 Today will be much less ranty and less physicsy, well, somewhat less physicsy. There'll be an increasing amount of geometry, hopefully. So last time I told you about what ADSEFT is, it's a conjectured correspondence or duality of physical equivalence between, on the one hand, certain quantum field theories, which are theories, as we discussed, that live on some fixed spacetime and a quantum mechanical. And on the other side of the correspondence, certain quantum theories of gravity we could call them. But really, they're quantum theories where space and time is dynamic. That's what one might most generally call a gravitational theory. And there are conjectured specific examples where you know what both the theories on both sides are. And there's a lot of evidence, in particular for this canonical example, this n equals 4 super Yang-Mills we talked about. There's a lot of evidence that this conjecture is true. It's still a conjecture. And as I said, it's very unlikely to be proven. Because to prove it, you would need to know what both sides look like. And understanding quantum field theories at a rigorous level of proof is already very hard for free field theories, let alone interacting theories. So I don't think it's reasonable to expect this to be proven. And as I said before, if you believe it's not true, I think by now you probably need to have some rather specific reason you believe it not to be true. However, that said, it's still important to stress. This isn't known to be fact yet. So I talked about relativistic quantum field theories, which maybe to many of you is very familiar and to many of you is a little bit alien, probably half and half. Today I'm going to tell you what a CFT is. And given what we've said before, I think we can go reasonably quickly. Conformal field theory is a huge subject. It's basically a statement. A conformal field theory is a quantum field theory with some additional symmetries beyond relativistic invariance of the Poincaré symmetry. It's a huge subject, particularly in low dimensions. And we won't be interested in low dimensions by which I mean one plus one theories where all sorts of beautiful things happen and will need very little from the general story. So I will only say things about what we'll actually need in order to define ADS CFT. So we'll start off by talking about CFTs. And hopefully we will also discuss ADS, which how many of you are familiar with ADS spacetime? How many aren't? OK, so hopefully we'll be able to discuss both of these things today. And then I'm hoping tomorrow I'll be able to state, actually, in detail how the correspondence works. We'll see. So let's just recap. In the usual coordinates on Minkowski, we have the Lorentz and translation symmetries of the Poincaré group, we can think of as being generated in this way. And we are going to now consider a theory that is not only invariant under those, but it also has a scale invariance. So we'll also require our QFT to have scale invariance. And scale invariance is an invariance of Minkowski spacetime, again as with the Poincaré group, where I just scale all my coordinates by a constant, time and space. So that's a scale transformation. And our fields that we build our theory from, if we want our theory to have a scale invariance, we should have that the action is scale invariant, and that we should have our fields have some definite transformation under this. And we require our fields to transform in the following way. So maybe there's other indices on the field that some tensor field, but let's suppress those. So the field transforms as an invariant as a scalar quantity up to some overall scaling, where delta is some constant associated to the field, and delta is called the scale dimension of phi. And remember, we've taken mass units. So we've taken h bar equals c equals 1, and we're thinking of all the units being given now in terms of mass. And this scale dimension, at least classically, is just the mass dimension of the field, if you have a little think about how this works. So classically, it's just the mass dimension. And the generator for this transformation, we call the dilatation or dilatation operator d. And it looks like this. So for an infinitesimal scale transformation, the infinitesimal shift in my field is governed by this acting on the field. This is how the field transforms under this transformation. So lambda is a constant. Yeah, I think we're butting up against physics versus maths in a very fundamental and deep way. And I'm not sure at this point, I'm not sure I can raise this to a mathematical level on the fly that's going to be useful and correct. So we can think, let's see, how do I think about this? So I can think of this as some diffeomorphism equivalently. And then this operator here under this diffeomorphism would be the Lie, I mean this diffeomorphism is generated by some vector field. And this is the lead derivative of this field contracted with that generating vector field. So that probably still wasn't high enough level. So then together, remember we already have our generators for the Poincaré group, which are the translation and Lorentz generators. And this new generator has some definite, I'm going to Well, it has some definite trans, we can now add it to our algebra, this is now our algebra. These are all the commutation relations of the Lie algebra together with the Poincaré algebra that we would require our generators for this scale invariance to satisfy. And just something to note, now M squared, which was a Casimir of the Poincaré group is no longer commutes with everything, so no longer a Casimir. And so in particular, you can't have a field with a mass as we're used to in relativistic field theory no longer is a good representation under this larger Poincaré plus scale invariance group. And in a scale invariant theory, all couplings in the action should be dimensionless, i.e. that the action should have no mass scales within it, because obviously those mass scales won't be scale invariant. So action should have scales. So a theory that's scale invariant is of this marginal type that's truly scale invariant is of this marginal type that we wrote down before. And in particular, roughly speaking, it will look the same at every scale. So it's a very special theory that's guaranteed to exist at high energies, at low energies, everywhere you like. So these are complete theories. These are good theories in the sense of if you want a fundamental theory that describes, that works up to arbitrarily high energy scales or short distance scales, these types of theories would be in that class. You might wonder whether there's a limit on what delta could be. Delta should be real and greater than some number that's greater than zero. And for those of you who know about quantum field theory, there are unitarity bounds given the, we might see this later, but depending on the spin of this field, there's some different number here that depends on the number of dimensions you're in in the spin of the field. But in particular, delta had to be greater than zero for unitarity reasons. Yeah, was there a question? Sorry, yes, I've suppressed indices, as I said. So these could be some spinner, tensor, fields. Yes, generally, generally. That's right. So indeed, for example, the Dirac field transforms in some analogous way. OK, let's have some examples then. So the free scalar that we've discussed, so I'm suppressing the Lorentz index there, this is a scale invariant, I'm sorry, of course, what am I doing? The free scalar, firstly, I can't have a mass term. I should have put a cross through that. I'm not allowed a mass term, because that will obviously break scale invariance. But if it's a massless free scalar, that is scale invariant provided my field has its usual dimension, delta is d minus 2 over 2. So provided delta is d minus 2 over 2, this action will be invariant under the scale transformation. It's certainly invariant under Poincare, as you all know, but it's invariant under scale transformation. The reason being that under a scale transformation, this will pick up d factors of lambda. The derivatives are inverse d by dx, so they will give me 1 over lambda squared. And so I'll require my field to transform with the appropriate numbers of powers of lambda to cancel all of that and leave the action invariant. Yes, that's at the quantum level, yeah, exactly. Yes, I mean, don't worry about this. That's for people who we might say more about it later, but it's not important. Here's another example. Classical, say, Phi to the 4 or Yang-Mills in four dimensions has a scale invariance. So if I write down a theory like this, let's have some coupling, let's call it C. Normally you call it lambda, but I realize I've used lambda, so I won't. So this thing, again, for the same, this is now in four space dimensions, for the same delta here, so delta would be 1 in four dimensions, this is also scale invariant. So this interaction term. Remember, this is a free theory. The action is quadratic. It's sort of trivial theory, but this is now an interacting theory. Classically, if you solve the wave equation sort of in modes, the modes would all couple together through this nonlinear interaction here. This term is obviously scale invariant because the four lambdas you would get by transforming this under the scale transformation are then canceled by the inverse four you would get from the transformation of phi itself. You may be wondering what this has got to do with Yang-Mills theory, but Yang-Mills theory structurally looks very similar. If I write it out, non-Avelian Yang-Mills in terms of its vector potential, roughly speaking, you'll get a kinetic term and you'll get some interaction terms, one of which looks like this and the other one will scale the same way. So this is true classically. Now we can wonder quantum mechanically, it looks like these actions scale as they should. One of the key points that I didn't, I can't remember if I said yesterday, but let me reemphasize it if I didn't, is that just because an action has a symmetry classically, doesn't mean when you put it into a path integral and think about your quantum theory that that quantum theory will still have the symmetry. It's a very subtle business and it's a very subtle business because in the path integral you have this horrific integral over all field configurations that you have to do and to make sense of it you nearly always have to introduce some way of getting rid of the infinite number of degrees of freedom in the field that you're integrating over. And in particular if that, when you regulate your theory by, for example, saying I'm only going to integrate over configurations of my field that have a say a frequency or momentum less than a certain amount, if that regulator breaks the symmetry, when you remove it, it's far from clear that your theory will still have a symmetry. It's far from clear your theory will even exist in a nice way, but if it does, it's far from clear it will have the symmetry. And in particular this scale symmetry is all about scale and the whole point of introducing these cutoff, the regulator and then renormalizing the theory by removing the regulator is to kill off very, very short distance behavior that is causing the problems with the path integral in the first place. And so it manifestly breaks this nice scale symmetry that our action may have. And it's then a very subtle question when you remove your regulator, whether you recover a theory that's scale invariant. And in fact in this case, for the free theories, it's sort of, it all works and it's fine, but in any interacting situation, it's very subtle and these theories are not scale invariant. Okay, sort of famously. So Yang-Mell's theory in four dimensions is famously not scale invariant. The coupling is dimensionless, that's true, but when you regulate the theory, remove the regulator, renormalize and so on, you secretly pick up some scale dependence in the coupling. And this is what's called the running coupling. So an example of this, a very physical example is that in our theory of sort of hadronic physics, QCD, the theory of quarks and gluons, the coupling in that is indeed dimensionless, but as you look at processes at different scales, the effective coupling changes. It's a very subtle effect. So scale symmetry is not something trivial in, the quantum level is not something trivial in the action, it's far more subtle. Nonetheless, this example of N equals four super Yang-Mills, again, looks like some sort of Yang-Mills theory with some scalars and so on. So it sort of structurally looks a little bit like this, it's classically scale invariant. That is a theory that is quantum mechanically scale invariant, but it's not a triviality that it's by any means that it's scale invariant at the quantum level. Anyway, so let me just say scale invariance is often of an action, is often broken, but not always UV regulation and the subsequent renormalization. However, as I said, there are certainly theories that retain this scale invariance, and then it's an interesting fact that in all cases we know of, unless there have been very recent developments I'm unaware of, which there may have been, but certainly up to recently, in all cases we know of, when you have a Poincaré invariant theory that is also scale invariant, in fact, you inherit an additional, slightly larger symmetry group which is the conformal group. So you land up with a conformal group or a conformal field theory. So to my knowledge, there's no proof of this. Well, there are proofs of it which require some assumptions about the theory. So I think that it's not clear that what the most general proof of this would be, but there aren't sort of relevant examples where you have Poincaré and scale, but not this slightly larger group. And what is the slightly larger group? There's an additional generator of this conformal algebra or conformal group, which is pretty odd and looks like this. It's called the special conformal transformation. I'm not sure why it's special, I should say, but maybe when you see it, you'll see. So again, using the usual Minkowski coordinates, the, it would be generated by a diffeomorphism that looks like this. Okay, so, okay, so where A is some D vector, so A squared is it contracted with itself and so on. So it's a rather strange action on Minkowski space. It is, if you're brave and plug this into the usual Minkowski metric, you can check that it leaves the metric invariant, the form of the metric invariant, and then you can look at its generator which I won't write down. It's not particularly illuminating, but once you have the generator, you can compute the algebra of its generator which we'll call K with the other generators we've written down and I won't write out it in gory detail. I'll just be schematic. This is the usual Minkowski metric and then there's some other terms. There's another term with a different index structure. And so there's some commutation relations with the other generators. Oh, I'm sorry, this should be, this should be mu. But anyway, I mean, I haven't, you know, it doesn't really matter the precise form here because I'm gonna write it more nice, well, more nicely in a nicer way. The algebra is actually, well, the group is SO2D. And it's not very obvious from the way I've written it down as it's been written down, you know, as we've sort of added generators, but if you repackage everything, it takes a much nicer form. So let's introduce indices AB, which are zero, one up to D plus one, where zero to D minus one were our usual spacetime mu nu indices. So we've got two more indices. And then if we write or make the following identifications, and you find that these are with, I should say J, so JAB is anti-symmetric, so the other components are determined. So these are now the generators of this SO2D in a much more familiar form. I'm sorry, I think I mean this. I'll leave it as an exercise for you to check the ordering of the signs here. So this looks like the usual rotation generators or Lorentz generators, but just with different numbers of minuses here. So that's why we have SO2D, we've got two minuses in this. So this is the group that leaves this invariant. So the conformal group, so if we have Poincaré and scale symmetry, we always land up with a theory that has a larger symmetry group, this conformal group, and it's SO2D. That's the important point. Now, obviously when a theory has symmetry is it has consequences, physical consequences. Let's just recap some of the physical consequences of the Poincaré symmetry for vacuum correlators. So we talked about how correlation functions can be computed from this path integral generating functional for them, but these are important. Quantities remember they look like, these are examples of correlation functions. They tell you in some sense about the behavior of your theory. This in some sense tells you what a field is doing at a given point. This object is telling you about how a field propagates from one position in space time to another. If I have more phi's, for example, that would tell me about how particles in this theory scatter, at least for Poincaré invariant theories. And Poincaré invariants already tells us that if we're looking in the, for the field theory in its usual vacuum state and we look at these correlation functions, this must be a constant. It can't depend on space and time and that just trivially comes from translation invariance of the vacuum. And this quantity here, well, whatever it is, it's only a function of the difference in the space time positions. And in fact, it's only a function, I should say, of the norm of the difference in the space time positions because of translation invariance and because of Lorentz. So already, that's quite constraining, but scale invariance now constrains things further or rather, well, so for a CFT and not just a relativistic field theory where these would be true, it constrains things a little further. This has to vanish. And that's basically because if the field has any scale and in vacuum, this was non-zero but constant, this would have some scale and it would break scale invariance explicitly. So the vacuum's not allowed to break scale invariance in a scale invariant theory and so this has to vanish. Note that if the dimension of this was zero, this wouldn't be true, but for reasons of unitarity that we said before, that's not allowed. There's a lower bound, deltas should be positive. Yeah, no, no, no, no, it's under the whole thing. This, yeah, the special transformations as well, but at least as of, I'm not quite sure what the state of the art is, but there aren't, I don't know of any examples of theories which are not, which are invariant on the scale, but not invariant under this extra symmetry. So, but in fact, some of what we're gonna say doesn't depend on the whole conformal group, it actually only just depends on the scale part. Anyway, the form of the two-point function, which is perhaps a little bit more interesting physically, is rather constrained. You can sort of understand intuitively why this roughly the scale of two phi's is, well, so this form here, which you relativistic symmetry basically just told you it was a function of the norm of X minus Y is now entirely constrained to a particular power of that norm. So I should say this is the norm of X minus Y to the two delta where C is some constant, which of course you can set to one by just redefining your field appropriately. And in fact, more generally, if I have, if you have fields, I'm now gonna put some indices on these, let's say A with scale dimension delta A and another field B with scale dimension delta B, the two-point function or rather, let me just say for some, I'm being silly for some A's. Let's assume I've got a few fields and I've labeled them with some index A. It's more sensible. And as a corresponding scale dimension delta A, then if I look at the two-point function of two of these things, it's relatively easy to see that it will vanish if the dimensions are not the same. And it will take a form like this if they are the same. So this is nice. So conformal field theories are nice because the propagation of particles is then completely fixed. Well, I should be more careful. Because these particles don't have a mass, it's very subtle whether one can't really interpret them. Conformal field theories, if they're interacting, always have very long range interactions and it's rather subtle. The normal manipulations you go through to convert a correlation function like this to a statement about propagation of particles don't really work anymore. But nonetheless, these correlators exist and they tell us roughly about how if you disturb the field in one place, information propagates physically through excitation of the whatever it is that however you want to think about what this field is doing. It's also true that the three-point function is constrained up to some constants. So it's functional form. So if I have some fields again like this, A, B and C, I look at the three-point function which morally speaking, although not for a conformal field theory, but morally speaking in a normal relativistic theory would tell us something about scattering. So this form is fixed in terms of X, Y and Z. I won't write it down because it won't be relevant to us. Higher point functions and I'm not constrained. They're in the sense that they have non-trivial dependence that you don't know a priori on the positions of where the fields are living. So they are constrained but they're not constrained enough to know explicitly all of this. Yeah, I suppose maybe it's, I mean, I'm not sure what it is. I don't know if I can give you a deep reason. Maybe think of it like this. If I think of the correlation function of two fields in relativistic theory with different spin and I think of the two-point function. If they've got the same spin, they can have a non-trivial two-point function. If they have different spin, they can't. Why? Well, it's because of the structure of the vacuum. Well, sorry, they, sorry, I should be more careful. They may be able to have a non-trivial correlator but they may not depending on the spins. I should be more careful there. But yeah, so, I mean, I could show you in more detail after how this works but it's not, you have to use properties of the vacuum being invariant under deletions and Poincaré. Let me see. Lastly, every CFT, well, if we, every quantum field theory, we expect to have a, we expect to be able to take our theory, put it on a curved space and in doing so, we can vary the metric and generate a stress tensor in the usual way. It's one way to think about the stress tensor for the theory which tells us then putting it back, putting the metric back to Minkowski that will give us our Minkowski stress tensor. And because of this structure, we see that the stress tensor itself must always have dimension D, okay? Because the action's obviously dimensionless, there's D, X is in the measure, the metric's dimensionless in my conventions here. And so the stress tensor has to have dimension D and that's always true. And the stress tensor can have some non-trivial two-point function which again takes a similar form. I should have said, what I should have said here is these really are scalar fields as I've written things. There are analogous formulae when you have indices say tensor indices, but now this object here will be, again determined by some constant but will also have some tensor structure, obviously. Let me give you an example of that here. The two-point function of the stress tensor always takes this form, but obviously we've got some index structure which has been suppressed there which comes out in some object here which I build from, I won't write it explicitly, it's not terribly illuminating, but I build from these objects which are scale invariant. So the metric and also the combination X mu X nu over X squared which is scale invariant. And so this is a slightly complicated expression in terms of these things but it's not difficult to write down. And this quantity here in a conformal field theory is usually, well it's often called the effective central charge, it doesn't matter in two dimensions, it is a central charge for the conformal algebra but in more than two dimensions it's by analogy it's called this but you can think of it as counting the degrees of freedom. This is roughly telling us how energy propagates from one place to another and roughly speaking the more fields you have around to propagate the more energy can propagate. But in particular let me just, because this may be relevant if we have time later, you can calculate what this is in this theory N equals four super Yang-Mills, this example we had before, and it's equal to some number which in fact is independent of the coupling interestingly although that's not usually the case. So it only depends on N, the rank of the gauge group which indeed counts the number of degrees of freedom in your theory, the number of fields or elementary excitations you could think of. But it's some number, yeah, was there a question? Yeah, well it doesn't matter. What is the dependence? It's something like, I'm not sure I can reproduce it on the fly, it's some combination of these objects. Well no, it's some, I'm not sure, there's a, this has got various symmetries, it's symmetric in these, it's symmetric in these. This is also a conserved tensor, one has to build in conservation, so imagine the most general set of terms you could write down using this sort of object and this sort of object that are compatible with symmetry under these two indices and the whole thing will be conserved if I act with a derivative of mu or nu or alpha or beta. And there's only one thing of that sort and I'll leave you to look it up or compute it yourself, it's not very hard, but it's not illuminating. Was there another question? No, okay. Good, so, and the last thing then I want to say about conformal field theories and then you're all experts or at least you know enough to carry on with what we need for ADS CFT is that in our path integral, remember we wrote down this generating functional for correlation functions, which if I functionally differentiate it will give me these correlation functions and schematically again, there may be many fields, I've just, I'm just calling them all if I'm suppressing indices, Lorentz indices and there may be many, obviously there are many local observables that I could write down. Let me just write one, but you could imagine lots of them and remember we call, so this is some local functional of the fields and in this we call a source for whatever operator it is and then we think of this as a functional of J and really because there are lots of different local operators, really this is a function of lots of different Js, one for each local operator I could write down and then if I functionally differentiate with respect to J appropriately I and then set the source to zero, I generate these sort of correlation functions we were talking about, but the only thing I want to say here is that in a conformal field theory because this will have some dimension, let's call it delta O, the source because the, when I add these sources the new action with sources should still be scale-imbarant otherwise I've explicitly broken scale-imbarant, the source should also have a transformation on the scale so that let's say the scale dimension of the source, the way it transforms will be correlated so it'll go like D minus the dimension of the operator so that this whole thing is scale-imbarant and that will again be important later and then the very last thing I want to say but I won't dwell on this is that I've really talked about a conformal field theory on a flat spacetime, on Minkowski spacetime just as with any, just as with a Poincaré invariant relativistic theory on Minkowski you can then put it on a curved spacetime but this, if you're on a curved spacetime you've manifestly broken Poincaré invariance but the action is still constrained by the Poincaré invariance although there are potentially new terms you can add to your action that will have vanished in Minkowski terms depending on curvatures, in the same sense you can promote a conformal field theory to a curved spacetime and then the theory usually will then have a symmetry under vial transformation or scale transformation of the metric where I change the metric by some but this is a bit like a scale if this was Minkowski this would look like a scale transformation but more generally for any space I can do a local scale transformation but like this, a vial transformation and if I take my CFT action and covariantize it appropriately and I can usually ensure that the theory will have some invariance or at least covariance under transformations like this it won't be important for what we have to say but it's just worth pointing out okay, so now that's enough so many of the things that I've written on this board or these boards we will see coming up later in very different contexts so do bear certain things in mind at least the form of these two-point functions for example, just keep this in your mind okay, that this goes like this we'll definitely see this within a lecture or so in a completely different context and it'll be very relevant, yep oh, I see, sorry, yes thank you good, I mean of course so now let's talk about the second part of ADS CFT the other bit of jargon which is ADS so many of you will know about ADS space time or have seen aspects of it so let me, but many of you won't so it's a peculiar space I'm gonna consider D plus one dimensional ADS so we were talking about D dimensional field theory this will be D plus one dimensional ADS and it embeds as a hyperboloid which I'm gonna take to have some radius let's say L in a sort of two-timed Minkowski space I'm not sure what to call it, R2D as so if I take these coordinates on this two-timed Minkowski space then ADS can be embedded so that the induced metric is the correct ADS metric in the following way as this hyper surface so just note that whilst the ambient space that we're gonna embed the metric in has two times because of the way the embedding works the induced metric will only, well, will be Lorenzian so note constant Xi are time-like closed curves in this embedding so if I sit at constant X I can still move essentially around this circle parameterized by U and V and if I ask what's the proper distance I move or the proper displacement I suppose I actually move a proper time it's a time-like direction so I'm moving, I mean a hyperboloid obviously looks something like this but I'm not sure I can imagine the two-timed directions in a nice way but anyway the point is you're moving around the hyperboloid but both the directions you're moving in a time-like so this is a closed time-like curve and so ADS is not this embedding alone it's the covering space of this so ADS, it can be embedded as that but ADS is the, I'll use the word universal cover but I'm not sophisticated enough as a geometer to really know what that means so if you know what that means that's great I'll be more pedestrian it's the universal cover of the hyperboloid as physicists we can be more explicit I parametrize the hyperboloid like this these thetas parametrize a D minus one sphere a unit and D minus one sphere you can verify that if you plug that into there using basic trigonometry it satisfies the embedding condition and furthermore covers all points on that hyperboloid and if we were just covering the hyperboloid tau would be an angle it would be the angle around this hyperboloid but because we actually want a space that doesn't have closed time-like curves ADS space time, for ADS space time we think of tau being a real, not an angle so as tau goes around I wind round and round my hyperboloid infinitely many times and the induced metric if you now substitute this into here to compute the induced metric so the metric and this is a global chart usually what's called the global chart of ADS looks like the following so it's got a simple form we can also write it in a schwarzschild-like form where this function f looks like this so you see the relation between rho and r is just some simple relation like this and you see that when rho or r is small it's an origin of spherical coordinates but when r becomes large this deviates from flat space this term becoming dominant over the usual one here that is all I would have in flat space reflecting the asymptotics being different now what are the isometries of this space the embedding we've used allows us to see the isometries immediately and they're SO2D they're obviously SO2D the isometries are just the equivalent of the Lorentz group for this two-time metric is SO2D not sure what you call it if there are two times but two, well, anyway SO2D and it also leaves this this is the inner product of two vectors if you like and that's left invariant again by this so it leaves the embedding invariant and therefore these will be this is the group of isometries of ADS it's not manifest, you know, this larger this group isn't manifest obviously in this particular chart which manifests only some rotations SOD-1 and an SO2 if you like or just translation symmetry in tau but from this we can see that there is a larger isometry group hiding now let's try and understand the asymptotics so as rho or R equivalently goes to infinity this metric is dominated by an exponential blow-up of the time and angular pieces like this so as rho becomes very large you can see the space is becoming very big in these directions and rho goes to infinity okay so this becomes obviously arbitrarily large now what what does that mean what sort of asymptotics is this well this is a space which is conformally compact so meaning there's an asymptotic region which can be compactified in the following sense so a conformally oh by the way the word conformal here is not the same conform I mean in geometry this is called a conformally compact space but it's not really related to conformal transformations well directly there is some relation but it's a slightly confusing terminology I mean or rather it's a fine terminology but don't be confused by the competing terminology as I should say so a conformally compact space can be written as in the following way so we have a we have a metric here a regular metric a manifold M with boundary that encloses M and Z is what's called the defining function and so Z is greater than zero inside M and vanishes on the boundary of M linearly let's say vanishes linearly i.e. DZ is not equal to zero on the boundary of M and such a space so G is a perfectly regular metric with a real boundary and the full space time and the full space time then has an asymptotic region due to this defining function one over it blowing up but it blows up in a sort of controlled way so the full space time has what's called a conformal boundary so and it lives let's say on M minus its boundary so the boundary now is some asymptotic regime region so the full space doesn't have a boundary anymore you can only think about its asymptotic behavior we say it's got a conformal boundary although there's no boundary at all it's a boundary only in the sense that you can conformally by multiplying appropriately you can then turn it into a real boundary of some other metric space space time and we say it has a conformal boundary metric which is equal to that induced on the boundary of this regular space by this regular metric okay so this regular space has some real boundary with a real induced metric on it and we say that's the conformal boundary metric of this full space really there's no boundary at some asymptotic region but there's some notion of it having a boundary you know having some metric that describes that that asymptotic region but and it's not going to be relevant for us but obviously it's inherited what I've said I have a freedom in how I can formally compactify a space time that has a conformal boundary if you give one positive smooth function on this space and recover a new defining function with it you know with a different regular metric so this whole construction is only defined up to multiplying this defining function by some positive now strictly positive function and therefore the conformal boundary metric is defined only up to a vial transform so again it won't play much role in what we're going to do but it's just worth noting and now we can see let me see I started at two fifteen right so ten minutes now we can see what this is okay this is a conformal boundary this is a conformally compact space and the conformal boundary metric at least a representative for it i.e. it's only defined up to vial transformation so we talk about conformal classes of the boundary metric as in the equivalent elements within the class being generated by vial transformations so this thing that the conformal boundary metric of this is actually just or a representative of it is just this it's just minus d tau squared plus d omega squared so let me do that more explicitly so for ADS over there if we take e to the rho e to the rho equals one over z then asymptotically just asymptotically we could we see the metric tends to this form and this I can write as one over let's call it big z squared this is our defining function it certainly goes to zero in a nice this is a regular metric I can say take z to be from zero to infinity and then or in fact I don't need sorry I don't even need to so I'm only doing an analysis locally anyway but z has a boundary at zero is what I want to say and extends some to some positive value here this defining function indeed goes to zero at the boundary and is positive away from it so this is a conformal boundary and so the conformal boundary metric let me write it as the the boundary of let's call it this here this space m and write it as the boundary of m even though it's not a boundary it's a conformal boundary let me just write it like that so this is what people would usually call the Einstein static universe but it's basically just time cross a sphere around sphere so that's the boundary of this ADS spacetime and we can then draw a nice picture of it in our picture we've conformally compactified it so we're sort of drawing a picture of it in this conformally compactified sense we should think of it as an infinite cylinder, solid cylinder where boundary of this cylinder so the ADS space is the interior the boundary of this cylinder if you like in this regular space is what will give you the conformal boundary of the full spacetime so this is the conformal boundary it's not a real boundary remember it's an asymptotic region the ADS spacetime lives in the interior and our tau coordinate goes up angular coordinates go around and if you like our radial coordinate as we've drawn it there's a sort of center of spherical symmetry and then our radial coordinate goes out and reaches infinity of this conformal boundary tends to infinity there and one of the key points that's very easy I mean it's a simple calculation one of the key points about ADS is that whilst this is an asymptotic region this conformal boundary is time like the Lorentzian manifold and in particular one can ask supposing I send a null ray in my space out towards the boundary does it reach it and in fact as I've drawn things this isn't really a conformal diagram I mean it's not a Penrose diagram but morally speaking light rays would travel at 45 degrees in this diagram so in particular light rays really do reach the boundary if I emit them from anywhere in the interior with my flashlight they travel out and they meet they hit the boundary at a finite time tau and that's a key physical point about ADS space it's actually not true if you take a space like curve and take it out to the boundary it's an infinite proper distance very easy to see that but a null a null geodesic travels out to the boundary and hits it in a finite time tau the key physical point there is that ADS acts like a box like a real physical box so even though this isn't a real boundary it's a conformal boundary it's not some asymptotic region actually for null rays they really get to the boundary in a finite proper time for observers in the interior so imagine someone sitting arbitrarily far out you know in a finite time in the interior you could send a message arbitrarily far out get your friend to send it back to you and you'll reach, you know, you'll recover it in a finite proper time so what that means physically is if I want to define dynamics in this space I must impose some sort of boundary conditions out here you know all my fields, all my dynamics high energy modes, short distance wave mode or you know high frequency wave modes will propagate out to the boundary and I'll have to deal with the boundary when I deal with dynamics so I can't ignore it and in some sense I will expect stuff to go out and with appropriate boundary conditions it may come back at me okay so it really acts like a box yeah there's a couple of questions, yeah if I've got a time like curve here could be a geodesic oh this is some null ray I just mean it will reach the boundary at some in the conformal boundary at some finite tau and then or if you like it will go what do I mean or sorry how do I answer the question shall I say so I think it is a I actually can't remember if it's a finite affine time along the curve you can do the calculation and tell me but it's not the important point and I think it's not a finite affine time along it's not a finite affine parameter along the curve and for the physicists that's because the affine parameter basically tells you about the blue shift or red shift of a photon if you like which becomes infinite so it isn't a finite affine parameter but nonetheless it will reach in a finite time tau and therefore someone arbitrarily far away can catch this before it escapes and send it back and for me in a finite time tau I will then see that see some information returned sorry was there another question okay oh yeah maybe I I'm not sure I'm not sure what you mean it's not a physical boundary no it's an asymptotic region it's an infinite proper distance spatially if I if I send a spatial curve out I mean you can just you know do the do the calculation for for the metric I gave you if you if you look at a curve that's on a constant time slice and ask how long is it it's got a proper an infinite proper distance length to the to infinity yeah so it's it's really not a not a not a boundary in the usual sense of boundary but the important point is it is a boundary for null rays um let me just finally in the one last one minute write down another uh chart which we will be using really or this is the way we'll think about ADS and it's the Poincaré chart and in terms of this embedding if I did everything right but you can go away and check there's a rather complicated way of parameterizing not all of the hyperboloid but half of it using a different parameterization than we had before so this is only covers half the hyperboloid and what half is it or well or rather um there's a there's a complicated map when you think about what portion of this space you're covering so you only cover half of the hyperboloid but remember ADS is the cover of the hyperboloid so what bit of this solid cylinder are we covering and basically we're taking a null wedges I think this is the way to think about it you take a null wedge that intersects the boundary or slices through this cylinder and then take another null wedge that just intersects the boundary here at the same point and basically the interior of this wedge is what's covered by this and so if we thought about a conformal diagram for what these what we're covering really there's a boundary or a conformal boundary and then we've got some Cauchy horizons here and here associated to the these null surfaces but the induced metric now takes a very simple form and just to be clear here A ran from one up to D minus one and now you know mu is our usual zero one up to D minus one okay so this is the metric of ADS in this Poincaré chart it doesn't cover it's not geodesically complete obviously I can extend it through these Cauchy horizons but note I'll say this again next time this manifests some interesting isometries in particular there's a Poincaré-like isometry which is the usual one associated to these coordinates this is the Minkowski metric so clearly if I do a Poincaré transformation on these X's I generate an isometry of this metric and there's also an interesting one here where I scale my Z coordinate and the X's in the same way okay that's obviously also an isometry it leaves this metric invariant but think about the conformal boundary now Z is a defining function in its own right now what's the conformal boundary it's at Z is zero it's just the Minkowski metric and these isometries therefore have an action on the conformal boundary and what's the action on the conformal boundary it's Poincaré and scale and in fact whilst this doesn't manifest this special conformal transformation that's also there it's just not manifest in these coordinates more complicated isometry in these coordinates is something nasty so as we'll say next you know as I'll just re-emphasize the beginning of next time we see that the isometries of ADS have an action on the boundary that looks like it's the conformal group action okay so sorry I ran over