 Hello? Ah, yes. So let me just recap where we got to last time. We introduced this ADS spacetime, which has a neat embedding as a hyperboloid into some two-time generalization of Binkowski's space. And we saw that the isometrers, which are best seen from this embedding picture, are an SO2D. This is highly analogous to when you embed a sphere into Euclidean space, a round sphere, into Euclidean space, and obviously get an SO isometry group that has an action on the sphere. It's just got some flips of sine in the metric. So this is now a Lorentzian metric and so on. Now, I told you that there's a global chart for the metric on anti-desitter. But in fact, the chart that we're going to predominantly use is not a global one. It's one that can be thought of as coming from a sort of null wedge cut out of this space. These should meet at a point. And it's called the Poincaré chart. I gave you some explicit parameterization in terms of the hyperboloid. And this is this Minkowski metric. And in terms of a conformal diagram, a Penrose diagram, for this space, it's not a complete space. So we have the boundary, which now lives at z is 0, the conformal boundary. But we also have some Cauchy horizons associated to these null wedges, the fact that you can just continue out perfectly happily. But from the point of view of field theorists, we'll see this. We already started to see last time this is a sort of a natural coordinate system to use. We also talked about how, as many of you with a geometry background will know very well, this is a conformally compact space and has a conformal boundary, which is defined up to some conformal rescaling. And z, in its own right, here is a defining function for the boundary, the conformal boundary of this space. And when we look at what the conformal boundary is, it's just Minkowski. So in this form, the conformal boundary in these coordinates, the conformal boundary or a representative for it is just Minkowski metric. Now, you may be confused that we already said from here we know that the conformal boundary is this Einstein-Static universe time across a sphere. And now I've got a conformal boundary that's Minkowski. How did that work? And the point is that Minkowski and the Einstein-Static universe are related by a conformal transformation or a vial transformation. It's not completely obvious, but it's a simple exercise to show. Now, before I go on, I just wanted to put a caveat to what I was saying yesterday. It was a good question after for the physicists among the audience. So if you're a mathematician, I suggest you now close your eyes and put your fingers in your ears. But please don't listen. I'm not going to explain it in terms of bundles. I have no idea how to. But for the physicists, just to avoid any confusion, this is a correlation function derived from the path integral. So of course, in Lorentzian signature, it would be time-ordered. What I've actually written here is the Euclidean continuation. If you want the Lorentzian version, that's an exercise for you, Fourier transform, use the i epsilon prescription, and so on. The point is when in all discussions of conformal field theories, it's just complicated to work in Lorentzian signature, and everything looks very pretty in Euclidean signature. And you'll see that again later in some of the other things I'm going to say. So whenever you see explicit expressions that look like this, there will be some implicit analytic continuation, as we nearly always do in field theory. So if there's a mathematician next to you who's still got their eyes closed and fingers in the ear, maybe you can give them a nudge. Otherwise, carry on listening. OK, so as we said last time, but I want to re-emphasize it because it's a very important point, this metric manifests certain of the isometries, certain of these isometries, very obviously. And so in particular, Lorentz and translations in the x-coordinates alone obviously leave this invariant and are part of this isometry. However, there is also. So these are holding z constant. But there's also, it's immediately clear from this form of the metric that if we were to do this diffeomorphism, again, this metric is invariant. And so this is again part of this isometry group. But it has a very interesting action on the conformal boundary. So these act on the conformal boundary as the Poincaré group. And this acts as a scale transformation. And what isn't manifest is the remaining sort of symmetry or the remaining part of the SO2D, which is this special conformal transformation. But nonetheless, it's in there. And you can show that it's action is the usual action on the conformal boundary. So what we learn is these act on the boundary as the conformal group. And by boundary, I mean the conformal boundary metric. So I'm typically going to just call the conformal boundary the boundary, but understand it's not a boundary. It's a conformal boundary, but just for simplicity. I hope that won't confuse. So the isometries of anti-disseter act on the conformal boundary as the conformal group acts. Now, that's pretty much where we got to last time. So ADS geometrically has properties that I'm sure you know. It's an Einstein space, which if I introduce explicit coordinates. And in my conventions, the Ricci tensor is given in terms of the metric in this way with what we would call in physics a negative cosmological constant. So it's a negatively curved space. And in fact, it's maximally symmetric. And the vial tensor vanishes, which means Riemann can be written in terms of Ricci in the usual way. But in particular, because of this, ADS solves the Einstein equation in the absence of matter but with a negative cosmological constant, obviously. Because this is the Einstein equation for just gravity with a negative cosmological constant. And we should interpret it as the vacuum solution for that dynamical system, for that theory. ADS is the vacuum solution or the sort of empty, the simplest solution for gravity plus a negative, let me call it lambda cosmological constant, by which I mean the theory whose Einstein equation is this. However, and so this is just an analogy with Minkowski being the usual vacuum solution for just pure gravity without a cosmological constant. It plays exactly that role. And from a physical point of view, what we're really interested in is not just the vacuum, but excitations, dynamics of this theory. So this is what we're really interested in. From the point of view of ADS CFT, we're really interested in this some gravity theory that live in spaces that are asymptotic to ADS. An asymptotic ADS spacetime is obviously one that solves some Einstein equation. Well, we will require it to solve some Einstein equation. So when I talk about asymptotic ADS spacetime, I'll be talking about it in the context of some Einstein equation with negative cosmological constant or some appropriate matter to give the right to at least look like a cosmological constant asymptotically and tends to ADS asymptotically. And I'll be more precise about that later, but let me leave that for now. So I will give you gory details in which in what way I mean, probably in the last lecture, in what way I mean that the spacetime should approach ADS. So our setting then, this is often called the bottom-up setting in ADS CFT, will consider gravity, GR, well, maybe Einstein gravity, plus some negative cosmological constant, but perhaps plus some matter in d plus 1 dimensions. This is the physical theory we're going to be interested in, such that ADS is a vacuum solution when we turn off all the matter. Why is this called the bottom-up construction? In the bottom-up way of thinking in ADS CFT, we are going to think of our gravitational theory in the simplest possible terms and try and understand the general phenomena that can happen in this gravitational system, as opposed to a top-down way of thinking where you say, OK, I believe ADS CFT works in this particular case for this particular field theory being due to some particular string theory whose low energy limit is a particular supergravity with very particular fields. And I should only work in that setting, so I've got to take some very complicated gravity theory with lots of fields and so on and so forth. These are very complicated theories, but they nearly always, you can consistently truncate the dynamics away for many of the fields and in the end reduce some of the dynamics, at least to something much simpler that will always look like this. In particular, in any case where you have a conventional theory of gravity, by which I mean two derivative Einstein in however many dimensions you like, and with all sorts of complicated matter fields, fermions, bosons, all sorts of things, you can always truncate everything away to land up with just gravity and a negative cosmological constant, and any solution of that you can re-plug into your full theory. So let's look at these very simple sectors of these much more complicated theories rather than sort of saddling ourselves with a tremendous task of understanding super gravities, which are very complicated. So it's a consistent thing to do this in general, but one has to also bear in mind that at the end of the day, the physics that you look at should be embeddable in some sort of theory that you really believe exists. You can't just make up completely baroque theories of gravity that happen to be asymptotic to ADS and imagine that they will necessarily be found in some version of ADS-CFT. That's not, it's questionable whether that works. OK, so just to reiterate this point a little bit more, in the case of n equals 4 super Yang-Mills, which we discussed, there's a very specific 2B super gravity, let me call it SUGRA as usual. And in fact, it's 10 dimensional. This is a 3 plus 1 theory. And the vacuum solution here isn't just ADS. It's ADS 5 cross an S5. And that's where the 10 dimensions come from. And it's a complicated theory with lots of matter. But this S5 doesn't really play a terribly interesting role. It's just some sort of direct product, and you can, in the vacuum, and we can consistently truncate to field configurations that are constant on the S5 and remove, well, we could remove all of the matter and just be left with Einstein gravity in a negative cosmological constant in five dimensions. And it's also possible to consistently truncate and keep some matter fields. OK, let me carry on then. So for now, as I said, we'll just consider Einstein plus cosmological constant maybe plus some simple matter, maybe like a scalar. And Einstein plus lambda I would normally call the universal sector of ADS CFT because it's there in every known example where you have an Einstein-like gravity dual. You could always reduce to, if you've got a d-dimensional QFT, a d plus 1-dimensional Einstein plus cosmological constant, we'll always sit inside whatever complicated gravity theory you have, which is nice because, of course, this is just geometry. So it's a simple geometry, the sort of geometry that geometers are interested in. So you're interested in Einstein metrics. So the theory of Einstein metrics inevitably sits inside any ADS CFT correspondence. Let's have some examples of things that aren't ADS but live in this universal sector and play important physical roles. So one of them is ADS Schwarzschild. And let me call it planar ADS Schwarzschild. There is a more conventional ADS Schwarzschild, which is asymptotic whose conformal boundary is that of global ADS, meaning Einstein-static universe. But you just take the usual Schwarzschild metric in Schwarzschild coordinates and you add an r-squared, your function, your GTT function, which is normally 1 minus m over r, you add to that something going like r-squared. That will solve the Einstein equations with the negative cosmological constant. However, because we're interested in this setting where the conformal boundary is Minkowski, because from the point of view of field theory, typically you're interested in field theory but not space first, the relevant black hole is actually a different one, which is physically inequivalent. It's not isometric to the other one I mentioned. It takes a very simple form. This function F is just a function of the coordinate Z and Z naught gives the position of the horizon. Z goes to zero is again the conformal boundary and we see again Z in its own right as the defining function, the conformal boundary as Z goes to zero, F goes to one. It's again just Minkowski. So we haven't messed with the conformal boundary. So this is indeed, as we will discuss later, asymptotic to ADS in the usual sense. But it's nonetheless not ADS, it's a black hole. It's an interesting black hole. For those of you who know all about black holes in the usual setting, this is a very different black hole. One point to note is that the horizon is Euclidean space. So it's not compact. All of your lovely theorems about black holes very often use compactness of the horizon somewhere. So all sorts of interesting things happen to black holes in ADS that simply don't happen in asymptotically flat settings. For example, well, actually let me not complicate things. So that's one solution. It's often drawn if you read books or lecture notes on ADS CFT. It's often drawn like this. This is meant to depict the boundary. Z is zero. This is meant to depict the horizon. Z is Z naught. And this business is meant to depict the warping, if you like, of the spatial slices, the constant Z slices by this factor. Of course, really the warping becomes infinite as you go near the boundary. But nonetheless, that's difficult to draw. You can calculate for those who are interested the Hawking temperature of this black hole. Well, I mean in various ways, but if you like the Euclidean methods, it's very straightforward. This is, by the way, the Hawking radiation associated to the killing vector d by dt, which is the natural killing vector of the space, and also the time on the boundary. And the entropy of this diverges. But there is a sense in which you can define a density. By the entropy, I, of course, mean the area suitably divided by 4g Newton with appropriate h bars in. So if you like, the area diverges, because it's the Euclidean metric on the horizon. But nonetheless, there is some notion of a density. So because it's infinite, whenever you have infinite systems, you always have to start working with densities. So there's black hole thermodynamics and all sorts of things as usual, but defined in terms of densities rather than extrinsic quantities. Sorry, was there a hand? Another, oh, an interesting point here, just to bear in mind, as z goes to infinity keeping z fixed, this returns to the form of AdS. And one way to think about that is that the Cauchy horizon of AdS, in some sense, is like a black hole horizon whose temperature has gone to zero. So in physics language, we could also call the Cauchy horizon of Poincaré AdS an extremal horizon, meaning it's a zero temperature horizon. And there's some theory of extremal horizons that have been developed. I won't go on about that. There's another interesting metric that perhaps people are less familiar with, but certainly is very interesting. And it's called the AdS soliton. So this metric, in fact, the conformal boundary here is Minkowski, but something I can do and leave this a solution trivially is to wrap up this Euclidean space into a torus just by some identification of these coordinates. So I can wrap this into, say, a square torus just by making these space coordinates angles. And now my conformal boundary isn't Minkowski anymore. It's time cross a torus. And in fact, this is a perfectly fine thing to do. And from the point of view of AdS CFT, we'll later discuss the interpretation of this. But just from the point of GR, this is a perfectly fine thing to do. It's a perfectly good. Locally, the boundary will look like AdS, but globally, it's different to the AdS boundary. So it has this different topology. And in that sort of setting, there's actually another nice solution which is closely related to this one, which is, in fact, trivially related to this one, really. You just let me call this now some other label swivel where this theta is some circle coordinate. So maybe I've identified all of these into circles, or maybe I've only got one that's a circle and the others are extended. These are all fine from a GR point of view. And this now isn't a black hole at all. You can see there's no horizon. But instead, it's more like a space. One could draw a similar picture. And this is a space where instead of it ending at z, is z0 at a horizon. Instead, the circle, this circle on the boundary. So here, at some point, we've got a circle. What's happened is it shrunken smoothly provided, let me see, provided I identify 1 over L, provided I pick z0 appropriately, I get a smooth shrinking of this circle as I go away from the boundary in what we call the radial direction, the z direction. And this is called the ADS soliton discovered by Horowitz and Meyers. I mean, it's a very simple solution. But in terms of GR, it's actually quite interesting. If we have time later, one of the interesting points is that these have positive, supposing I wrap this up into a torus, now I can talk about the energy rather than just energy density. It's finite. And wrapped up on the torus, this has positive energy, just like Schwarzschild normally has positive energy. It's got some energy which goes to some power of z0. Unsurprisingly, in this case, this actually has a negative energy. So in some sense, this is the natural ground state of GR when the conformal boundary has circles in it. So it's quite interesting. From a GR point of view, you're thinking about proving positive energy theorems and so on. Depending on the situation on the boundary, it may be that ADS isn't the lowest energy solution that you might want to, well, yes. Already there may be lower energy solutions out there to consider. OK, anyway, those are just a couple of solutions that are not ADS but are interesting physically. This would be the vacuum solution. This is some static solution in this theory that's got some energy in it. Oh, yeah, OK, sorry. Yes, the Fs are exactly the same. I mean, if that solves the equation, this does trivially by just analytic continuation twice. I just take t to i theta. And so this t is not that t. But if you see what I mean, I mean, it obviously solves the equations. Yeah, well, you could identify here as well. So I mean, you see, this is an example of a case where the boundary at least has one circle. It may have more circles. But now there are multiple infilling Einstein metrics, maybe from a geometric point of view. We have the usual ADS one. Now, the usual ADS metric, if you identify a circle, actually has a cusp-like singularity. And then this is the more natural vacuum state, which is perfectly smooth and has no singularities. So the natural cusp-like singularity you get when you make this identification on the conformal boundary of ADS, it sort of resolves because there is actually a nicer and lower energy smooth solution. Anyway, let me not go on about that. There was one more question. Sorry, can you? Yes. Yes. Yes. Yeah. I mean, you have to, there's a horizon. I mean, it's just like Schwarzschild. There's a horizon you then have to. So let me see, what does the conformal diagram look like? I think it looks like this. So this chart, as usual, is only covering the exterior of the horizon. And then there's some other charts to see the singularity. But, yeah, so maybe something to say is that this is locally asymptotic to ADS in the sense that I don't have the full infinite conformal boundary anymore in this case because of the singularity. So maybe to say this is, maybe even in this case, we should be a bit careful when we say it's asymptotic to ADS. It's actually removed a lot of the conformal boundary that ADS has. But of course, let me just say the general solution to this sort of gravity theory then looks very complicated. Let me draw it in this sort of setting. Dynamically, we're going to imagine we've got this ADS boundary or asymptotic ADS conformal boundary. And we've got, we should think about dynamics in this as requiring some asymptotics to ADS near the boundary. But also, we've got these Cauchy horizons. Well, maybe we don't even know what we have here. But certainly, we can start with some characteristic data on this null surface or Cauchy horizon. And then see what happens. Evolve the Einstein equations. Maybe we put a black hole in here. Maybe we have, you know, maybe we put some little black holes into this data. We shoot them together. They collide and form a bigger black hole. And who knows? There's some radiation given off, bounces around. And important, you know, as I emphasized last time and I should have re-emphasized, ADS is like a box. You know, stuff hits, reaches this conformal boundary in a finite time. So this dynamics definitely knows about this boundary. It's very important. Even though it's infinitely far away from a spatial point of view, you know, radiation may go there. And you, with the boundary conditions we're going to choose, it will bounce back. And there'll be some very complicated dynamics. OK. So just because you write down only a couple of exact solutions, I mean, it's a full theory of gravity in principle. Something to note is that we mentioned this before. The time-time component, you can see from these static solutions, the time-time component, which tells us about redshift physically, in ADS, as we go away from the conformal boundary, as we go to the conformal boundary, it diverges as 1 over z squared. And as we go to the horizon, it goes to 0. Or in the black hole case, as we go by horizon, I mean, Cauchy horizon, in the case of ADS Schwarzschild, it diverges at the conformal boundary and goes to 0 at the black hole horizon. So if you think about, if you're sitting in the interior of the spacetime at some constant z, and bear in mind that that's not geodesic, so you'll have to be accelerating to do that, exactly in analogy with usual Schwarzschild to sit at constant radius, you have to accelerate. If you're accelerating sitting at constant radius z, let's say, you will see physics happening near the conformal boundary as being very high energy, very blue shifted. Whereas if, as usual, physics near the horizon will be very redshifted. So the interpretation of these sort of radial-like coordinates that take you away from the boundary is that there's also an energy. Somehow, stuff near the boundary is very energetic. Stuff far away from the boundary is low energy. And it's just in this very simple sense. So we often, in physics, which is probably confusing, would say this is the UV part of the geometry, the stuff near the boundary, and this is the IR, the infrared part of the geometry, which is far from the boundary. But just terminology in case you hear it. So now we're going to do some calculations. So it is complicated to study the dynamics of gravity, as you're all well aware. And so we're just going to study the dynamics in some detail for a scalar field in ADS. And I will need to do this so that you can see somewhat precisely how the ADS-CFT correspondence will work. And then probably in the next lecture, I will hopefully today, I'll state what the ADS-CFT, how it actually works, the correspondence. Perhaps it'll be the beginning of the next lecture. And then we'll discuss how it works in the case of gravity rather than just a scalar field in some detail, mainly because geometrically, that's probably the most interesting case, but physically not necessarily. So let's now talk about, so we're sort of moving towards ADS-CFT now. But in order to get there, we're going to consider a bulk scalar. By bulk scalar, I mean a scalar field living in ADS, it's going to obey the usual, it's a massive scalar, so to obey some wave equation. And again, we're going to take this Poincaré chart, when we think about its dynamics. We're going to think about its dynamics in this chart. And physically, imagine that you've got one of these simple, consistent truncations of some complicated theory. If you want a physical picture of this, where you had gravity, cosmological constant, and you managed to truncate everything away except one scalar field, which is entirely possible. And then you look at small excitations of the scalar. Now they will, of course, back-react on the geometry, so it won't be perfect ADS. But if the fluctuations in the scalar are small, the back-reaction will be small. OK, so this is physically relevant. It's not just some toy model. You're looking at fluctuations of a scalar about the vacuum geometry. Now it's just sort of simple calculations. Now there's an action, which you all know. There is a subtlety here, which I'll come back to later. So I probably shouldn't have mentioned it. So we have the usual action. And now we're going to consider, we're going to Fourier decompose our general solution. So our general solution will now be a function of the x. Let me suppress the indices. There will be a function of x and z. But will Fourier decompose and think of the general solution as a sum of modes, Fourier modes, in these directions, which are obviously translation. There's a natural isometry due to translations. And that naturally means we should Fourier decompose to simplify everything. So we still have non-trivial dependence on z. And then we can sum up all of these modes if we figure out for a given Fourier mode what f is. We'll have to solve some equation. We'll talk about that in a minute. We figure out for a single mode, well, we figure out for a general mode. And then we can add them all together with some coefficients to get the general solution as usual. So now let's focus on a single mode. Let's try and determine what this f is. So focus single mode. And now when we plug this into the wave equation, you'll find just a differential second order ODE for the non-trivial bit of the solution f. And if I write f as if I pull out a factor of z to the d over 2 and then write it in terms of some function of k times z, where now I should just be clear when I've written k, what I mean is this is a vector. Now if you substitute this in, I won't write it out, but h obeys a Bessel equation. So it's simple solutions. So the solutions of this are some powers of z times Bessel functions whose arguments are arguments of k and z. So you find this h of z, well, sorry, let me call it h of mu where mu is k z. H of mu is in general some sum of two Bessel functions where their parameter is given in terms of the mass of the field and the dimension. So it's a simple calculation to go through. Now we can then analyze. There are two sort of interesting regions of the spacetime. OK, yeah, go for it. This is ADS. Yes. Oh, this you mean? Yes, yes, sorry. This is the yes. Yes. Oh, yes, indeed it does. Yes, exactly. Yeah. Sorry, I'm not sure of. Sorry. Oh, thank you. Yes, OK, let me make it explicit. Yeah. In fact, I can be very, yeah, that's right. Thank you. And so by this I mean, thank you, sorry, that was too quick. Yeah. As I said, that's exactly what we're doing, yeah. So I mean you can ask just the problem of a scalar in ADS, but physically this is like small fluctuations of a scalar in a theory of gravity plus a scalar plus a cosmological constant. But the fluctuations are small enough that there is some back reaction, but it'll be quadratic in this field. So whilst it will affect the equation, it'll affect the motion of this field via the connection in here. It will only do it at some higher order in the amplitude. So we're getting the leading behavior, yeah. Yes, so that's right. So K could, depending on whether it's time-like or space-like, that's right. But if you want, let's just, we could just analytically continue time to imaginary time. This is now just the Euclidean metric, and then we don't have to worry if that's concerning you. But indeed, it could be imaginary, but not complex, either real or imaginary. OK, so let's see. So if you go through, think about the asymptotics of Bessel functions, you will find that z is 0 or mu is 0 is a regular singular point of the Bessel equation. There's some Frobenius expansion in the usual way. And putting that in here and rewriting, just defining, in fact, let me modify this just for now to make it slightly simpler. There's the leading, well, let me say, there's one power series that starts with this power and then has an expansion in powers that go up in z squared. And the other part of the power series solution has this leading term and then coefficients that go up in z squared from it. This A is this A, and this C is, in fact, some linear combination. So this is a linear combination of A and B. And just to be careful, you could have a situation for specific nu here where there's some log terms as usual for Frobenius expansions. But let's, for simplicity, consider the generic cases that there is no log case. So you can deal with the log case, but we're not going to. And then I'm just going to define this. I'm just going to redefine, instead of working with this eta, or sorry, nu rather, I'm going to define this to be d minus delta. And then this power becomes delta. So I've just defined delta to be d over 2 plus nu. And delta then, what is it? If you figure out just from here, think about what delta is. Delta solves this equation. Now, in principle, I could take any values for m here. This nu could be imaginary if I take sufficiently negative. If I took m squared to be, I'll take, if I think of m squared being real, not necessarily m. Because of course, actually, m doesn't enter anything. It's only m squared that enters the equations. I should think of m squared as being real. But it could be negative, in principle. And then if it wasn't sufficiently negative, I would get an imaginary eta, an imaginary delta. And I'm not going to allow that. I want to restrict my masses so that this is real. Sorry, that delta is real. And in fact, for simplicity, and you can do better than this, I'm going to assume, so for simplicity, I'm going to want delta to be greater than d over 2. Although, in fact, it turns out for well-poseness of the equation, you only require delta to be greater than d minus 2 over 2, greater than or equal. So that is a condition on m. So let's draw a graph of this. It has zeros at 0 and d and looks like this. This, I can depict as this curve. And so this point here is d over 2. So I'm going to assume that my mass squared is greater than this minimum value. So let me call this L squared mbf squared, where mbf squared L squared, if you work it out, is minus d squared over 4. Remember, again, just to emphasize, it's only m squared that enters the action, the equation. So it's only m squared that should be real. And I'm going to assume that delta is in this region here, which there is this restriction on the mass, but it's also a further restriction on delta, because I could have chosen delta to live. In fact, the minimum it could be is sort of here. This is this condition. And there is some bit here that one can deal with, but I'm not going to deal with it. It's more complicated. So let's just assume it lives here. That's fine for us. Note, you may be confused that the mass squared could be negative. That looks concerning. It's what we might call a tachyon in physics. Although it doesn't make the equation ill-posed, it's just an instability. But its instabilities of this nature, a negative mass squared, are instabilities on large scales, large wavelengths behave in an unstable way. Short wavelengths are still stable because they're dominated by the kinetic terms. The kinetic terms dominate this negative mass squared potential. And so because ADS acts like a box, it shelters us to some extent from long wavelength instabilities. And that's why in ADS, you're allowed to have negative mass squads as long as they're not too negative. If they go too negative, then you get complex solutions here. And then it turns out everything is ill-posed. But you can tolerate a certain tachyonic behavior in your field. This was all explored by Breitner and Friedman many years ago, hence BF. Right, so we can think of, so for simplicity, what we're saying essentially is that delta is the larger root of this equation. And because of that, and just to emphasize again, it is possible to have logs in this expansion, but they're not generic, so we're not going to consider them. You have to do slightly more fiddly things. But because of the fact we've taken delta to be the larger of these two roots, this gives the leading behavior of the field near the conformal boundary, and this part is subleading. Supposing you sold a wave equation in a box, a real box, what could you do near the boundary of the box? You would maybe Taylor expand or power series expand your field, but it would just be a regular Taylor series-like expansion. So it would just be a simple expansion. There would be a leading piece, which is constant, and a subleading piece, which was linear in the proper distance away, the normal distance from the box. So assuming the box had this translation symmetry in the wall direction, and you decompose that way. So you could think of this as the constant piece for a normal box, and this this, so what you might call the Dirichlet, if you were fixing Dirichlet boundary conditions, that's what you would fix. And this would be if you would like the Neumann bit of the behavior, if you wanted to fix Neumann boundary conditions. It's a bit more subtle here, whether you can fix Neumann boundary conditions, so we won't go into that. But we are in the end gonna be thinking of this as the part of the data that we're gonna fix in this problem. So as we said in ADS, this boundary really acts like a physical boundary in some sense. You know, high energy modes can get near to the boundary, arbitrarily near to the boundary. We have to fix some sort of boundary conditions there. What are we gonna do in the end? We're gonna want to fix this, the leading behavior. So it's not, again, it's not necessarily the value of the field because that will typically go to zero, or may go to zero, but we're gonna fix it, the coefficient of this leading behavior. That's the analog. Now, if we think about going back to real space, and not just looking at a single mode, but looking at some general sum over modes, what this is telling us is that the structure of the solution asymptotically looks like this. Where this is associated to this constant of integration, but this was a single mode. I just had a constant of integration here. But of course, when I sum over all modes, okay, when I sum over all modes with some orbit, maybe I shouldn't have called this A, so let me call it something else. No, I've used B, C, D, E, F, no, G, no, alpha. Let me call it alpha. If I choose some arbitrary alphas, I can turn the behavior, you know, choosing appropriate arbitrary alphas, I can arrange any phi naught as a function of the X coordinates that I like. Likewise, choosing any constants here, I can arrange any phi tilde D, as I've called it here, function of the X's that I like. So these are free functions of the X's corresponding just to this bit of data and this bit of data for a single mode. And everything else is determined in terms of these, and in particular, so all of these terms are determined by phi naught, and all of these terms are determined by actually both phi naught and phi tilde D. Remember, this is the sub-leading part, so these do in the end care about what's going on here as well as what this bit of data is doing. You can just plug this into the equation and explicitly see that this form is true. It's not enormously complicated. And in fact, these have a beautiful structure, these terms, the way these are determined by this is just by derivatives. So for example, phi two is some, roughly speaking, it's the Laplacian or the wave operator, the d-dimensional wave operator acting on phi zero. Phi four here, the next term, would involve four derivatives, a local expression involving four derivatives on phi zero and so on. And these, again, are just more and more derivatives acting on phi zero and phi D. Note something really trivial. Under our trivial, yet important for what will come, under our isometry, which from the point of view of the boundary generates scale transformations, but it's just an isometry. I mean, it's just an isometry of the interior. So this, from the point of view of the interior, I can think of as diffeomorphism. This is just a scalar field, so it doesn't care what coordinates I express it with. But because of this expansion in z, if I want to preserve the form of this expansion, it will induce, if I scale z, it will induce a change in finite and it will induce a change in all of these terms in a particular way. So in particular, if I want to preserve this form, there's then this natural action here from the point of view of this reduction to d-dimensional data. So this is something we've seen earlier when we were talking about conformal field theories. I should have said, I mean, I picked this letter delta. Again, it was the scale dimension of things in the past when we were talking about conformal field theory. Here it was just some rewriting of various coefficients that came up in the scalar equation in ADS. But there will be a reason that I've called it delta. It's not entirely without reason. Now, we've discussed the behavior in the boundary and I've even said that what we'll want to do is fix this leading behavior. It's like fixing a Dirichlet condition at this boundary. But we also have to think about what to do at this past Cauchy horizon. I said, generally, we want to fix some characteristic data there. And what we're going to ask is that the field is trivial there. There's no outgoing radiation from this past horizon. You can take other boundary conditions and that's fine and you can solve what happens. But we're just interested in the simplest situation. So what we're interested in, physically, is having no scalar field radiation or energy flux through this. Instead, we're going to fix the scalar field to do something on this boundary by basically fixing this to be some non-trivial function. So we're going to excite some scalar field dynamics by making this do something. And then that will induce some response in the solution. But we're not putting energy in initially. It will come, for example, maybe we'll choose this phi 0 to be 0 here and then not 0 here. And there'll be some radiation. There'll be some complicated dynamics. It may be trivial here, but there'll be some dynamics later. So let's try and do that. And when you go through the analysis, you see that this mode function for an individual mode, as z goes to infinity. So as we approach these Cauchy horizons, there's some definite behavior that comes from these Bessel functions. But if we require that we only have outgoing radiation here, nothing incoming, what that means is we have to kill this behavior. So we have to set B as 0. So we had this second-order differential equation. There's two constants of integration for every sort of mode. And this will be one of our boundary conditions. So physically nothing's coming into this past horizon. And then the other boundary condition will be the one that fixes what we'll do on this boundary here. So now following that through back into real space, that relates this data to this data in some way that you can compute. So in particular, thinking about F again, just to be somewhat explicit, I think it's something like this. The form here is really not terribly important. I'm just trying to convince you that it's something you can calculate. So instead of having an arbitrary coefficient here, this coefficient is now determined in terms of this coefficient. That's just putting this condition into what we had earlier. And then again, going back to real space, so Fourier transforming back to real space, or inverse Fourier transforming back to real space, you have to inverse Fourier transform this. And when you do that, that gives you the relation between this and this. And it tells you that phi d, what I've called phi d tilde, let me, I put a tilde on it here. Let me not put tilde's on it. Otherwise I'll forget them everywhere. I don't know why I did now, I think about it. When you inverse Fourier transform, this data is determined by this leading data integrated against some kernel, which looks like this. Again, for the physicists in the audience, there are issues about, you have to think a little bit carefully when you're in Lorentzian space about precisely how this is defined and so on, retarded, Green's functions advanced, and so on and so forth. Anyway, let's think from a Euclidean point of view where everything is very straightforward. But for the mathematicians, don't listen to that, please. Now, what we're now going to be interested in is the action evaluated on our solution. We've now solved this problem with the physical boundary conditions we want. We're now going to solve, well, we've solved it. We are now going to be interested in the action for reasons that will be apparent when we discuss ADSEFT. So the scalar action, and let's call it onshell, by which I mean evaluated on our solution. Onshell is a sort of physics term. So let's write our action. This is now an action in the d plus 1 dimensional space. Let's integrate by parts. Let's imagine we don't take the full space. Well, we need to think about what happens at the conformal boundary, and also what happens at these null surfaces here and here. So let's imagine just taking the action evaluated in some region, M, and then we'll try and take M to be the full space. So if I integrate by parts, the action, I get this. And because we're evaluating it on a solution, this is just the equation of motion. So this whole term drops out. So the action evaluated on a solution is just given for the scalar field by boundary term. So all we have to do is now evaluate this boundary term on our solution and see what we get. So let's define i of z to be this. In fact, I think I've defined it. So explicitly, this is just an integral over the d coordinates, the x coordinates, d z, phi, where phi is this function of x and y that we've computed. And now let's think about what it does at the various boundaries. Now, the first point is that the behavior of the scalar at z goes to infinity, we see here goes like z to the d on 2 minus 1 half. So it goes like z to the d minus 1 over 2. And if you plug that in here, that should vanish as you take z to infinity. So it turns out there's no contribution from these null surfaces to the action. So the action, in fact, will just be determined by what happens near the conformal boundary. And what happens there? Well, let's plug in our solution, which is this. Let me leave that up. As z goes to zero, i of z, if you plug things in, I mean, phi is given in terms of some power series expansion. And so this i, this boundary term that's going to determine the action, or in fact, it is just the action, but it, well, it's evaluated at some z. We're looking at how it depends on z. So this is at some specific value of z provided we're near enough to the boundary. So the asymptotic expansion is good. It will take this form. And then we see it's disastrous, complete disaster. And it's complete disaster because delta is greater than d over 2. And so this is negative, this is negative power of z. And we're interested in z, trying to take z towards the boundary, z is zero. It's a complete disaster. So you get a whole load of, depending on the dimension, there'll be some number of horribly divergent terms, and then there'll be finite terms. But these are all divergent. But there's something very interesting about the structure of how awful it is, which saves us. And the point is, the divergences, it turns out, come entirely from this part of the expansion, which doesn't care or doesn't know about this data here. And it doesn't, yes, so these divergences, whilst they're awful, they depend only on phi zero, not this other bit of data, phi d. So the basic idea, why not get rid of them? We're going to fix, we want to fix phi zero. It's like a Dirichlet boundary condition. We get to fix it. We know what it is. Okay, we get some divergences. But let's just subtract them off. We know before we even solve our equation what the divergences are going to look like. They don't depend on the solution, they just depend on the data. So we'll just remove them. And what we're then left with is some finite stuff, which does depend on the actual solution. And in particular, the boundary conditions we've imposed elsewhere on our equation. So let me just, in the remaining, let me have about two minutes, let me just say this and then we're pretty much done for today. So what we do, so we remove the divergences, how do we do it? We define what we call the renormalized action. The renormalized action is the limit, as epsilon goes to zero, of this action where we cut off our space at z is epsilon. So we're going to take m to be ads up to or down to, maybe I should say, z is epsilon. So we're not going to include right down to z is zero, we're just going to cut off the space. So it's got a boundary at z is epsilon. So we'll take this to be our usual action, or let's call it this i of epsilon. And then we add to that some, something we call counter terms, which are going to remove the divergences. We've got to fix our boundary condition to say what we're going to do, and we're going to fix our boundary condition in the following way. In this, in here, before we take the limit, we're going to fix our boundary condition by saying that the scalar field must behave like some fixed function times some appropriate power of epsilon. Okay, and this basically fixes in the limit, it fixes j to be phi zero. That's all we're saying. So we're basically fixing phi zero. But we don't, you know, if we want a good boundary condition, we don't want to talk about, we want to phrase it in terms of the field itself, not some asymptotic expansion. So phrase in terms of the field, we impose this in here. So this will be a functional of j, and then some unknown stuff. We evaluate this in this cutoff. And these terms here take the form of local functions. I realize, just see. I think I've, one of my pages has gone astray. The disaster, here it is. Apologies. So these are some terms with some specific coefficients. These are local functionals of, local and intrinsic functionals of the scalar field. Again, on the surface at constant z. So these will involve z derivatives. For example, they may, they may be higher. You know, there's some finite number of terms here. They may involve derivatives of z, sorry, of phi, but only in the intrinsic directions of this slice. Okay, so think of projecting the field onto the constant z slice, and then you can write down any sort of local functional of that field involving derivatives living within the z equals zero, the z equals constant slice. So this, and if you arrange these constants correctly, what they basically do is remove these terms. And you're left. There's just a finite number of them in any given dimension. You're left essentially with this bit. And, sorry, this is the very last thing I'm going to write down. And so in the end, when you, when you go through that procedure, you pick out just this one term, and the answer is then that this renormalized action, let me just call it this, doesn't really matter. Coefficient is, you could think of this as phi naught. I've just now called it J. This is the unknown bit of data. You know, basically you pick out this term. There's some minor subtlety when you actually go through the calculation, but pretty much you pick up that term. And given what we said, we can think of this. We know what this is in terms of phi naught or J equivalently, so I can write this like this. And that's where we're going to end today. And then we're going to do stuff with that tomorrow. I'm going to, I have got exactly enough to now tell you exactly how the, the ADSEFT correspondence works. How to map data from the CFT to the, to the gravity theory. But we've now, yes, so we've evaluated our scalar field. We've got a terribly divergent action, but it was divergent in a sort of trivial way that only depended on our boundary data. And there's a nice way to remove that data, just by adding some terms that live only on the boundary. So they're essentially extra boundary terms. They don't change the equations of motion. We've added some boundary terms that are constructed to remove the divergences, and we now get this finite action. That's called holographic renormalization. There are nice lectures on it you can find if you want more information. That's enough for now.