 I'll start with a quick review of what we had so far. And so we started in the first lecture by talking about the connection between entropy and area for black holes. We talked about the example of the Schwarzschild black hole. And we said that in ADS CFT, we could understand this connection between entropy and area by saying that the area of the Schwarzschild black hole was actually equal to the microscopic thermodynamic entropy of the CFT. And then we had a different way to understand that as entanglement entropy, so on the geometry side, there is a maximally extended version of the Schwarzschild black hole. Maldesina proposed that this maximally extended black hole could be understood as being dual to a specific purification of the thermal state of a CFT, known as the thermal field double state. So this is where you have two CFTs, and they're entangled together similar to this formula. And so in that case, the thermal entropy of the CFT is, we can think of that as the entropy of the single CFT subsystem, or equivalently, that's the entanglement entropy between the two CFTs. And then we moved on to say that if you now consider just a single CFT, what we understood is that even the vacuum state of a single CFT, so as an example, consider the CFT on a sphere, so even the vacuum state, which is a pure state of the single CFT, if we consider a subsystem of this, and specifically I want to consider a ball-shaped region on the sphere, so we consider the subsystem B and its complement, then there's an entanglement structure, which is just like what we had for the two CFT state describing the extended black hole. So this is now just the ordinary vacuum state, but we can still understand it as an entangled state between this ball and its complement. So the fact that it's entangled corresponds to the fact that here we're summing over some set of product states. So it's not a, even though it's the vacuum state, there's lots of entanglement across the surface. So specifically what this means in this case, so we can write it in a way that looks like this thermofield double state, but these energy eigenstates, they correspond to eigenstates, not of the full Hamiltonian on the sphere, but of some symmetry generator that generates, this is a symmetry of the CFT. It's associated with a conformal killing vector, so a time-like vector that acts within the domain of dependence of this ball-shaped region. So there's this time-like coordinate that I'll call theta generated by some Hamiltonian h, theta, and I'll tell you exactly what that looks like later on in this lecture. And so what we're talking about here is some set of eigenstates of that Hamiltonian. And over here, similarly, we can define states of this side, and those would be eigenstates of Hamiltonian generating a symmetry in this region, which is the domain of dependence of this complement of the ball, or this other ball, which is the complement. So this is all happening on the boundary. For now, we can ignore the middle of it. So the structure of the vacuum state, if we make this arbitrary choice, and we say this is one subsystem, this is the other, it looks just like the structure of this thermofield double that was dual to a black hole. And so we can carry over a lot of what we learned about the black hole. We can now think about this in the holographic context and say, if we make this choice and we split up the boundary into these two parts, we can think of the density matrices describing those two parts. So each of these are density matrices that look thermal. And we can say, well, the whole vacuum state is dual to this ADS. But now we can ask, what is just this density matrix dual to? What part of the geometry does this contain the information about? And so even though there's not, from the start, any obvious decomposition of the geometry, once we make this choice and focus on this region, we can, there's a naturally associated wedge of the full geometry that we could say is dual to this density matrix, or we can argue is dual to this density matrix. So this wedge is basically just the region of the geometry. One way to define it is the region of the geometry that I can communicate with. So I can start at some point in this domain of dependence region for B and send a signal and then receive the signal back. And so that defines some wedge of the geometry. It's actually also another way to understand it is it's a Rindler wedge of ADS. It's the region of ADS that if you have some accelerated observer that starts here and ends up here, so this is some constant acceleration trajectory, this is the region of the geometry that that observer can see or that can communicate with, can send signals to and receive signals back. So this is like a, it's like the ADS version of Rindler space. So OK, so it's natural to say that this density matrix for the region B, that's sort of, that's dual to the gravity in this wedge region. And similarly, the other density matrix is dual to the other region. And then I was saying that these third and fourth regions, well, they must be, the information about those must be contained in, must be the information in the state which is not contained in these two density matrices. So that information is in the details of how the two are entangled with each other. OK, and so finally, so all the similarity with the black hole case suggests that maybe the entropy also, so the entropy of the subsystem should have some relation to an area in the bulk geometry. And in the black hole case, that area was the area of the horizon, which is the area of some, so some spatial surface separating the two sides. So this is a space-time diagram. But if I just draw the space for the black hole space-time, then it was these two asymptotic regions connected by some wormhole. And the area is just the minimal surface, an extremal surface separating the two sides from each other. So the black hole entropy was just equal to the area of that surface divided by 4. And similarly here, then we might guess that the entropy of this subsystem would correspond to the area of this surface separating the two sides of ADS from each other. And in this case, we can actually calculate what this entropy is in some CFT. We use a cutoff to get a finite answer and compare it with this area. And one finds that indeed, this formula is true. And so then this motivates further generalization, which is by Ryu Takayanagi and a covariant version by Huberni-Rangamani and Takayanagi. And they said, now consider some more general state of the CFT. So some state psi, which is not the vacuum, but not the thermal state either, could be something more general, which is, we assume, is dual to some asymptotically ADS space-time. And now instead of the ball-shaped region there, consider some more general region. So this is the space that the CFT lives on. And A is the subset. And so again, we can consider the density matrix. So starting from the state psi, we can describe the subsystem A by a density matrix or by an ensemble. And in general, this will have some entropy. And so Ryu Takayanagi, the entropy of the subsystem, would correspond to the area of something in the bulk geometry. And their conjecture is that what you want to do is define a surface A tilde in the bulk. So we can think of this as living on some spatial slice of the bulk. But actually, in general, it's not clear which spatial slice. It's not clear initially what spatial slice you should pick, because now there might not be any kind of special symmetry or any special time fulliation. But what they say is you want to find a surface here, which is an extremum of the area functional. So I should just write that down. So the area functional is just this. So this spacetime M has some metric. And then it's the usual definition. You would pull back the metric to the surface and take the square root of the determinant, integrate. So explicitly, this is the functional we're talking about. And so you want to find a surface A, which divides the space into these two parts with boundaries A and A bar, where A tilde is an extremum of this functional. It might be that there are more than one. It might be that you find multiple extrema. In this case, the prescription is that you should choose the one with the least area. Generally speaking, you will just find one extrema. And so their conjecture is then that this is the surface whose area corresponds to the entropy. Yeah, yes. Right, so the question was, in order to check this in detail, do I need to know how field theory parameters are related to gravity parameters? So I mean, I guess there's two points of view. So I think the original point of view was just suppose you have some existing ADS-DFT, like n equals 4, and super Yang-Mills theory with large lambda. And we think this is dual to type 2B string theory on asymptotically ADS. And in these existing examples, we would have some specific relation between parameters of the field theory and then parameters of the gravity theory. And then on top of that, so this would then just be an extra thing. I mean, then we could just check this directly, because we already know, based on other calculations, how these parameters are related. If this is the only thing, I mean, another viewpoint that I'll take a little bit later on would be to say, just start with the CFT, and then you can ask. You can calculate entanglement entropies, and then ask, is there some geometric way to represent those? And then I guess in that case, you could do one particular calculation to fix this relation between G Newton and the field theory parameters. So say I calculate the entanglement entry for some specific ball. OK, then maybe I need to fix the coefficient using that one, but then I can check the infinite number of other possible regions and see if it works. I said it last time, but I'll say it again here. This is a little bit of a funny formula, because in the way that this is written, both sides are divergent quantities. So when you calculate entanglement entropy, I can think of a field theory as being regulated somehow, where you say as a limit of some discrete system, balls and springs or something. And in the regulated theories, you can always focus on some subsystem and calculate the entanglement entropy, and it's some finite quantity. But typically then when you take a limit where the regulator goes away, in quantum field theory, this entanglement entropy is divergent. In a 2D CFT, for example, in the vacuum state, if I compute the entanglement entropy of some interval, there's actually a universal result, and you get a logarithmic divergence. And then higher dimensions, typically, then you have a lot more choice of the shape of the region. So then there's a divergence which is proportional to the area of the boundary of A times, so area of the boundary of A times the appropriate power of the UV cutoff to make this dimensionless. So one way to interpret this formula is that you should calculate things with an explicit UV cutoff in the field theory and then compare them with quantities on the gravity side, which have an IR cutoff. So on the gravity side, the area here is infinite because it's an infinite distance to the boundary of ADS. And so you could try to work with a UV cutoff in the field theory and work with a cutoff in the gravity side and then compare things. I find it a little bit more sensible just to do what we normally do in field theory and try to define quantities which are actually finite. So a less ambiguous kind of comparison would be between not the entanglement entropy itself but other quantities derived from the entanglement entropy. So I'll mention a couple of things if you have various combinations of entanglement entropies, so the simplest being this combination. So we consider some state and the entanglement entropy of this subsystem, the entropy of this subsystem, minus the entropy of the combined subsystem. Then what we notice is the divergence here, which is proportional to the area of the boundary of this plus the divergence there, cancels with the divergence here because the boundary of A plus the boundary of B is the boundary of A union B. And so this quantity is typically finite for well-behaved field theory states. This is the mutual information. This is called the mutual information between A and B. So it's literally how much entanglement there is between A and the rest of the system plus the entanglement between B and the rest of the system minus the entanglement between the union of A and B and the rest of the system. So roughly speaking, what this picks out is some entanglement between degrees of freedom of A and degrees of freedom in B, entanglement or correlations. So that's one finite quantity. Another one is just to look at vacuum subtracted quantities, entanglement entropies. So if we're interested in some state psi, which is not the vacuum state, then you can calculate the entanglement entropy for some region but then subtract the entanglement entropy for the same region but in the vacuum state. And this quantity is also typically finite. So this is often what we would do with, say, the stress energy tensor in field theories. I already mentioned some of the evidence for this comes directly from calculations for ball-shaped regions in the vacuum state of a CFT. You can check that this agrees. In 2D CFTs, there are more detailed calculations that you can compare. You can do the calculation for intervals in thermal states or for multiple intervals and see that it's correct. Other pieces of evidence are that this definition of Ryu and Takinagi obeys various properties. It can be shown to obey various properties that entanglement entropies are known to obey. So for example, this mutual information, in quantum mechanics, you can show that it must be positive, not negative. And there are a list of similar inequalities. And so one can check. One can ask, given a reasonable spacetime, if you define the entanglement entropies using this formula, if this formula is true, do the entanglement attributes you calculate obey the various constraints. And so there are a number of non-trivial checks that have been done of that form. And I'll probably say more about this later. And then actually in the case where the spacetime is static, then there's actually, I mean, more or less a proof, maybe not a rigorous proof, but some fairly convincing argument by Lukowitz and Maldesina. And that starts with the idea there is that in the field theory, there's some way to calculate Renyi entropies. I think I mentioned in the discussion yesterday about some path integral way of calculating, of representing the density matrix. So there's some path integral way of calculating these Renyi entropies, or more specifically, trace of rho to the n as a quantity that one can give a path integral description of. So this thing, using the standard ADS-CFT correspondence, you could map this over to some gravitational path integral. And so we essentially know a correct answer for how to calculate what this corresponds to on the gravity side. And so starting from that connection, Maldesina and Lukowitz made an argument. So to relate these to entanglement entropies, you have to take some analytic continuation and take a limit where n goes to 1. So Maldesina and Lukowitz made a formal argument that using this ADS-CFT connection between the two path integrals and taking this limit, one can actually recover this result. And that's so far for the static cases. So I'm going to talk about some implications of this. One more point. So this formula as written, it's understood to be, so this is the CFT entanglement entropy. It's understood that this is a large n type formula. So if we have holographic theories, there's typically a parameter which is the number of degrees of freedom that's large. So it's just the rank of the gauge group in n equals four or the central charge in 2D CFTs. And so this CFT quantity here is understood to be, so it has some expansion in the inverse of the parameter, a one over n expansion. This area here is understood to be supposed to be equal to the leading large n, so the leading term in that expansion. And there are other, so other corrections have been discussed if you look at the sub leading terms in this one over n expansion of the CFT entropy, then they correspond to other things. So just mentioned specifically Faulkner, Lukowitz and Maldesena said that the next term, the first term is the area of this extremal surface and then that gets corrected by a term which is actually a bulk entanglement entropy. So a quantum correction in the bulk which is the entanglement of the perturbative bulk fields on one side of the entangling surface and on the other. But mostly I'm going to focus on this leading, this sort of large n limit of this expression and think about the consequences of that. Okay, so implications, okay. So the first one is that now that we have this connection between entropy and area, not just for simple states or the whole system but for arbitrary subsystems. So this gives us a huge number of relations between entanglement and between the geometry. And because we have now so many connections there, I think we can plausibly reconstruct, so given a state which has some holographic dual space time, we can plausibly reconstruct the geometry at least, at least the part covered by some of these extremal surfaces calculating entanglement entropy. So so far there's not a general formula for how to do this but you can imagine that prescription is that someone gives you a state of which they assure you has some gravity dual geometry. Okay, and now you want to figure out what geometry is encoded in this state. So you just go ahead and calculate the entanglement entropy in the field theory state for lots of different regions. So you think of all sorts of different regions and you calculate all these entanglement entropies. And then you just ask is there an asymptotically ADS space time where the extremal areas reproduce all your answers? So is there a space time where the areas of the extremal surface with boundary, the same as the boundary of A, matches with all these entanglement entropies that you've calculated? And the point is that this is a highly over constrained problem. So if we think about the space of functions, so you think about well how many possible regions are there that I can compute the entanglement entropy of? I mean there's some enormous space of these regions. This is subsets of Minkowski's Euclidean space or subsets of a sphere. The space of functions on that space of subsets is much larger, the space of asymptotically ADS metrics. Okay, these are functions of some small number of variables, so this is a handful of functions of some small number of variables. This is a function of subsets of a space. And so the picture is basically like this if I'll draw it as a low dimensional space asymptotically ADS metrics. So given any asymptotically ADS metric, you can compute all of these areas for the various subsets and that will map you on to some point in this space of functions S of A. But it's a tiny measure zero subset. So let's say someone just gives me an arbitrary state of some arbitrary quantum field theory and I go ahead and calculate all the entanglement entropies for all the regions. So I'm going to end up some place in this space of functions, but it's very unlikely that all those entanglement entropies could be reproduced by some geometric prescription like this. Okay, you won't in general, if someone just hands you one of these functions or some set of entanglement entropies, you definitely won't be able to find some geometry which reproduces those. Okay, the space of geometries is much too small for that correspondence to be one to one. Okay, so the fact that you're told that this is holographic from the start. Okay, so if I know I'm on here somewhere, then I should be able to find, I should be able to find the spacetime. Okay, so the, I mean, one caveat that I have in brackets over there is that there could be parts of the spacetime where none of the entanglement, none of the extremal surfaces reach, and then I'm not going to be able to reconstruct those. Okay, so just to draw one more picture, the idea would be, if I want to, here's my, think of the geometry, if I want to reconstruct the metric in a region there, I mean, there's, I think of some surface passing through there and then I can make all sorts of perturbations to the space. So there are many, many, many, many surfaces that pass through this region, and if I know the area of all of those, then I can imagine inverting. Of course, it's a challenging mathematical problem to do this inversion. Okay, so just summary only, very special entanglement structures will be consistent with a geometrical description. So this is actually a really interesting question, just to understand better, I mean, what properties of a field theory state will guarantee that I can find a geometry that captures the entanglement entropy? If we can understand that better, I mean, this is, this should help us understand which field theories can have a gravity dual. So if we can understand which states have their entanglement entanglement entropies captured by geometries and then understand, say, you know, what kind of Hamiltonians do those states, are those states, say, the ground state of or the low energy states of, that could probably really help us understand why ADS-CFT is working or which theories have gravity duals. Yes, that's right. So specifically, specifically, you could definitely have a part of the geometry, sometimes people call it the entanglement shadow. You could have a geometry where some region in the middle is not covered by any extremal surface. So this is kind of interesting. So actually, regions behind black hole horizons, for example. Okay, so that's why I said that. You're obviously not going to learn about this part of the geometry because these areas don't give you any information there. By the way, in two dimensions, for two-dimensional spaces, it's a mathematical theorem that you can do this reconstruction and the only obstruction would be parts of the geometry where the geodesics don't cover. Okay, so, yeah, any more. But yeah, the question was about the quantum corrections. And, right, so if we think of the gravity, what do these quantum corrections mean? If we think of the gravity theory, so this is supposed to be, the CFT is to some quantum theory of gravity. And so on the gravity side, there's a classical limit, which is what I'm mostly talking about. But then we can think of there being quantum corrections to that classical limit. And the simplest quantum corrections you can understand by thinking of perturbative quantum fields living on a fixed spacetime background. So you have some geometry and then on top of that, I guess the next approximation would be, instead of just having a geometry or some classical fields, you can think of there being a geometry and then also some quantum state of the various supergravity fields. So the first correction that I was talking about, so instead of just learning about the geometry, if you look at the entanglement entropies and also care about the one over end corrections to those, then you'll also learn about the state of the fields in the geometry as well. So the connection is not just with geometry. Once you include these quantum corrections, you learn about the bulk fields as well. Okay, well since there's this more or less complete information about at least parts of the spacetime contained in the entanglement information of the field theory of, sorry, yes. There was a paper by Aaron Wall and someone else, maybe Netta, Engelhardt, that specifically addressed this question. It's called, maybe entanglement shadow was in the title. Oh, sorry, yeah, Extremal Surface Barriers, okay. And so I would recommend that you have a look at that because they have, yeah, they've spent the whole paper kind of looking into that. And I think it's a little bit more general and they have some general results about when it can happen. Okay, so since we're understanding the geometry as being contained more or less in the entanglement structure in the field theory, we can be even bolder now and ask whether you can, so understanding how gravity emerges, I'd be part of it as understanding how spacetime is encoded in your field theory. But then we can ask, I mean, can you actually see some sort of spacetime dynamics or can you learn about something, about gravitational physics by understanding properties of entanglement in the field theory? And so let me, oh, I erased the picture. So let me reproduce what I just erased. So we had this map between asymptotically ADS metrics. And then this was the much larger space of functions S of A. And so, and that mapped into some small measure zero set of this space. Okay, so there's an interesting point which is that not all of these functions S of A could actually correspond to the entanglement entropy of various regions in a consistent theory. So the entanglement entropies satisfy various, must satisfy various constraints. And these are constraints that you drive directly from quantum mechanics. They're just linear algebra properties about eigenvalues. So it's impossible to violate them. Okay, so I think I mentioned one already. You have two regions, A and B. I defined this mutual information which was S of A plus S of B minus S of A union B. And it turns out this has to be positive. This is actually a good exercise. So this has to be positive for any quantum theory. There's another one that's similar but involving three regions. Okay, so you can show, and it's much more difficult that entanglement entropies for combinations of these combinations of three regions have to satisfy this inequality. This one is called sub-additivity. This one is called strong sub-additivity. And then there are more that I'll talk about. There's one called the positivity of relative entropy and the monotonicity of relative entropy. So there's a number of these constraints. And on this picture, what it means is that part of this space of functions, let's say, these are typically inequalities. So there's only certain functions, S of A that could actually be entanglement entropies. Okay, there are some, if you have some function of regions where you find one of these to be violated, then that function can't possibly be the entanglement entropy, describe the entanglement entropy for regions in some consistent quantum theory. On the other hand, if you start with a metric, so if I start with some asymptotically ADS metric and then compute the areas of extremal surfaces using the Ryutaki-Anagi formula, I don't always end up in this good region. For some metrics, I end up in the region that can't correspond to a consistent quantum state. Okay, so these ones here, these violate entanglement constraints. And what you find is that some of the space times, some of the asymptotically ADS times I start with, I calculate all of these extremal surfaces, I find that they're in the allowed region, they satisfy all the constraints. But other ones, I write down the space time, I start calculating extremal surfaces, and I find that this function violates the constraint. Okay, so what does that mean? I mean, it means that this geometry here, it can't really be physical. It can't actually, if it were physical, some physical consistent asymptotically ADS geometry, which is an allowed state of quantum gravity, which would be dual to some allowed conformal field theory state, well that must satisfy all these. And since this one doesn't, it's an unphysical geometry. And so by kind of mapping out this space, the green region, we'll be able to learn about something about constraints on geometries based on these fundamental quantum mechanical constraints. So the question was what do I require for these to be true? So most of these are just true about any quantum systems where A, B, and C are arbitrary subsystems. So I think that even, I mean, I have in mind local quantum field theories, but it's likely that even for more general theories, as long as I can isolate some subsystems, as long as I, where they, where you have some degrees of freedom and then some independent degrees of freedom, then they should be satisfied. Yes. Yeah, I think, let's see, I mean, you could choose them. I think in this way of writing strong subjectivity, normally we imagine them to be distinct, but I think I could write this in another way where they would be allowed to be overlapping. Yeah, actually I don't remember, maybe this is more, maybe this is okay, generally. Yes, yeah, we'll see, I'll go through some of these results, yeah. So I'm running, let's see, I'm running out of time. What I'm gonna do is set up, so the next lecture, basically the plan for the next lecture is to explicitly derive it and tell you about some of these constraints. So I'll just set that up by mentioning some of the specific constraints that I'll use. So what I'll do is start with geometries that are close to ADS. So ADS, for pure ADS, then this is okay with all the constraints, if I just have that, but even if I perturb away from ADS a little bit, then what we'll find is that some of the geometries actually violate some of these entanglement constraints. And so if we start, okay, so if you think about these small perturbations, we can ask what are the constraints on states which are close by the vacuum state? So if we start with small perturbation to the vacuum state of a CFT, then there's actually a simple constraint on entanglement entropies that you can derive directly from the definition. So we can recall the definition of entanglement entropy in terms of the density matrix. And now we just imagine taking the state of the whole system. For us it's going to be the vacuum state, but this is so far it's more general. So take the state of the whole system and perturb it and see what happens to the entanglement entropy. And so what we find is that we have the variation of this that gives minus trace of delta rho, dog, rho. And then there's another term that you would get by varying the logarithm, which would be basically just this, but it vanishes because these density matrices should all have unit trace. So if I vary the state, then the trace of delta rho is just going to be zero because all of the rows always have trace of one. So this is a formula for the variation of the entanglement entropy. And now we make a definition. I'm just going to rewrite that by defining the modular Hamiltonian. So HA, the modular Hamiltonian, is just defined to be minus the log of the unperturbed density matrix. It's just this quantity that appears. And the reason that we make this definition is, so then now I can rewrite the right hand side here as the change in the expectation value of that operator. So the change in the entanglement entropy for my small perturbation is equal to the change in the expectation value of this operator. And this is useful, I mean, in many cases, this HA would be very complicated, but there are cases, if we have a starting point where we know exactly what this density matrix is, then this will give us a useful formula. So I'll just end with one example that we're going to use next time. Okay, so if we have a system where the density matrix takes a thermal form, so if rho A is equal to one over Z, e to the minus beta H for some beta and some H, then this one gives us that delta of SA is equal to one over T or beta times delta H. And so this one here is this whole thing, it's just like the first law of thermodynamics. I mean, in fact, it's sort of a quantum version. It is the first law of thermodynamics, but it's actually a nice quantum generalization. So what it says is that if I start with some system, I mean, looking ahead, what I'm going to have in mind is a ball-shaped region of a field theory in the vacuum state. And we remember that we could calculate the density matrix explicitly for these ball-shaped regions and they took a thermal form. And now what it says that if I make a perturbation, if I vary away, say from the vacuum state in this case, that for any variation, the change in entanglement entropy is going to be equal to one over T times the change in the expectation value of this Hamiltonian operator. So it relates the change in entanglement entropy to the change in energy. In the field theory, that's going to turn out, this side will be able to write in terms of the stress energy tensor. Okay. And so this is the basic constraint that you get for small perturbations to the vacuum state for entanglement entropies. And next thing, we're going to translate that to the gravity side and get a constraint on geometries, which are close to, which are close to ADS. And what we'll find is that this exactly gives the linearized Einstein equations. So you actually, directly from this thermodynamics of entanglement, sometimes people call this the first law of entanglement entropy, we're going to see that you just translate it mathematically starting from the Ryu-Takyunagi formula over to a constraint on geometries. And it says that the consistent physical geometries are the ones that satisfy the Einstein equations to a linear order around ADS. So we'll do that. And then we'll talk about a non-perturbative, well, we'll talk about constraints at second order and then kind of non-perturbative constraints. And then you learn things about sort of non-linear, non-linear gravity, again, coming directly from quantum mechanics. And that'll be on Monday. You have a question? Yeah, this is, so I had this Rindler Hamiltonian for the wedge. And then you map that using a conformal transformation to a Hamiltonian that generates a certain conformal transformation, a conformal killing vector in the domain of dependence of the ball. And next time I'm going to write explicitly what this modular Hamiltonian is. Yeah, okay, so I'll stop there.