 Okay, so good morning everybody. Let's get started. So, Tadashi will continue with his third lecture. Good morning everybody. So, I'd like to now go on the topic of holographic calculation of entanglement entropy, namely, holographic entanglement entropy. And in my first lecture today, I'm going to give some basics about holographic entanglement entropy and some recent developments with some properties of this entropy. And in my second lecture, I'd like to give some recent applications which we are working quite recently. So, this is section three, the holographic entanglement entropy. So, we just simply call HEE, Holographic Entanglement Entropy. But before that, I just want to give some short summary of ADS safety, because that's quite useful for my second lecture today. So, we want to specify a basic coordinate system. This is ADS safety, or discard by Maruda Sena, 1997. And so, the claim is just, as you know, it's gravity, or maybe string theory. It's originally derived in string theory on anti-Doshita space. And let's call this as ADS D plus 2. So, this is my convention. Maybe people usually use D plus 1, but this is quite useful, because in this case, ADS safety argues. So, it's like a homomorphic theory. Yeah, so it's a simple case. It's like R D plus 1. So, D plus 1 dimensional homomorphic theory. In some case, we consider R times S D, the sphere. But anyway, D plus 1 dimensional homomorphic theory is dual to gravity in D plus 2 dimension, 1 dimension higher, because this is so-called holographic principle. And this has a negative cosmological constant, namely, anti-Doshita space. So, this is a basic claim of ADS safety. And so, typically, let's call this as some manifold M. And typically, this CFT is defined on this boundary of the spacetime. So, if we have some bulk space M, then boundary, we have some holographic dual. So, here, we have gravity, but as we can see from this Peckenshtein-Holkenholmerer entropy, the degree of freedom in gravity somehow looks like one dimension lower. That means just the degree of freedom is proportional to area, not to the volume. Entropy is not proportional to volume, but it's proportional to area. So, that suggests that it's... Gravity theory is equivalent to some non-gravitational theory. Non-gravitational theory is like counter-fuse theory or some quantum mechanics, many more quantum mechanics, which live on one dimension lower space. And ADS safety can be expressly understood in terms of the string theory, even though rigorous proof is still missing. This is a very important future problem. And so, here, if we want to get some classical gravity limits, namely some generativity or supergravity, then we have to take a particular limit of quantum-fuse theory, so which is a large N, N is the degree of freedom for the SUN gauge group. Some gauge theory, we have typically this CFT, the gauge theory, and this is the rank of gauge group, and the large N limit and the strongly coupled. And in my talk, so I'm basically assuming this condition, but we can also talk about some one-over-n collection and coupling constant, strong coupling collections in a perturbative way also. And sometimes I have also mentioned that. But so anyway, so this anti-Doshita space, we are going to focus on two different anti-Doshita space. It's the most typical one. First one is so-called global ADS, and second one is called Po-ankal ADS. And we take hypersurface definition of anti-Doshita space. So in my convention, I take this way, D plus two, equal to some hypersurface, one square, and blah, blah, blah, X plus D plus one in some two-time space, right? R to D. So there are two-time directions and these spatial directions. So metric looks like this. These two guys have a negative, so time-like, two-time, and other guys are positive. So this is the definition of anti-Doshita space, and obviously this has a symmetry. This has a symmetry, S-O, oh, maybe I got it here. S-O. So you have two? Time two? No? S-O, S-O, S-O. You're right. Thanks a lot. Yes. I should put it R-square, so this is very important. And which is identified as this regroup, identified as conformal. Conformal symmetry in CFD, and which lives on D plus one. And then, so we introduce basic coordinate. Yeah, so it's good to put D plus two. That's D plus one. So these are, okay, so it's, anyway, yeah. So it's like R cos rho and cosine tau, and R also cos, and sine tau. And other guys are proportional to R, sine hyperbolic, rho, and some spherical coordinate, some polar coordinate. It's called omega one, and it's like similar way. Then that's D plus one. So these are, guys, two equal one. And then, so we can write metric, so just induce from this metric, so we can just reduce this metric on this hyper surface. So this is a quite well known metric. It looks like this. And this omega state, the omega state is just metric of S, S-T. So this is D plus two dimension. This is one radial direction and one time. And you can write this picture. It's quite standard picture. So we have a low direction. Ah, sorry. We have tau direction. This is a time direction. And we have some radial direction. Here it's a low. And this spherical direction, we'll write it omega i. So it looks like this. And we see this boundary of this, this is called global radius. So this global means it covers whole part of space time. And this is quite important later. Discussion of local excitation, which I am. But anyway, this is global ADS-3. So this is called global ADS-3. And this boundary is, we can see that low goes to infinity. So the boundary is here. So that means we have tau direction and omega direction. So that means r cross S-D. And then, so this is related to this ADS-T correspondence. And it's bulk theory, but gravity is dual to boundary theory, which we have conform of your theory. And the second coordinate is a Poincare ADS. So this transformation is quite important for my later discussions. Poincare ADS-D plus 2. So this is called global ADS. So for this, we somehow rewrite these coordinates. These are basic coordinates. And in terms of different parameterization. So these are one parameterization rotor. This is very beautiful parameterization, but there are another beautiful parameterization. So we combine this in the right corner. And we call this z plus z. X mu is some low-rentian coordinate. So mu takes a value. It's 0, 1, 2, and d minus 1. Sorry, 0 minus 2, up to d. So this d plus 1 dimensional performance theory appears here. And if we take difference, then r is the ADS radius. So this is the ADS radius. So it's obvious, because it's like all metrics go with proportional to r squared. This measures size of anti-Doshita space. And also there are other components. d plus 2 looks like rx0. This is the time direction, z. And xi, which means i to 1 to d, 1 to d, they look like rxi and z. And it is very easy to confirm this constraint. This is the way to solve this constraint. And this guy gives very beautiful metric, very simple metric, which is known to be Poincare ADS. So here, good thing is that we have this Minkowski space here, standard flat Minkowski space. So it's like r1d. This is the radial direction, z is the radial direction. So if we write picture, we find this. And this is the z direction. And this means we are squeezing. If we go to the large z, that means that squeeze the space time, metric of space time. It's proportional to always 1 over z square. And it's get divergent at some boundary. So also the same thing is true for as log of infinity is a boundary, but it's there where we have divergence of metric. And this is identified with uv divergence of conformal fuselage. In quantum fuselage, as we know, we have ultravalve divergence. And that's also essential in the computation of entanglement entropy. So at z equals 0, it's metric is divergent. So we put some, normally put some cutoff here. Option is very small quantity, but I'm going to mention this more later. But anyway, so this is a time direction x0 and this is a space direction xi here. And the z direction is warped. There are some warped dimensions. And boundary is here. So boundary is here. This is a boundary. So in this case, one of these Poincare ADS U plus 1, 2 is obviously R1D. So this is R1D. These two guys are basic setup. Of course there are many others, but this is quite relevant also my discussion later. But at the same time, we should ask what's the connection between this two space. This should be the same, because it's come from the same definition. ADS covers only part of the space time. And if we look, so this global space, global ADS covers everything. So we can embed this Poincare patch into this global part. So that is given by, maybe like this way. This way, and we write some wedge here. Global ADS is of course extended infinitely. So we are particularly focusing on this wedge. So this is, we cut along this line. This is just null direction. And this part is like tau is half of pi. And here it's like tau is minus half of pi. And we look this region. So this shaded region is correspond to full region of Poincare ADS. And the, for example, one good pro is some particle. We can imagine some particle is passing through the center. And the center of global ADS, that means just rho equals zero. Just rho equals zero. So particle, this static particle, just trajectory of static particle. This particle don't have any motion in this space. But we can map this trajectory here. This is very easy, because we can regard this rho equals zero condition. It just means all of these coordinates are zero. So that means, for example, this guy is zero. This guy is zero and this guy is zero. So that means these two guys are same. Sorry, we set these two guys are same. So that gives d square and xx mu equal to r square. But also this guy is zero. So that means this guy is zero. That means we cut it out x i equal to zero. Then this just gives t squared. And this means that z equals to square root of t squared and r squared. So this is a trajectory like some moving particle. So it's like a falling particle. So this is a kind of horizon. It's called black brain. So there are heavy objects here. So that means the particle are attracted by strong gravitational force. And even if you start with some particle state here, it's pulled by gravitational force and it's falling into the horizon. So it looks like this, but however, this trajectory goes to minus infinity to plus infinity of time of the observer who lives on the boundary of one credit. But in this coordinate, you just see it's only part of trajectory. The observer really sits on this particle. It looks like more global motion. This global motion is covered only partially here. So this issue is quite important in later argument. Also many parts of this area is safety. So I just want to also mention about UV cut-off. So this is also very essential in the computation entanglement entropy. So we basically in ADSFT, the z direction, so you can see some nice symmetry. You can rescale z goes to lambda z, some scaling. But at the same time, x mu goes to lambda x mu, the metric is invariant. So that means this is a kind of scale transformation and this scale transformation is somewhat equivalent to this shift of z. So because of that, z is identified as some length scale in CFT. So this is basically, this is called UVIR relation because this here is a metric divergent in this limit. That means it's quite an infrared in gravity viewpoint, but this infinite metric actually corresponds to the UV divergence in conformity, I should say. This tells us basically, this z direction is a length scale and this is kind of a renormalization group flow. So this is trivial renormalization group flow because we are talking about conformity theory. But if you have some mass deformation, then you see non-trivial renormalization group flow and if you have a mass gap, end up with just some capital geometry, nothing here. So that means degree of freedom in large z is missing. That means because of mass gap, no degree of freedom in the infrared. So that way, this is quite useful information. And at the same time, we don't write, we don't want metric divergent. So we put some cut-off. We put cut-off here. It's a geometrical cut-off in ADS viewpoint, ADS gravity. But however, this is duality because of ADS CFT, duality, so this is actually UV cut-off CFT. And in that sense, epsilon is regarded as a lattice constant in lattice regularization. Or a momentum cut-off is like one over epsilon. So this is a basic issue when we play with ADS CFT. And also, basic principle, this is a more very, very important thing, is that bulk to boundary relation. We're going to use this later to derive holographic entanglement entropy. And so this is worked out by a famous work, Gav Zakhryov-Vitain, 1998. And this says that's a very simple thing. So if we compute partial function in gravity, then this is equivalent to partial function in CFT. From here, you can say also entropy in gravity is equal to entropy in CFT, free energy in gravity is equal to free energy in CFT. And also you can add some external fields, like some source or some magnetic field, as I mean, electric field, magnetic field and also gravitational field. Then you can take derivative about external fields and then you also find the correspondence of this correlation function and so on. So I'm not going to tell you this quite, I mean, in some sense, classical, classic well-known facts, so I'm not going to detail, so partial function is equivalent. So these are basic preparation for this next topic of this holographic entanglement entropy. So yeah, so now we're going to, next topic which is a holographic entanglement entropy. So I would like to explain this. First I give some, especially how I'd like to give how we can formulate calculation of entanglement entropy in CFT in terms of gravity and give some quick derivation later. But it is good to start with a simpler situation, namely, if we have some static system, this is a holographic entanglement entropy for static space time, static background. So this case, point is that you can also do some Euclidean analytical continuation without any problem. So Euclidean space, Euclidean space, equally. And this is first formed by me and Ryu, 2006. And I just first give some basic calculation. So you can imagine this ADS CFT setup. I write it here again. So it is easier to do it in Poincaré, but you can do it on any different coordinate system. But if you remember the definition of entanglement entropy in quantum field theory, basically, so let's take some time slash t equals zero to specify the Hilbert space, some state. And we decompose Hilbert space into two parts, right? Always bipartite A region and B region. Maybe you can also flip B and A. This is fine, like this. And once you decompose space into two parts, then you can define entanglement entropy, which we wrote SA, right? And in this case, maybe this is a pure state, like I said. But for mixed state, this is no longer true. And so this point is that this is a quantum mechanically impressive define, but in the ADS CFT gives some quite geometric calculations. So you can imagine this black hole entropy. That is given by area of black hole horizon. So the same thing happens for this holographic entanglement entropy. So the idea, so we have some extra dimension, which is z direction already introduced there. So we take some surface which covers region B. Anyway, this boundary is very important. This is obvious. We explained something called area law and so on. It's always proportional to boundary area and so on. So reading the divergence should be proportional to area. And anyway, so we pick up some surface which called gamma A. This surface, you can call also gamma B, but in this case. But let's call it gamma A, which means that, so this gamma A satisfies the condition that boundary of gamma A is boundary of A. Well, maybe in this pure state, but let's not talk on this. We can talk about also mixed state case. Yesterday I explained that why entanglement entropy is not so good quantity to characterize quantum entanglement for mixed state. But nevertheless, we can define entanglement entropy and actually in a holographic set up that's actually quite nice quantity. But though it's not count number of EPR pairs and so on. So this we impose this condition. And we also need to impose some topology condition. Like these two guys are like, you can imagine this A surface. Maybe it is good to think A, but in this case you can flip this because of pure state. So A region is homologous to each other, to the gamma A. So this is topological condition. But anyway, so up to this condition, we find that this entanglement entropy is in CFT actually equal to some geometrical quantity which is like minimum area surface. So we compute area of this gamma A, I4G Newton. And we take all choice of gamma A, but gamma A satisfies these two conditions. Gamma A is same as boundary and gamma A is homologous to this homology condition. Not homotopy, homology, homology condition. And so we can compute. So this is a basic claim of holographic entanglement entropy. And then we can check various properties before we derive this formula. So first of all, we know the basic property of entanglement namely area law. So we should be able to reproduce area law. So this is very easy to see because you can compute this area of gamma A, which looks like anyway. So we have to take a minimization, but obviously the metric divergence here should be perpendicular. The surface should be perpendicular to the boundary. So that means if you integrate metric and if you remember this Poincare metric, which I write it here, so it looks like this. Some R, these are radius, radius, some power of D because always this is D dimension. So this is very important that this is D dimension and that means co-dimension too. Entropies always associated some co-dimension to surface because so that dimension cancels with Newton constant. So it's like this. I will be careless about numerical constant. And we take integral R of Z, but this we have Z here. And so we know we have cut-off. Right now this cut-off plays an important role. So in principle this metric divergence here so it gets just divergent. We put some cut-off here, epsilon here. So integrated epsilon to some value, which I don't know, some value, we don't care because we are interested in most reading contribution which is really localized as a boundary. Metric gets divergent here. And of course we have X integral. And this just gives an area of boundary of A, D minus 1. So in this way we find roughly speaking this guy and the integral of this gives some epsilon to D minus 1 power and the area of boundary of A. And some sub-reading terms. But this is exactly what we expect for area law. This is area law in the behavior. And this constant is actually even dimensional conformal feature, this is like central charge. It's quite natural. It's proportional to degree of freedom. And odd dimensional case also states some good quantity to characterize degree of freedom. And related to F-function and so on. So first test is somehow very easy. And now we go to another issue of mixed state. Modern issue in the mixed state versus a pure state. Yes? Do you really mean central charge or I mean in four dimensions for example? What do you mean? Is it roughly the central charge? Yeah. Do you mean A or C? Yeah. Because this is a leading order in a classical gravity limit. A and C is equal. I see, I see. Everything is degenerate. To see more details we need to take into account collections, higher derivative collections. And another issue is this one. And we can imagine some black hole, finite temperature. Finite temperature black hole, ADS black hole is dual too. So just application generalization of ADS-FT tells us this is dual to finite temperature TFT. Temperature is the same. Just some black hole has a temperature, a Hawking temperature, and that temperature is dual to conformity temperature. And we can think about a situation. Now I switch to global ADS. So it's like round shape boundary. So we have black hole horizon here. This is a black hole horizon. And we talk about entanglement entropy for particular region A. And let's take other region B. B is a complement. But however in this case, low AB is not pure. Not pure. Actually it's like low AB is equal to low AB is equal to canonical ensemble, beta H. So it's not mixed state. So that's case, something special happens. So here also this homology condition plays a very important role. So we can imagine there are actually two different minimal surfaces which covers A or B. So this is one surface which covers A, but we can have another surface which really ends on the same place like this way. But this homology condition tells us this surface should be dual to A and this surface should be related to B. So we call it gamma A and gamma B. They are different. In this zero temperature, this is a zero temperature, vacuum CFT is described by Poincare ADS. That's case, these two guys are same. But this is no longer. Levels, this region is A. This region B. Yeah, inside is B. So this region is B. So we take ADS-3 setup maybe for simplicity. And then time direction is this direction. And this is one circle direction and the spatial direction in CFT. And this is the radial direction. And these two guys are different. So entanglement between A is just computed by area of gamma A divided by 4D newton. And this is different from SB which is given by this area of gamma B. So this is very special. And indeed this is consistent with what we know. Only for pure state SA equals SB4s. But for this kind of setup, so state is not true. And the reason for why gamma B looks a little bit larger is just, it's also count number. It also has a contribution from Beckystein-Folkin formula. It's like almost wrapped. So that is actually very classical correlation. And it's not related to containment entanglement as also discussed yesterday. But that's the reason we have some extra control. But nevertheless, this is quite important quantity. And we can see some interesting transition behavior here already. I'll come back to this later. If we assume A is very small, A goes to very small. Then A is just this region. And the other is B. Then the surface is here. But you can imagine this kind of surface. Up to some region, this surface exists, but there are much smaller area because we always have to take a minimum. If there are several candidates which satisfy this condition, we have to pick up the minimum area one. So this here is the situation. There are these surfaces. But actually we should pick up another one which just, namely, it wraps here and also just pick up this guy. So let's call this a gamma black hole. So in this setup, this is a gamma A. This is just a small surface of gamma A. Gamma B is identified this gamma black hole union gamma A. Gamma A is gamma A. So if so, we will see that entanglement entropy for B minus entanglement entropy for A is same as black hole entropy, which is just area of black hole horizon and divided by 4G newt. This is the one expressed connection between black hole entropy and entanglement entropy. This is equal to thermal entropy, thermal dynamical entropy, thermal entropy. So this is one concrete way to say the deep relation between entanglement entropy and thermal dynamical entropy. So in the sense entanglement is a sort of generalization of the standard thermal entropy. But it's sort of a huge generalization. And this relation is always true if we take a limit but in holographic case there are some even though A is not completely shrink to zero size, it's some finite but it's very small already this transition happens. This is a very sharp transition and it's like sometime called phase transition. This is very specific to holographic theory, like some conformal few series dual to classical gravity limits of anti-dorshita space. I will come back to this issue later. Yeah, good question, but actually minimal stuff you don't get into the horizon. Yeah, but this is the reason that we are talking about static space. If you think about time-dependent black hole, you know like collapsing star, then actually it's penetrates apparent horizon of black hole. And one more thing is some inequality. Like some strong stability which also we explained. This is very quickly, we can easily show this is let us remember strong sour activities like this, A, B, C. This means A union B union C, this B union C. So this is very easy to show and we can just schematically write it here. Everything just project down to two dimension, but you can imagine the same thing also to go for higher dimension. So we pick up some region A, B, C like this and then this guy, S, A, B is some minimal surface area of minimal surface here. S, B, C is the minimal surface here. But this is obviously so smaller than A, B, C. So we can imagine this surface, you can think pick up only this surface somehow pick up this surface. This is really artificial. There are even casps in reality, but there are actually much smaller surface. Area is smaller. Smooth surface. And also you can imagine this guy is like very funny surface, but there are obviously smooth and surface which are smaller area. So this guy is just this guy and this guy is just this guy. So obviously this inequality holds in holographic theories. And then one more test is 2D CFT. So yesterday we derived the formula, the famous formula. So of this entanglement entropy, 3, C and log L over epsilon. And so if you take an infinitely long total space and subsystem is interval, subsystem is some interval ring cell and epsilon is a lattice constant. There's a bias also cut off here. And we explained that. So we want to derive this in ADCFT, this holographic calculation, but this is very, also it varies. So we know actually first of all this length is let's take 2 over L and 2 over L. And this is a region A. This part is a region A. And then end point is here and we have to find some minimal surface, but this minimal surface is not surface, actually geodesic. It's always co-dimension 2 and here we have in mind area 3 divided to CFT 2. So it's like co-dimension 2 guys are lying, right? And we just want to think about geodesic links, geodesic line. So this geodesic line is actually turned out to be just a very beautiful one, just a half circle, semi-circle. So this is just described by so this is the z direction and this is the x direction and time is again fixed. And so this geodesic turned out to be z for L square and x x square. And in a Poincare metric, in Poincare metric it looks like this. And we fix time equals 0 so just 2 dimensional. It's Poincare risk. And then this guy, we can rewrite this into using this surface it's like this L square and z square and times d z square. So this is just induced metric on this geodesic. Then we integrate so area of this this geodesic gamma a gamma a is just integrate this metric. And z takes value from epsilon, we need a cut-off epsilon again and up to this L over 2. And we just integrate this guy. RL divided by z and L square for z square. And then this is very easy to do and then we have log formula. The reason we get log epsilon divergence is just here. So this is kind of constant when z is very small so we have d z integral anyway. So that gives precisely also no other term no other constant term. And then RL I thought you got a good point. Excellent. We have to do because this integral just computes one part of the contribution and we need one more contribution. Thank you for comments. And then so finally we take s a is like area of gamma a divided by epsilon which is looks like R over 2g newton and log of epsilon. But now we use the famous relation something called Brown-Eno relation between radius radius and central charge. And then this is just 3 over c and log L over epsilon. So this way we can reproduce what we know from CFT side. This is one of the simplest test check. So this is a story in static case. So now I also give some formula for more general case which is holographic entanglement entropy in time dependent time dependent setup time dependent space time. So this case one problem is that we cannot take a simple and a weak rotation into Euclidean space. If we do naively then we get valued metric which is not good. Also on this kind of low range and space time here we restrict everything down to time slice. So in that case we can have a nice minimal surface. But if you are thinking full low range and space if we go in narrow direction then area is just collapsed to zero. This minimization maybe just give this singular surface like narrow surface is not good actually. Doesn't do nothing related to entanglement. So we have to be very careful about the choice of some surface. We cannot just say minimum area surface. And instead correct formulation is this one. This is a formulation which I worked with Veronica Fubény and Mukund Rangamani and one year later than that one. And so this case is really kind of argument. First we do extramarization. Not necessarily minimum, not necessarily maximum. So we take area of same surface gamma a. This is the same condition. So here boundary of gamma a is boundary of a and gamma a is homologous to a. And we take divided by 4G newton the same as Beckenshe and Hokeyman again. We compute this. Anyway there are many several candidates for this guy. Then we pick up just minimum guy. Let's call this a solution. So there are some solutions. This is a kind of differential equation partial differential. We have to solve it. But you get some discrete number of solutions and we have to take a minimum among them. So this is a formulation. We cannot initially start with some minimum or something. So this is the extremar surface. This is called extramar surface. So that means just we compute area functional and just require its variation in 0. And the actually the reason why we have extramar surface is like this. So you have in mind some we have some Poincare metric. Let's imagine some area 3 for simplicity. We have some geodesic. In area 3 extramar surface is just geodesic. Space-like geodesic. Space-like geodesic. Space-like. So if we think about some kind of time slice, then if we move this way, then indeed this is a minimum surface. But if we move in time-like direction it's a Laurentian space, it's like metric is different. It's like this way. If we move in t-direction it's minus. So it's an area actually minimize. Actually area increases in this direction. But this direction increases. So that means we have to maximize in this direction and minimize in this special direction. So that's the reason we need an extramar condition. And this is quite manifest in another formalism, which is so this is t, there are equivalent but another formalism which is formed by example, which is another formulation. So it's different. So this is we call it also covalent holographic entanglement. Another covalent. This is equivalent from 2000. This is actually quite useful to prove strong survivability for time-dependent backgrounds. This formula is not, from this formula it's not easy to derive strong survivability like this because if we think about this two geodes equal to surface they are not always in the same time slice. But another formulation helps us to show this. So this SA maybe we put it for W. And this is another two-step computation but first we can do is minimization. So anyway we want something minimal surface. But we have actually a minimum of a particular time slice. So we have in mind we have some Poincaré, let's take Poincaré, but we can take some particular time slice. So which covers end on this boundary of A. Let's call this the A. This region is A. Some surface, you can choose many different surfaces. You can also choose this kind of surface but always end on A. You can choose if it remains. But let's fix one of the time slice. Then we can on this time slice we can talk about this kind of minimal surface. Minimal surface. So this is this condition. So we have some surface but which only always lives on some particular time slice. This sigma is time slice and compute and it's the same condition. So this is the same. So we go to gamma A. Homological condition. This is quite a stable calculation because always in a space like manifold we just minimize surface. And after that of course the result depends on the sigma. Choice of time slice and we take a maximumization about sigma. And this gives holographic entanglement between covariant and formulation. So these two guys are also proven equivalently. But the quick understanding is that you have to maximize here. This maximizes here. This sigma and the minimization is here. So the maximization of this guy by changing sigma time slice. And this guy is just fixed time slice. And we have a region A. And we have some surface which cover region A and you minimize area. But this part tells us. And these two steps proceed here. And the reason why this is quite useful for, I mean, to prove strong flexibility is maybe I should like show this here. Yes. Sigma is any time slice which end on the region A. So we have some time slice. These are ADS space. But let's fix one of the time slice. Then on this time slice we can minimize the area. We can find minimal surface. But the problem is that result depends on the choice of time slice. And then finally take a maximum against any choice of sigma. Proof of SSA in covariant case. We can do this in the following way. Using this second formulation. This is also obtained by Aramor. And so we have some we can imagine some Poincare setup for anything you like actually. So as a boundary it's very easy. So we have some A, B, C. We have three region called A. A, B, C. End a little bit more. C and here. This is some time slice and we have A, B, C. But problem is so if we think real surface looks like this two guys that looks like this is gamma A, B and this is gamma B, C and they are not on the same time slice. It's not easy to not possible to find some slice which includes both of them. Because here it's in some sense they intersect not doesn't completely intersect. It's like a slightly separated. But the idea is to start the opposite way. So we start with the answer. The answer is this guy. Let's start with this guy. So anyway, this is something good. The reason is so you can find some surface of B and we can find some surface for A, B, C. But they don't intersect in some sense if we look from the top. So that means we can always find some time slice which includes both of them. So we can always, for formal proof of this, you can find some sigma which includes both of them. So sigma includes gamma B and gamma A, B, C. We'll be owing to that. So anyway, we compute area of gamma B plus area of gamma A, B, C. And on this sigma slice, we can easily prove this same as this static case. We can easily show A, B on this sigma and the area of this B, C on this sigma. This inquiry just saved us that one. We fix some spatial time slice. So we have some surface so we have some surface which looks like this and like this. So this is on the same surface sigma. So that's the reason I put sigma. But actually the real surface is different. The real surface is something similar but it's a little different. Maybe it's going this way and this is a real external surface. But nevertheless, because if we remember that formulation always correct one is a maximum against choice of sigma so that means correct one should be satisfied this thing quality. So this way, in the end anyway, we prove this strong quantity. So these are basic checks which we have to do to believe in this entanglement. This quantity is the entanglement. But next I'd like to give more direct derivation by using ADS-CFT. So this is the final C is some derivation holographic derivation this by Ruko Itzan and Mada Sena. But as there are some derivations which are pointed at a very good point just after our proposal. But this Ruko Itzan gives the most correct derivation. So here we just use this bulk to boundary relation. Partition function is equal to each other in CFT and holography. So we start this CFT partition function is equal to gravity and gravity looks like simple. That exponential minus action is gravity and action looks like I'm talking about Euclidean formulation. So we have minus sign and just G and R minus 2 cos constant and some maybe some time we need to include given but that doesn't contribute actually. So I don't write it. Also there are, you can assume scalar field or fermions but they don't contribute. Only metallic is important to compute this. This is also closely related to the fact that somehow only gravity theory if we think of a scalar field theory and fermion theory and maybe gauge theory. So there are no classical contribution to entropy but only at spin too. I mean gravity theory has kind of mysterious we have some classical contribution of entropy. This is a typical example of Beckenstein-Folkenhoemmler and more general consequence is holographic entanglement entropy. We just come from classical GR. So that is we can neglect also other contributions and then if we remember this replicatoric, so we are interested in this guy and take derivative N and this is we call ZN ZN and Z1 to N's power because of normalization. So we want to anyway compute this guy in CFT but this is equal to gravity. So let's do it. Let's compute this quantity. But this quantity, if we remember the basic picture, so we do some path integral calculation. Then we have some let's take a region A here so we have this path integral and region A is here like previous yesterday's computation and then idea is to put some negative deficit angle such that we have N seat angle is 2 pi N. So we have 2 pi N angle. A naive holographic dual is this is actually original argument by Fulsai but so we can just naively extend this deficit angle into the bulk and we can evaluate action in this pattern but this is not exactly correct because in general relativity we don't like this kind of singular geometry. There are actually actual solutions actually for any integer N there is some nice solution which is a smooth metric always there are such solutions. Actually we should use such a solution so we have some smooth solution smooth solution with 2 pi N periodicity on boundary, only on boundary not in the bulk there are some smooth solution let's call it so we have some but however this solution and this naive solution it's deficit angle solution there are some 2 pi N bulk maybe bulk deficit angle which is given by delta equal to 2 pi 1 minus N I mentioned this yesterday solution this is not exact good solution singular solution but naive we can come up with this but actually it's like conical singularity you can just write this way low square d theta but theta is periodicity is 2 pi N this we say it's like conical singularity but let's pretend that actually we get the correct answer the reason this difference is in some sense small so if we plug this solution in this action but we know so already at any there are no singular let's call this is solution let's call solution g this is metric let's call this g2 and g1 and g2 is of course coincide and also they are solution to Einstein equation so that means that means if we consider N greater than N different from 1 and achieving some analytical continuation we find this action gravity action gravity so let's gravity action with this g1 minus g2 is actually N minus 1 square order not there are no linear contribution the reason is that linear contribution cancels because of the solution to Einstein equation so you can imagine some perturbation of IZ but the leading term is delta g and Einstein equation and already 0th order satisfies Einstein equation this term is gone so always this delta g square so delta g is proportional to because of this argument N minus 1 so that way there are no linear contribution that is quite fortunate because only linear term contributes to entropy because we you remember this we have one first derivative of N and that N equals to 1 so this term only affects only affects linear entropy for Neumann entropy for Neumann entropy we can neglect the difference and of course we have to be more careful about this kind of argument but this for details you can look at but this is intuitively quite confusing and then so anyway we can just pretend that everything is fine for this naive solution and indeed this naive solution is used to derive holographic format by the side then this is very easy so for for this rectangle we are going in deficit angle delta equal to pi minus N fine rich scalar in some manifold this manifold with the deficit angle it's like always looks like for pi 1 minus N and delta function so let's call this surface as gamma digit angle surface gamma it's localized singularity is just localized on this surface so it looks like delta function but of course there are other contribution but they don't contribute to entropy and then we compute this gravity action for N seed case then we just put this factor then it's 4 pi so minus 4d newton that means 4g newton N minus 1 and area then we get area of gamma N because we have a curvature but this becomes 4 pi 1 minus N and delta function and so and then we can take derivative about N so this we just compute entropy is like derivative of N and log of the N the 1 to the N that means we can delta N and it's like minus so it's like let's put plus then iN minus N i1 but the i1 is trivial because it's just 0 so we get derivative and then 4g newton area is that appears but this is not the end of the story so next we have to say this is the minimal surface but that just come from Einstein equation you can compute it directly basically so you can regulate so this is a singular surface original surface like 2 pi N but you can slightly modify this part as some smooth function which looks like low square in a little bit larger region and it becomes okay so it's like low square in low is a little bit larger but it's epsilon then maybe N square low square you can choose such an interpolating function then there are no singularity and then just compute it then you get it but you can also understand it's kind of if you have some N copy so you can think that's kind of a lot of sphere if you have N copy of sphere and paste with each other and if you apply Gauss bonus theorem you can also derive this there are many ways to derive this form and yeah okay so it needs time but just okay starting out so mention one more yeah so let me mention holographic CFT just want to quickly finish so here we assume some classical gravity classical gravity gravity and dual for ADS in ADS and this is dual to something called holographic CFT so this is the name is large N large N or large central charge and strongly coupled so this is characterized by the spectrum as we often say there are some black hole in 2D CFT let's think of 2D CFT so above some energy scale then we have black hole states there are quite a lot of states these are black hole states but however for below the state there are some gap and below the gap so there are a number of space it's quite sparse sparse spectrum so this sparse spectrum characterizes this holographic CFT and there are now there are many many computations like Hedrick Hedrick's work 2010 and Hartman's result 2013 and there are many many works including ours so there are some using this characterization we can analytically compute reading contribution to entanglement entropy also as a quantity like partial function correlation in this large central charge CFT sparse spectrum I'm not going to detail but just want to summarize the consequence of this thing this is actually a phase transition actually I already mentioned so this phase transition phenomena simplest case happens when we have in mind is like when subsystem A consists of two intervals and in my lecture I always assumed that subsystem A is just one interval why not we can consider some disconnected union so two intervals so this case there are two contributions two possibilities so if they are far apart and if they are very close to each other A1 and A2 if they are far apart minimal surface looks like disconnected this is the gamma A1 gamma A2 in this case they are connected this is A1, A2 most union of these two and in this case mutual information A1 and A2 defined as SA1 plus SA2 SA1, A2 is actually zero because this guy this SA1 here we just show the computation of SA1, A2 and just disconnect the same as summation but here A1, A2 is non-trivial, non-zero it's positive, always it's positive because this guy is different from the other guy this guy is large and such a phase transition happens and this phase transition occurs because of large C large N limit so if you take a large N limit even originally smooth transition smooth crossover behavior becomes a very sharp phase transition and this is indeed this kind of a phenomenon doesn't happen for standard CFT but if you take large central charge CFT so you can derive these properties these are connected these works and one more thing is this in quality called monogamy of mutual information so I just finish with this one this is one characterization this is some similar argument similar to strong stability but strong stability known to be true for any quantum mechanics monogamy of MMI mutual information this is a mutual information so this is given this is done by Heidelich Hayden Heidelich and Maroni this is a very interesting result that we consider mutual information between ABC is always larger than mutual information AB and AC mutual information roughly measures the amount of correlation between two systems and this is basically related to correlation function between A1 and A2 universal way to formulate correlation function in quantum information language and this is always this is not necessarily true for any standard quantum state but this is always true in holographic CFT holographic entanglement and reading order as we can quickly show this so this is ABC and this basically says this summation is smaller than this guy it's very similar to strong stability proof A and C these guys obviously this surface cover CB so it's obviously this is smaller and this surface covers here very similar this is just equivalent this statement so definition of mutual information like this so this means SA plus SB plus C we write this in terms of entropy AB it's like similar to BC CA I think this is negative this is probably negative yeah this is non-positive so this is equivalent and this is a clear issue but if you see some particular states something called GATG state for part type GST that means square root of 2 and 0000 and 1111 and obviously you can see this is bioretic this state is opposite because everything is the same but log 2 positive sign is larger than negative sign so that way this is somehow good characterization of the quantum state which we realize in holographic some classical gravity limit of holography yeah okay so I just wanted to finish here continue next time