 So if you can take your seat, we can start with the second lecture by Hiroshi. Okay, thank you for coming to my second lecture. So yesterday we finished a discussion of primer of information theory. So today I'd like to review aspect of ADAC of the correspondence that's relevant for the rest of my lecture. And hope to get into the beginning of the discussion of holographic entanglement. I guess this is how it is. Is this better? Okay. So ADAC of T correspondence. So this is a correspondence between a gravity theory, quantum gravity theory, in D plus one dimensional antiliter space, times possibly some compact space. And this space you can, for example, introduce some metric where x is Minkowski and coordinate, which goes from zero, which is time, to d minus one. So this has a standard Laurentian signature metric, which is minus dx naught square plus i goes from one to d minus one of dx i square. And these extra coordinates that basically measure the distance to the boundary of this space. I'm going to set this, what you may call the curvature radius of ADS to be equal to one. So I'm going to remove this. Okay. And so this is supposed to be equivalent as a quantum theory to D dimensional conformal field theory. And it's important to stress that this is actually isomorphism of Hilbert spaces and operator algebras on them. So in CFT side, it's a standard quantum field theory, so you can define Hilbert space and define operator algebra. On the ADS side, we don't have by itself a non-perturbative definition of the theory, but you can perturbatively quantize it. And the perturbative part should match up with this. You don't have a non-perturbative definition on this side, but you can use this. If you assume this isomorphism, then you can use this as a definition on this side. Okay. And then you can discuss various consistency check of this proposal, which I'm not going to do. Conformal field theory, as the name suggests, has a conformal invariance. And in particular, this conformal field theory, so this has a conformal invariance, which in particular contains scale invariance. So what it means is that, oh, by the way, so under this correspondence, we have to identify this coordinate on the boundary at z equal to zero to the coordinate of this conformal field theory. So I'm going to use the same x. So and then suppose I analytically continue this time coordinate so that we are now in Euclidean signature d-dimensional space. And then conformal transformation is the rescaling of these coordinates by some factor. So this is a conformal scaling transformation. And in this conformal field theory, there are lots of operators. And these local operators, each of them are characterized by the scaling dimension. So delta is called scaling dimension. Okay. And one of the very important concepts in conformal field theory and which plays a very key role in this correspondence is the so-called state operator correspondence of conformal field theory. So there are two correspondences, so I should not get confused. This is a correspondence within conformal field theory, not correspondence to this side. There is a correspondence between, within conformal field theory, and it is a correspondence between local operators having this fixed scaling property to states in conformal field theory. More specifically, if you have a local operator O of x with scaling dimension delta, then it's supposed to be equivalent to a state. So I denote the state just to remember which state we are considering. Use the same symbol as the operator in the hillbill space of conformal field theory. But then this conformal field theory, hillbill space, is defined on the cylinder. We are placing conformal field theory on the cylinder. So you have the d-dimensional space time where the conformal field theory is defined. You have a time direction here, and then you have d-1-dimensional sphere. So spatial section of this conformal field theory is d-1-dimensional sphere, and then you can propagate this in the time direction. And the way that the correspondence works is the following. So if you have a local operator here, if you have a local operator, you can place it, say, at the origin of this coordinate. It can be anywhere because it's a translational invariant, but suppose you place it on the origin in this original Euclidean, a Poincare coordinate, flat Poincare coordinate. Then there is a scaling variance. The scaling variance generates the scaling away from this origin. So this is a scaling, and then delta is an eigenvalue under scaling operation if you place the operator over here. Since the theory is the invariant and the conformal transformation, we can perform conformal transformation. And in particular, there is a conformal transformation which maps w into z, like that. And if you perform this conformal transformation, if w is a coordinate on this plane, then z is going to be a coordinate on the cylinder. So I'm actually doing this in the case of d equal 2, but you can do it in any dimension. So you can see that z coordinate is actually coordinate on the cylinder because this coordinate is invariant. This relation is invariant under shift of z by 2 pi i. So this z coordinate is a natural coordinate on the cylinder. And this scaling transformation now becomes a translation in this direction. So if you insert this operator here, now you have a state here which propagates in this direction, and eigenvalue under this propagation is exactly delta. So this is basically the reason for this relation between the local operator with conformal dimension or scaling dimension delta to state, normalizable state of conformal user in hillbill space on the cylinder. So this is a relation between flat Minkowski space, local operator, and state on this cylinder. So this is a very important concept and very useful in many occasions that it's very important to understand this. Is this clear? So now this hillbill space of CFT on this cylinder. On this hillbill space, we have representation of conformal symmetry, where the theory has conformal invariance. So state in this hillbill space of conformal theory should be decomposable into some of the unit representation of this conformal symmetry. Namely, we should be able to decompose it into representations of conformal algebra. And so under this correspondence, what happens is that if you have a primary field, namely that you can decompose this, you can classify this operator under representation of conformal transformation, and you have a highest weight state, which we call primary state. So primary field correspond to the highest weight state on this space like sphere. So this is the correspondence between operator in conformal theory and state in conformal theory, and in particular highest weight state under conformal symmetry correspond to what is called primary field in CFT. You can think of this as a definition of primary field, if you like. Yes, please. Okay, I'm coming to that. Actually, I'm going to discuss non-local operator or non-normalizable operator? Oh, non-local operators like wisdom line operators, yes. So those are invariant under different subspace of conformal transform. These preserves different conformal transformations. So, for example, this one preserves conformal transformation which fix this point, for example. And therefore, it's the eigenstate of the scaling transformation. If you have, for example, extended wisdom line operator like extended things, it has to be invariant under scaling along this direction or transverse direction, but not rotation, for example. So it will preserve different symmetry. So these will not correspond to state. These will correspond to something else, like the domain wall or some other operators. So I'm not going to talk about extended operator in this talk, in this lecture. So I'd rather not get into that. I'm going to discuss with you about this subject later, if you like. Okay? Yeah, I could do it, but maybe in discussion session, because I don't think the purpose of this lecture is not about conformal idea shift. I'm just reviewing the relevant part of this. But just briefly, so conformal algebra include translation and Lorentz transformation, namely the standard Poincare symmetry of this Minkowski space, plus scaling transformation, plus you have this thing called special conformal transformation. And there is actually a deep question of whether if you assume Poincare symmetry, unitarity, and scaling variance, it can be upgraded to conformal invariance or not. Last year, at the same school, Zohar Komaragowski, I think, gave some set of lectures on this subject. And yeah, so that's basically the conformal symmetry in general dimension. In two dimensions, this can be further enhanced delasero symmetry. I don't think I have time to discuss more, describe more than that. I'd be happy to discuss with you personally or at the discussion session. Unitary representation of these algebras are completely classified and known, and those are the things that appear over here. Now, so this coordinate that I introduced here is called Poincare coordinate. But in order to discuss this type of state operator correspondence, more useful coordinate is global coordinate, which is given by, so I'm setting the ADS scale to be one. So this is the time direction. T is the time. Rho is again measuring the distance to the boundary of ADS. So you have d rho squared plus sine hyperbolic square rho d omega d minus one squared. And this is actually the metric on unit d minus one dimensional sphere. That's why I wrote omega d minus one squared. So pictorically what it looks like is again cylinder, but including the interior. So this conformal theory is defined on the cylinder, the surface of the cylinder, but the ADS space includes this interior of this. So it's sort of solid body. So it's a solid cylinder if you like. So this is the time direction goes in this direction. So these shaded regions are space like section of this global ADS coordinate. And this space like section look like Poincare disc. It's a hyperbolic space in d minus one, sorry, d dimension, excuse me, d dimension. The total space is d plus one dimension. So this is hyperbolic space. Euclidean signature hyperbolic space. So this metric here is metric of the hyperbolic space where rho is infinity at the boundary. So the distance to the boundary is infinite. And the time goes in this way. And of course the omega parameterizes the sphere d minus one dimensional sphere on the boundary. Now this coordinate we have right here only covers part of this space, which is sort of you can consider some kind of a square tilted in 45 degrees and patched over here. So on the boundary and then so this boundary region correspond to z equal to zero in this coordinate. So at z equal to zero we have this square region. So this is z equal to zero in Poincare coordinate. Constant x naught slice goes like this. So this correspond to x naught equal constant. And then constant space point goes in this way. So suppose for example you have fixed point in x1 to xd minus one. And then go into time direction along x naught you will be traveling along this direction. So this is Poincare coordinate squashed into this square region. So you can imagine you have a cylinder. And then on the boundary of the cylinder you paste this in this way. So that's the region which is covered and of course you have interior of this space. In the global coordinate the entire hyperbolic disk is covered by this coordinate. Whereas in this Poincare coordinate basically the region covered is inside of this area. I can describe this more precisely but this is roughly the idea. Any question? That's clear? So Poincare coordinate covers only some part of this entire ADS space. But in this overlapping region we can use both coordinates. But if we want to describe, so for example when you talk about state operator correspondence the time that we are talking about for conformal field theory this time is to be identified with this T. So that's why I'm introducing these two sets of coordinates. Is that clear? So now, so I told you that there is a state in conformal field theory. There are state in conformal field theory. There are operator in the conformal field theory. Those are concepts on this side. And then we need to identify the corresponding concept on this side. And we can identify at least the subclass of, you have a question. Pick up. I cannot hear you. The microphone is not on. You can speak louder. Global, is it? No, no. In Poincare coordinate is same? Yeah. These are just coordinate transformation. So we are talking about the same space time. Anti-witter space time is the same. Lorentzian signature manifold. I'm just talking about two different coordinates on this manifold. And these two different manifolds covers different part of this anti-witter space. The global coordinate as the name suggests covers entire anti-witter space, fully extended anti-witter space. Whereas Poincare coordinate only covers part of it. That's clear? Yes, yes. So I was trying to describe exactly where each of these coordinate covers. Poincare coordinate do cover the boundary. This is the boundary. And that's pasted over here. So imagine, so Poincare coordinate covers, so this is the CFP side. Z equals 0 in Poincare coordinate covers this part of the cylinder. And you can work out, for example, explicit coordinate transformation. For example, in 1999, I wrote this review article with four authors. And so if you look it up, that is a physics report review article. We have shown explicit coordinate transformation between these Poincare coordinate and global coordinate. You have a question? Yeah. We'll use that. So you don't have to worry about the example because I'm going to use these in the rest of the lectures. Okay? Any more questions? Okay. Well, maybe I should have spent a little bit more time reviewing it. I wasn't sure about the level of students. But if you want me to talk more, expand more on this, I'd be happy to do that. But if you rather want me to go on to some more advanced subject, I'd be happy to do that as well. Okay? Maybe we can continue on. If you have a question, you can come see me or discuss further on the discussion session. Okay. Now, so let's discuss ADS side further. So suppose you have some field in the bulk of ADS. And it can be scalar field, gauge field, spinar, graviton, et cetera. For simplicity, let me just talk about scalar fields. So in the case of scalar fields, suppose it satisfies the Klein-Golden equation. And I discussed that in, first of all, in the global coordinate. And the first thing that you may ask is how you can consider normalizable solution to this. Normalizable solutions are the one that decays sufficiently. So it's important to note that anti-deter space is non-compact space. So you have to consider normalizable state in order for the state to be corresponding to actually the solution, the wave solution to correspond to state in anti-deter space. And one of the criteria is whether you can make action finite. Suppose you find a solution, you substitute that solution into the action and evaluate the action. Is that finite or does it diverge? Okay, so that'd be a criteria for normalizability. Okay, so since the geometry has time-translational invariance, so it's useful to consider a solution whose time dependence is stationary like that. And if you substitute, and then the rest of the thing depends on, these coordinate with omega dependence here. You can substitute this into that and then solve it. And basically you can solve that exactly using hypergeometric functions. I'm not going to go into detail. The answer is that omega is actually delta plus n, where n is non-negative integer. And n just correspond to descendants of conformal algebra. So primary field has omega equal delta, and then you act conformal general algebra generators, and you're going to generate omega which differ from the ground state one by some positive integer amount. And then what is delta? Well, if you substitute this into that, what you find is a relation between the mass and delta. And the relation is that basically this box, if you act on this, turns into minus delta minus d over 2 squared plus d over 2 squared. And then you have to add m squared to it, and then you have to add m squared to it, and then it has to vanish by the Klein-Gordon equation. So this is a Klein-Gordon equation in this basis. And this is a quadratic equation, so you can solve it. And then there are two solutions because it's a quadratic equation. So you have two solutions, delta plus minus, which is equal to minus of d over 2 squared plus, or what am I writing? I'm sorry, d over 2 plus or minus square root of d over 2 squared plus m squared. Okay, so just as if we go back to junior high school, let me draw some picture. So this is delta, this is m squared. m squared is given by quadratic expression in delta, and so it looks like this. So this is a relation between m squared and delta, where the minimum, can you read this here? Minimum is located where delta is d over 2. Okay, and then let me for the later purpose also draw two more places. Let me draw this in red just to distinguish this. And then you have d over 2 plus 1 and d over 2 minus 1. Okay, so there is a relation between m squared and delta. You can see that if I choose plus branch of this solution, it covers this side, because if you choose plus branch, then this will be an increasing function of mass squared. If mass squared increases, delta increases, the minimum value is d over 2, because when m squared, so this is where the m squared is equal to minus d over 2 squared. So when m squared is minus d over 2 squared, this vanishes and this is d over 2, that will be the minimum value of the plus branch, and then it increases like that. If I choose minus branch, it would increase as mass squared decreases. Right, so it goes this. So you have two branches. Okay, now, so one interesting feature is that mass squared negative is allowed. Mass squared negative is allowed. So in Minkowski space, mass squared has to be positive, but in ADS, mass squared can be negative. As far as it is greater than or equal to minus d over 2 squared. So mass squared has to be greater than this, and this is known as Breitenrohler Friedmann band. Okay, so that's one observation. Another important observation is that if you look at this and if you think that you should choose delta plus as a correspondence between this highest weight delta, the conformal dimension and the mass, there is a puzzle. Because in conformal field theory, in d dimension, there is so-called unitarity bound of conformal dimension. That there is actually, if you require unitarity of representation of conformal algebra, there is the smallest value of delta you can have and delta has to be greater than that. And often in many conformal field theory, you have operator at or close to this minimum value. So those should be allowed. But those minimum value, the unitarity bound, is not d over 2, but actually d over 2 minus 1. The reflection of this. So in fact, therefore, this region should be allowed. So in order for the ADSCFT correspondence to work so that you have operator, you have a state, you have field corresponding to state, this range should be allowed up to here. So that means that for this range, you have to choose delta minus branch. Anyway, so that's sort of a relation between normalizable state and the corresponding state in conformal field theory. So you asked me to use Pankare coordinate, so I'm now going to use that. So this is a description of correspondence between field, in this case scalar field, massive scalar field in anti-interspace, and operator with conformal dimension delta. So this is a relation between mass in ADS and conformal dimension in CFT. So how do I do this in Pankare coordinate? So if you use Pankare coordinate, I forgot to say one more important thing before I go to Pankare coordinate. So let me do that over here. So I told you that we are considering this solution to Klein-Gordon equation, and we are considering solution where phi goes like e to the minus i omega t plus some factorized form with rho and omega over here. And we found that omega goes to delta plus possibly some integers. So let's just consider highest weight state where omega is delta. If you analytically continue, if you do the weak rotation, so that the time becomes the Euclidean time tau, then this dependence will give you e to the minus omega tau, right? So what does it mean? So that means that if you consider cylinder and now Euclidean time, and then this omega is now delta, so it goes like that, in the Euclidean time tau, and if you have cylinder like that, then its dependence is like that. Now if you conform or transform that back to Pankare coordinate, where you have a boundary point, and then you have scaling transformation like that, remember that there is a relation between the scaling direction and tau direction. This radial coordinate r is equal to e to the tau. So that tau goes to minus infinity correspond to this point, tau goes to infinity, scales away this. So that means that you can map this type of transformation property to the scaling by r to the delta. So this leads to the following observation, that if you actually describe this type of solution in Pankare coordinate, where you have a metric given by this, then again you can solve the Klein-Gordon equation. This is a solution to Klein-Gordon equation in the global coordinate, and this gives you a stationary solution like that, and I told you which one is normalizable and which one is not normalizable. And we can repeat the same exercise here. If I use this coordinate, then near the boundary where z goes to zero, you have two solutions, because it's the second of the differential equations, so typically you have two solutions, and one goes like that, and the other goes like that, where a and b are some function of x, and then z dictates how this scales as z goes to zero. Okay, so if you compare this behavior where you have z to the delta power here to the behavior here, where if you have a normalizable state with a conformal dimension delta, z to the scale like rescaling by delta, you would see that this should correspond to normalizable state. So you have two solutions to describe golden equation because it's the second of the equation. One is normalizable and another is not normalizable. This corresponds to non-normalizable solution, and then this one corresponds to normalizable solutions. You see that there is actually a map between delta to d minus delta, and this is exactly the map that exchanges delta plus and delta minus. So if delta is delta plus, then d minus delta is equal to delta minus, and vice versa. So that means that if the delta is in this range, the delta plus corresponds to normalizable mode, and delta minus corresponds to non-normalizable mode. If you are in this range between Breiter-Rohler-Friedman bound and the unitary bound of conformal field theory, there is the other way around. The delta minus is a normalizable mode, and the delta plus is non-normalizable mode. This range is a bit funny because it turns out that if you just use the normalizability of the action as a condition for normalizability, then both of these are normalizable. So you have to make a choice. And the choice is made by the requirement that it matches up with the conformal transformation property. So this is a normalizable state, normalizable solution. So this must correspond to a state in conformal field theory. So that means that if you have B of x, then you must have corresponding operator, and then B of x should be given by expectation value of that operator for the corresponding state. On the other hand, this does not correspond to state because this would not be normalizable. State should be normalizable. So this does not correspond to state in conformal field theory, but rather this would correspond to the perturbation generated by source. So namely, you have a Lagrangian of conformal field theory, the original conformal field theory. You can consider perturbing this conformal field theory by adding some source with the corresponding operator and integrated over the conformal field theory space. So this A corresponds to the source. And this scaling behavior and this property matches up consistently because here you look, all has a conformal dimension delta. So if you rescale the coordinate, this would rescale like lambda to the delta. So if you rescale x by lambda delta x, this operator would rescale like that. On the other hand, if you do the rescaling, same rescaling for this coordinate, in order for the metric to stay invariant, that, sorry, I'm sorry. If x is rescaled like that, I should do it more carefully. So if x is rescaled like that, then all is rescaled by this conformal dimension. But in the ADS side, in order for this metric to stay fixed, if x is rescaled by lambda, z has to be rescaled by the same lambda. So that means that b is rescaled in the same way. This would induce the rescaling of b by this. So you see that both sides scales in the same way, so that's consistent. If you go to here, you can also see that there is a similar consistency because now all is scaled like lambda of delta. But since we are talking about conformal field theory, whole thing should be scaling invariant. So this should transform in such a way that it would compensate for the scaling behavior. But we should remember that coordinate also rescales. So that would give you additional power of lambda to the d because there are d coordinates. This is d minus 1? Yeah, yeah, d actually, I'm sorry. You have d dimensional conformal field theory. So this would scale like lambda to the minus d because it's coordinate, so it scales in the opposite way. So then in order to cancel that, you need to have this type of behavior. Okay? So this scaling behavior matches up with the fact that b correspond to expectation value of the operator, whereas a correspond to source that you are introducing in order to turn on this type of perturbation to the conformal field theory side. So I have until 10 or 5 or something like that since I started at 5 minutes late. So I have like 20 minutes left. Okay, so I have to skip some discussion, but then, so this relation suggests that if you have an operator in, so suppose you consider the following thing. So suppose you have ui ads and d plus 1 dimensional space, so that means that in global coordinate you have this kind of geometry and then you have solid cylinder if you like. So this is ads space and then suppose you have this Klein-Gordon scalar field in ads. Okay, so among other field. And suppose you perturbatively quantize system, gravitational system including this scalar field. To the leading order in perturbation you treat all the field as free field and you quantize them and you construct focus space. And then there will be a Heisenberg operator corresponding to this scalar field phi and that Heisenberg operator is a function of t rho omega and it acts on the focus space of this field phi, right? So you have a focus space, which is a Hilbert space, part of the Hilbert space of this bulk gravitational system and then there must be an operator phi hat which creates this particle corresponding to this field anywhere inside the 1-theta space. So let's say here. So you can insert this operator, multiply this operator to the vacuum state of ads gravity and that will create particle at this location of t rho and omega. So here we have time t for example. Okay. So you have such operator and then if you compute expectation value of that it should give you expectation value of this corresponding operator O. Now I have said that one of the fundamental statement of ads gravity correspondence is that Hilbert space of conformal field theory quantized on the cylinder which is a boundary of this geometry should be isomorphic to the Hilbert space of gravitational theory in the bulk. So in the limit where you can do the perturbative quantitation of gravitational theory where you have focus space, et cetera then the focus space states should correspond to state in conformal field theory and indeed they correspond because if you create one particle state that would correspond to primary field primary state of conformal field theory of conformal dimension delta. Now if you have state correspondence between the state you also have to have correspondence between operators. So that means that well since you have this operator which creates a particle at this location in anteater space there must be a corresponding operator in conformal field theory. And that operator must have this consistency condition. So that suggests and this led to many people starting from late 90s to think about the relation between this local operator and the in the bulk and the conformal field theory part. And this is I should say is to the leading order as I said in the perturbative quantitation of gravitational theory in the bulk of ADS space and there are in general corrections due to interactions ADS and other consideration for example here I'm talking about scalar field but if you have a gauge field or gravitational field you have to also make appropriate connection to maintain the gauge invariance and diffeomorphism invariance which I'm not discussing right now but you have you must have some kind of linear relation between operator in the bulk and operator in the conformal field theory. And there must be there must be also similar correspondence between between them in Poincare coordinate which I'm going to discuss. So let me discuss this in the context of Poincare coordinate so then I can write this yes and of course there are gravitons which are also quantized which correspond to some Gaussian fluctuation of the geometry. Yeah so we can similarly quantize gravitons and then there are corresponding operators which are related to the energy momentum tensor on the boundary I'm not doing the gravitational part because it involves some additional subtlety because of the gauge invariance and diffeomorphism invariance it's easier to discuss in the case but there are literature especially people like who have studied extensively about the subtlety coming from this type of gauge invariance so we are interested in understanding how this relation works out there must be a real linear relation between the local operator on the boundary and local operator corresponding to creation of particles in the bulk up to interactions and this function G is called the smearing function and in order for this to be consistent with this property it has to satisfy the condition that since we want all hat phi hat of z of x as z goes to 0 goes to this local operator in the conformal field theory so that means that correspondingly this smearing function has to have the property like that as z goes to 0 and this should also satisfy the Klein-Golden equation in the bulk with respect to undotated coordinate and you can find a solution to such a set of conditions and I should note that if you look at the ADS-CFT correspondence literature then there is a notion of bulk boundary propagator this is not the bulk boundary propagator that satisfies a different boundary condition but you can actually find in certain cases you can actually find explicit form sometimes it's not a smooth function as you can see it involves delta function so you have to actually extend the notion of function to include the generalized function to define the smearing function properly but that's fine because it's actually integrated over with this local operator so for example, even if this contains delta function this integral makes sense okay, so I was going to discuss a little bit about how to construct such smearing function but I don't think I have time for that I'm just going to mention one reference so there is a paper by Hamilton Rift sheet 0606 141 I forgot this now in this literature, do you remember? yes, excuse me so for this reason in particular the construction of G along this literature is called HKLL and so this is actually there is a long series of references and this is one of them but this is actually a particular one that I'm going to use and so in particular we can so what they showed is that you can actually choose G in such a way that it has support on only part of the boundary so here I'm assuming that I'm integrating over the entire range of the boundary but actually it turns out that in order to construct a local operator in particular part of the bulk you don't have to integrate over the entire boundary part you just have to integrate some sub-part of the boundary so let me spend the remaining 10 minutes in specifying which part of the boundary that this particular smearing function requires to integrate which actually leads to some kind of paradox which is sort of one of the main going to be a main theme of this lecture but let me make a preparation for this so actually today it looks like I'm actually running a little bit slower I'm covering less than I had prepared to talk about because I realized that I had to tell you a little bit more about ADS-CFT correspondence but that's fine so let me in order to specify where to integrate so here is what I want to do so I have this anti-deter space and so what I'm going to do is that I'm going to tell you that suppose you have you want to construct bulk local operator at some particular point in the bulk I'm going to show you that it's sufficient to integrate this equation because you know that if you have free field theory so you probably did you take course in quantum field theory right? so the first thing you learn is how to quantize free field and the quantized free field, the Heisenberg field for the quantized free field satisfies the equation of motion you solve this equation and then you find some partial wave solution, each of that has a creation and analysis operator on it that's this I have that's why this satisfies this equation because I'm just doing the perturbative quantization of the local field in the bulk the first thing to do is to quantize free field which is the procedure is to start with solving linearized free field equation and then take the coefficient to be creation and analysis operator that's what I'm doing here and I'm saying that that should be expressible as a linear combination of local operator in the conformal field so this is interesting because this means that you can actually reconstruct local operator in the bulk in terms of linear superposition of local operator in the boundary CFT so I just wanted to decode this dictionary a little bit more detail now so this operator phi hat is located here and I want to construct this local operator as a linear superposition of this boundary operator in principle I told you the up real reason why you can construct it as integral over entire boundary now I'm going to tell you that you can actually you don't have to do that you can only consider some subspace of the boundary so let me specify which subspace we will be talking about so this boundary is z equals 0 so let's open up this z equals 0 so you have z equals 0 over here so this is just a boundary of ADS and this is time direction and then this is space of CFT suppose you consider you consider some subspace of this space-like direction is this clear so you consider this subspace here we are considering this subspace a here and now I'm going to define what is called domain of dependence of a so domain of dependence of a is defined as follows so suppose you pick any point in the domain of dependence and then take any curve that is time-like then it has to pass through a either in its future or in its past so collection of such point the set of points which have this property is called domain of dependence so it's typically of this kind of square region and these lines are light-like because for example if you pick a point here you can actually go extend the time-like curve indefinitely without crossing a so this point is outside of the domain of dependence this point is inside of the domain of dependence so this is d of a so this is a boundary so this is d of a this is my definition of d of a so now we have this region so we have this anti-litter space so now I'm going to tell you that if you have this so now I have this domain of dependence on the boundary so I have this domain of dependence and then I'm going to tell you that how much which part of anti-litter space by doing integral like that so what you can show and what was shown in explicit example by Hamilton Kabat-Liff system now is that the part of anti-litter space that you can reconstruct is called causal wedge so in the remaining two minutes I have to draw a picture of causal wedge so what I'm going to do is a following so so let's reduce everything by one dimension so that the boundary is now line so so far the boundary is drawn just like two dimensional cylinder it's actually d minus one dimensional circle sphere times the time but let's just shrink everything along this line so then the domain of dependence sort of now presented as a point and then domain of dependence is somewhere between here and here so you have domain of dependence here I hope you follow me so I'm just considering this domain of dependence but now I've represented it as one dimensional line segment the causal wedge is defined as intersection of future and the past of this domain of dependence in the bulk of ADS D plus one so here you have CFTD so you consider all possible future of this domain of dependence these are the future of the domain of dependence and then you have a past of the domain of dependence there is an intersection of this what it means is that if you pick any point inside of this causal wedge you can draw future pointing time like curve and you can reach D of A you can draw past pointing curve and you can reach the boundary so that's your causal wedge so the statement is that this construction works provided that Z provided that integral is restricted over D of A and Z of X belongs to the causal wedge of A so previously the integral was supposed to be all entire boundary of ADS the entire space that CFT is defined but it turns out that you can actually restrict this integral so the domain of dependence of A provided that the point you are interested in reside in causal wedge of the same region A so this is a very interesting statement that you can reconstruct operator in the bulk by using only part of the operator algebra of the acting on the Hilbert space not the entire space now this leads to some paradox that I don't have time to discuss today and then resolution to the paradox reveals that the state in this conformal theory dual to the bulk gravitational theory where we can describe which we can describe in terms of this perturbative concentration has particular entanglement property that's where actually the notion of entanglement comes in the title of the set of lecture is entanglement and geometry and this fact that you can reconstruct this operator by just using part of the Hilbert space the subspace associated to this region A leads to paradox whose resolution uncovers very non-trivial interesting and deep entanglement for state describing this kind of gravitational system and we can exploit that to learn various things about quantum gravity theory in the space so that's the subject in the next two lectures okay thank you very much time for one question because we are