call asymptotically local ADS spaces, I just mean things that have this Phefman Graham-like behavior, Phefman Graham-like asymptotics. So they don't have to be asymptotic to ADS. They have to be asymptotic to something that has maybe with some different conformal boundary metric, but nonetheless has this asymptotics I wrote down before. Dynamics, as I said, in field theory is absolutely disastrous to think about doing strongly coupled field theories dynamically. And so anything is there about dynamics is really quite interesting. There was a big, there was lots of excitement or lots of discussion and controversy from very nice work. There was early work by DeFermas, and I'm not sure if it was just Mihalis. I can't remember your paper. You had said that something along the lines of, the point about boxes is energy can't disperse. So in flat space, when you put in a little bit of energy, it will just tend to disperse. You have to put in a reasonable amount and focus at a reasonable amount in order to form a black hole. On the other hand, if you're in a box, there's a chance that an arbitrarily small amount of energy could, it can never escape. So maybe in the end does form a black hole and there was a conjecture that it, that that's indeed what happened in ADS. And these people did some actually very nice numerical work, sophisticated numerical work, in fact, and showed that indeed it seemed like arbitrarily small perturbations of ADS that didn't immediately collapse and form black holes. But if you wait a very, very long time, in the end they do. And so that's interesting. So small data in global ADS, say for a scalar, tend to collapse to a black hole. So it can't disperse, but somehow the energy gets more and more focused over time and any end collapses. For example, if you deform the boundary of this space to not be the ADS boundary, there's lots of debate in the literature about what sort of thing would happen there, whether this phenomenon would still happen or not happen. No one really knows, but it's a problem in PDEs, which I think is very interesting. Or it's an example of one that would be very interesting for ADS CFT, but more generally anything you can say about dynamics would be interesting. The one caveat I would say is if you're thinking of embarking on some sort of ADS CFT-like mattercy thing, the sort of statement one's interested in is a statement that translates into a statement in the field theory. And that's a very important point. So what you want to be able to say is, you know, if I take some non-vacuum initial data, like, you know, supposing you could prove this generically small perturbations of ADS did form black holes, you'd want to prove it in some form like, you know, arbitrary deformations of this data here for a trivial, you know, boundary, global ADS boundary collapse inevitably to form black holes. What you don't want to have is something like if I put a condition on the Riemann tensor in the interior that it's greater than some something or other, then I'll form a black hole because that doesn't trans, you know, that's not a statement that could be made purely in the CFT. So that's the one caveat I would make is the statements that you would want to make would be ones that can be said entirely in CFT language. They would be the really interesting ones. Let me just say you can consider, we get to choose the boundary metric. So we can consider the CFT on a Euclidean space. Maybe that's the space, the boundary, a conformal boundary metric or representative of. So this is now some Riemannian metric by Euclidean, I mean Euclidean signature space. So this is some Riemannian, maybe more mathy. So this is the, I know what this means actually. Okay, so, and then of course this problem is really the problem of infilling Einstein metrics, negatively curved Einstein metrics. Of course you could add other matter as well and then it's more complicated. As far as I understand, I mean very little seems to be known about, well, I know that very little is known about what type of solutions there are. That's certainly true. What I do know in the math literature is you know many fewer solutions than the sort of things we know in the physics literature. Although we don't necessarily know them analytically but we're pretty sure that they exist. So we have lots of examples of things that maybe, I should say, this Riemannian setting is the same as a static setting. If I have a static space time I can always trivially continue time to Euclidean time and it becomes Riemannian. So static Riemannian is the same for us. But I think a very interesting question rather than examples of solutions is actually existence. I personally think this is a very interesting question. Why is it a very interesting question? It's a very interesting question because the CFT, partition function, on some metric is given by this path integral which reduces to this classical, you know, if there's a classical solution that reduces to be given by e to the i renormalized classical action on the solution, on the infilling solution, if one exists, if one doesn't exist, there isn't a saddle point. And what that means is that the path integral is not dominated by a saddle point. It doesn't necessarily mean all is lost and the theory is somehow bad. It just means that there isn't a classical saddle point. So the behavior is inherently quantum. That's really interesting. And that's something actually that we don't know much about is when do we have, when can we expect, for what data on the boundary can we expect classical saddle points or infilling solutions? Even in this just metric setting or with matter fields. And there have been various studies with matter fields. I'm thinking of recent work by Gary Horowitz, George Santos and people who've shown that actually once you deform by matter fields on the boundary, you can lose existence of solutions in the interior. And that's physically very interesting. But even in the context of just geometry, it would be very nice to have a handle on this. Now there was work. And this is where I should reference this paper. This is an oldest paper by Michael Anderson who's thought quite a lot about problems of Einstein metrics and has a nice review about some aspects of the geometry of ADS CFT. Okay, so he doesn't really discuss. I mean, it's more a discussion of various, some geometrical problems or review of problems he's worked on rather than the full review. But nonetheless, I think if you're a geometry, it might be interesting to look at. But in particular, he talks about this existence of infilling metrics. And one of the claims is, and I'm not sure to what extent this is, I don't know what the state of the art is on this. But one of the claims is that if you have a boundary metric that has a positive scalar curvature everywhere, then there's an infilling solution. That's interesting. What happens if you don't have a positive scalar curvature? I don't know. So in a sense, having an infilling solution is nice. But what's perhaps more interesting is when you don't have an infilling solution. And I don't know if, even demonstrating that there is a case where you know there's no infilling solution would be interesting. You have to be slightly careful. You don't necessarily want to show that there's no infilling solution with a particular bulk topology. That would fall into this class of theorems that would be less interesting from the point of view of field theory because the field theory, you want to make the statement entirely from the field theory point of view. So you don't want to say, if I assume my bulk topology is such and such, there's no solution. Because the field theory doesn't care about the bulk topology, it just cares that there's some, at least some solution. What I should have said is if there are multiple solutions and there may well be, it's the one that dominates the saddle point analysis that's the one that is then equal to this. Okay, so you can have multiple ones, that's fine. What's really interesting is if there isn't one. So if you could make some sort of general statements about, for, even if you could say for some boundary metric, there exists no topology in the interior where you can have a smooth infilling geometry, that would be nice. It's also true that certain singularities are tolerated in string theory. So one could even, one could work in the class of smooth metrics, one could even wonder about what happens if you weaken that. And the very last thing I want to say, just in passing, as many, just because if you start looking online at the HET-TH, you'll see tons of papers talking about this sort of thing, but that in the static setting, so let's draw the boundary like this, I'll just be 30 seconds, in the static setting, so this is a constant time picture, this is the boundary, this is the bulk over here. If I draw a closed volume on the boundary, there is an object called the entanglement entropy, so here's my CFT living in the boundary, there's an object called the entanglement entropy, which is a very interesting physical quantity, not a correlation function, but it's again some other physical quantity that's interesting associated to that closed volume on the boundary. And in ADS-CFT, there's a very beautiful geometric interpretation for what calculates this entanglement entropy, and it's actually, at least in a static setting where it's easiest to state, it's the area of a minimal surface hanging into the, into this Einstein interior, okay? So some minimal surface, and it's got a divergent area for the same reason that everything's divergent towards the boundary, but there's a renormalization procedure, there's a renormalized area. And this is an interesting physical quantity, making statements about this sort of thing is quite topical in physics at the moment. Anyway, so I hope I've told you something about what ADS-CFT is, and I hope I've, yeah, I hope I've given you something to think about, whatever area of maths you've lived in or physics you live in, and that's all I have to say. Thank you. Thank you. Thank you.