 gravity, but in quantum gravity, but in gravity, most of the time, the time is meaningless because time is just a coordinate which can be changed by arbitrary different morphisms. Yeah, time, time different morphisms. Okay. So, so that caused lots of problems when we try to interpret what evolution means in gravity, in particular in quantum gravity, etc. So, in ADS, here I say most of the time, because in some situations, for example, in ADS, the situation is actually better because in ADS, there's an absolute asymptotic time because ADS has a time like boundary and from our standard definition of gauge transformations of different, yeah, the so the standard definition of the gauge to morphism, they should go to identity at infinity. So, so the time at infinity is actually reparatualization invariance under the standard, the gauge transformation. Okay, so that time actually has absolute meaning. And of course, that is identified with the boundary time. And this underlies our standard story for ADS AFT. So that's also the reason the quantum gravity in space times of other asymptotics appear much more difficult, because there's no such asymptotic time we can try to talk about evolution or try to talk about quantum mechanics. So, but even in ADS, there are many puzzling questions. So the obvious question is that can the asymptotic time be sensibly extended to the interior? By sensibly means we mean in the way which is different morphism invariance. Okay, of course, you can always introduce some kind of coordinate time in the bulk. And we believe often, yeah, we believe certain physics reflected by that coordinate time should be somehow different morphism invariant, but the question is whether there's a well defined way to see that. Anyway, so if the bulk space time is time translation invariant, if there's a symmetry, say time translation symmetry, and then indeed, we can sensibly extend this asymptotic time to the bulk, because now we can define a preferred time slicing in the bulk geometry. But for general time dependent the case, the situation is far from clear, because now there's no symmetry to give us a preferred slices and what time is meaningful then it's become not very clear. So here we want to, so here, so interesting intermediate example is internal black hole in ADS. Okay, so, so, so this is a familiar story in ADS, which are internal black hole in ADS should be described by two copies of your CFT in the, in the sum of your double state. And there's no interaction between the CFT and but they entangled. So this geometry is interesting. It's because for this geometry, there exists a time like cleaning vector outside of horizon. So because of this cleaning symmetry, because of this time translation symmetry outside of horizon, which we can actually sensibly extend this boundary asymptotic time to the region outside of horizon. But, but this geometry does not have a global cleaning time like cleaning vector. Okay, so you can only so the extension of the boundary time actually stops at the horizon. Okay, because of the region inside horizon, there's no time like cleaning vector. And so, yeah, so this is an interesting intermediate case, which we have part of the space time, which we can actually extend the asymptotic time to the interior, but not the full space time. Okay, yeah, of course, this feature actually, yeah, so there's a time like cleaning vector outside the horizon. So this feature, of course, is the source of many mysteries we have regarding internal black holes in ADS. For example, what are the, what is the interpretation of this F and the past regions, say in the boundary, can we give a intrinsic boundary theory with description, say of such geometric regions. And what's how do we describe cross-collect time. Okay, so we can extend this washout, we can extend the boundary time to the box washout time. But, but if you want to explore behind the horizon, then you need a emergent cross-collect time from the bound. Okay, then and also how do we see actually, there's a sharp horizon and associated causal structure, say intrinsically from boundary theory point of view. And also there's hardening questions regarding you can send, say if you have, say, excitations from the right and the L regions. And then you can send, send excitations and they can fall into the black hole and the intact with each other. So even the original CFT, they don't seem to be intact with each other. Yeah, there's no interaction between the CFT L and the CFTR, they're just in the entangled states. Okay, so how do we interpret this kind of interactions? So, so all these, say, kind of mysteries are related to the, the mystery of how to, how do we understand the box time. Okay, so, so let me just quickly mention. So, so what's the long description, say, of the bulk evolution, using the boundary. So the simplest one is of course the we can evolve the boundary CFT by HR minus HL. So HR is the Hamiltonian, say over the boundary theory. So the R corresponding to the Hamiltonian for the right theory and HL is the Hamiltonian for the left theory. So this operator have a sensible description in the bulk as just time translation in terms of the social time. Okay, so if you have a T equal to zero time slice, then I some finite T, then, then, then action, yeah, then time slice become like this. So this evolution is completely smooth in the bulk. Okay, so you can also imagine an evolution, say with HR plus HL. Okay, so he realistically, this HR plus HL acts on the black code geometry like this. Yeah, so, so this HR minus HL, of course, no matter how you evolve it is always outside the horizon. So you can try to do it HR plus HL. Actually, this action is not well defined because of the action actually create a kink near the horizon. Okay. And so this is related to that when you look at, yeah, this is related to the issue that when you go to in the larger limit, yeah, because the geometry is described in the larger limit of the boundary theory. And in the larger limit, the, the, when you go to the sector which involving black code states, neither the Hilbert space or the Hamiltonians were defined. Okay. Yeah, so that's a source of this kink. But we would like to see, so in order to explore the region behind the horizon, yeah, even if you allow this kink, the HR plus HL, HL only explore the region outside horizon, which is just, yeah. So, so what we would like to see some kind of smooth evolution. Okay. So from T equal to zero slice into some other Cauchy slice, which can actually cross the horizon. So that we can actually really see the region behind the horizon. And so, so likely this kind of evolution can only be emerging because of the, we already used the Hamiltonian given to us. And, yeah, yeah. So, so the goal of this talk is to explain actually how to use the boundary theory to construct intrinsically this kind of crucible like time evolutions. Okay. And also we will explain from the boundary theory point of view, how does the horizon and associated causal structure rises. Okay. So, so I want to emphasize, so in the past, of course, there are many, many work to explore the region behind the horizon using the boundary theory. And for example, look at various correlation functions, entanglement entropy, complexity, all those observables, they involve, say, when you compute them using gravity, they involve, say, a geometry behind the horizon. But, but none of those observables actually intrinsically tells you how those geometry behind the horizon arises. Okay. And also, they are hkl kind of construction, which if you put the operator, say, say inside the horizon, then you can express it them in terms of boundary operators. Yeah, for example, hkl and the curly arcs and the Raju, they have described that. But our purpose is different. So, so our purpose is not say just to express the boundary, yeah, this bark operator inside the horizon express in terms of the boundary theory, which, which in their previous constructions, they actually used the input of the bark time. So here we want to really see where does the bark time emerges. Okay. So, so we want to do everything. Yeah, to understand the bark time itself from the boundary theory. Okay, so here's the plan of my, my talk. First, I will outline the main results. And then I will talk about, say the, we'll, yeah, some kind of a background behind those results, which is the, which I will remind you a little bit, the entanglement structure for relativistic quantum field theory. And in particular, the structure of the type one volume algebra in describing entanglement in the, in quantum field theory. And also, I will describe this some special structure associated with this type three volume algebra that's called the half sided module inclusion translations. And then we describe how we actually construct this emergent time from the boundary theory. And then I will end with some conclusions. So, so before I proceed, do you have any questions? Okay, good, good. Then I will just describe the main result. So, so the main result is the following. So we will show there exists evolution operators on the boundary, which we can say they exist. Some operators say Hermitian operator G, which we can construct in the boundary. And then we can construct the, the one parameter unitary transformation associated with this parameter, this commission operator G. And then this G is satisfied the following properties. First, that they actually it involves the both R and L between them in a highly non tubular way. So it's not like HR plus HL, which you just, yeah, just, just add them a fact. Yeah, so, so you don't have the factorized structure. And the second way to more important property is that this G is actually bounded from below. Okay, so G, that means G actually behaves like a Hamiltonian. Okay, so normally, so in quantum mechanics, how do we distinguish say time translation from a spatial translation, or, or other internal symmetries. So, so what distinguishes the time translation is that the generator of time translation, which is Hamiltonian, has a spectrum which is bounded from below. So in contrast, say if you have a spatial translation, which goes back into momentum, or some internal symmetry, you certainly don't have this property. Okay, so, so, so, so this property, which makes this S, we can qualify it as a time, okay, we can qualify it as a time. So now you can, now you can act this unitary transformation on the operator in the, for example, in the right region of the black hole spacetime. Okay, so, so here, our input, you said we will assume that using the standard, yeah, because we can extend the boundary time to the region outside the horizon. So we have a straightforward description of the region outside the horizon using standard ADSAFT dictionary. And in particular, any, any local operator outside the horizon, okay, for example, in the R region, we can just treat this as, as some kind of boundary operator. Okay, so this, you can all either think of it as hpll, or think of it as the way, say, just think about the mode expansion of the spark fields, say in R region. And then the A and the a dagger, when you write the mode expansion, so that a dagger is essentially the boundary operator. Okay, it's the, the boundary operator for the generalized free field in the, in the boundary series. So, so anyway, the, so our input is that we can treat the any operator outside the horizon as some kind of boundary operator. Okay, and so this, and so this, and now we can just evolve such a boundary operator, okay, which does have a geometric interpretation in the box using this evolution operator. Okay, and then we will see actually this, such kind of evolution can actually show us the sharp signature of a horizon, and, and essentially generate for you the region inside the horizon. So, so yeah, so take it inside the horizon for sufficient largest, and it can be sharp signature of the horizon and generate the, the future and the past region from the right hand side. So, Hong, I have a question. Yeah. So, in GR, the time evolution can be space dependent, right? So here you're, yeah, what is the constant time slice here correspond to in the bulk? Yeah, right. So, so here we, we just consider, so this evolution operator, you should imagine it as the evolution operator which shifts the whole time slice. Yeah, yeah. But what is that time slice? Is the, oh, the time slice will, I will show you actually the infinite number of such U.S. you can construct, and they will you show you some examples of such time. But it's, it's, yeah. But presumably you can construct also local time evolution or not? That's right. You can. In principle, you can. Yeah. Yeah, just say which is not homogeneous in the, yeah, which, which we say have some non-chip dependence on the spatial coordinates. Yeah. Okay. Sorry, Hong, maybe I'm dropping ahead a little bit, but connected to a thesis question, this U of S will not necessarily be local in the bulk, right? It will not, it cannot be represented as a local evolution of the slice. Is that, is that correct? Yeah, we will see actually in certain regime they can be, they can be represented as a local slice or the local evolution. And in general, indeed the evolution is non-local. And by in certain regime can appear as a local. Yeah. Okay. And the, and the, yeah, I want to emphasize this G. You should view it as some kind of global Hamiltonian, which involves the whole slice. It's not some kind of local. Yeah, it's not some local action. The G just is more, G is something you have already integrated over the whole, the whole core surface. It's a global operator. Yeah. Okay. Can I also? Sure. So if I evolve for long enough using this G, would I also see the singularity? Indeed. Yeah. Yeah. So we should, yeah, we should. So, so, so, but to see the singularity explicitly in our construction actually, it's a little bit tricky. So, so I would say that aspect is not completely settled. Yeah. Yeah, just technically to see it, to see a, see a sharp signature of the singularity, it's a little bit tricky there. So, yeah, I believe we should see it, but we haven't really pinpointed very precisely yet. Thanks. Okay. So, so in the case of BTZ, you can actually work it out very explicitly. Yeah. Actually, say if you, we, now we believe if you do the JT gravity, say if you do two-dimensionals, the BTZ is three-dimensional. If you do two-dimensional, actually, maybe the evolution is always local. So, yeah. So here I'm just only describe the BTZ case. So for BTZ, actually, you can construct such kind of U.S. explicitly. And so again, we will look at this kind of evolution. And then it turns out, say, yeah, let me just describe what happens for one example of such kind of U.S. Okay. So one example, which we can work out. So, so you find that they actually exist as zero. So, so as equal to zero is just the identity, and then you evolve it with either as greater than zero, as smaller than zero. And so for this particular choice, for some particular choice of this U.S., we find that they exist, say, some finite value S0 greater than zero. And for smaller than S0, this operator remains in the right CFT. So remember this operator, we start from the right region of the black hole. So by definition, this is the operator in the right CFT. So we find that the, so for smaller than S0, this operator always exists in the right CFT. But then beyond S0, suddenly the degrees freedom in the left CFT appears. Okay. And so we interpret this as a signature of a sharp horizon. Okay. So indeed, geometrically, that's what you would expect when you take an operator in the right to cross the horizon. Okay. So imagine you have some kind of evolution, take a right operator from here, cross the horizon, and then you evolve. So before you cross the horizon, this operator is still in the right region. And so this operator will only involve right degrees freedom. But then there's a critical value when you cross the horizon. And once you cross the horizon, you are in closely connected with the left region. And then you must involve the left degrees freedom. Okay. So this is precisely what you expect. Yeah. So this feature is precisely what we expect, geometrically corresponding to a sharp horizon. Okay. But this formulation is actually general. We don't actually refer, we don't actually need to refer to any geometry. So you actually formulate this feature for any, say any quantum system, any two quantum system, you can entangle them, say in some similar field, double state, et cetera. And if such U.S. and S0 exist, you can say somehow that these two series actually are secretly connected. Yeah, these two series secretly have some kind of emerging horizon. Okay. And a structure. So we are saying that they're closely connected. Okay. So that means that actually, you can actually cross the horizon and then they become connectable beyond the horizon. Okay. So, so now let me say a little bit about this value of S0. So for this particular example. Can I ask you something? Yeah. So somehow this construction of generalized free field, which you are using here implicitly, right? Yeah. Yeah. Yeah. Yeah. That doesn't exist for just random. I mean, if you have a CFT without any large n limit or something like that. Good, good, good, good. Yeah, indeed. You cannot really say that. Right. You cannot even start this construction, right? So, so, yeah, so this is a very good point. So, so to make this definition, you don't need large n, to make this definition, you don't need large n. You can just say, let's consider two quantum systems and then take an operator from one quantum system, from one copy, from one quantum system. Do I exist operator which actually satisfy this feature? Okay. So, so this can be defined for any quantum system without large n without any holographic description. But indeed, then you can prove, which we will not describe here, because I don't have time, but, but Sam will, in the second talk, he will describe a proof that indeed this feature, this sharp S0 can only exist in a large n limit. Yeah. Yeah. Yeah. So even though this definition, you can define it in general. Yeah. Good. Yeah. So this would be a proof. They do this, at least would be a, I won't say a proof, at least would be a strong argument that the horizon can only emerges in the, in the angle to infinity limit. I mean, it's a, it's a larger artifact. That's right. That's right. Yeah. Yeah. So this is not should be a universal in some sense. Is it what you mean by sharp, by sharpness, meaning that if I use different operators, I will always cross the horizon at the same time. No, no, no. The horizon at the same time depends on your initial, this time will depend on the location of your initial operator. I put insert different operators at the same initial time, then I should always get the same as not, right? Yeah. Yeah. That's an interesting question. That answer, I don't know, because we only say, yeah, just for technical reasons, we only look at the scale operator, we didn't scan, say all other operators, but I expect it should be the same as zero. It should be the same as zero. But here by sharp horizon, I mean something different. So, so sharp horizon, I mean, there's a precise value of S zero. Okay. There's a precise value of S zero. So, so, yeah, so that means the sharp horizon, because it's a diacrophase transition. So before the S zero, there's no CFTL, but then after S zero, suddenly CFTL emerges. Yeah. So that's what I mean by sharp horizon here. And if you go to finite N, then you find such a sharp S zero does not exist. Even for any value of S zero, there's always some tiny tail of CFTL. Yeah. Okay. Thanks. Good. I have a question. So you said that for BTZ, there exists such a US, which presumably means that there exists such a such an operator G. My question is, how genetic is this operator G? And whether with some different operator G, there would, there wouldn't be any horizon anymore. Yeah. Yeah. I believe this, yeah. So this G is very generic. So, so, so we didn't, yeah. So, so our construction is very general. Does not, yeah, works for any series, say BTZ, any dimension, et cetera. But we worked it out explicitly in BTZ because this is the simplest non-trivial example, which is not controlled by symmetry. So if you look at JT gravity, then you merge in the US where we control the by symmetry. And yeah. So, and then you can also look at how many, so BTZ is the simplest non-trivial example, which is not controlled by symmetry. And yeah, but, but the construction is very general. Yeah. I see. Thanks. Okay. Good. Sorry, let me, sir. Yeah. The phi of X is a bulk operator, right? X. Yeah, phi is a bulk operator. Yeah. So if you don't have some kind of a generalized refuel construction, yeah, how do you relate it to a boundary operator? Yeah. So, so, so here, if you just talk about abstractly, of course, the, we can just formulate this, say you can take any operator in your CFT to see whether this kind of phenomena happens. Yeah. Just say whether they, yeah. For general quantum system, you can just ask the question whether there exists such operator in your right CFT that such, such phenomena happen. So if there exists such operator in your right CFT, such phenomena happen, and then you may say there's an emerging horizon. But let's see. X includes also the holographic coordinate, right? X is not capital X. Yeah. So what is the analog of that in general? I see. Oh, yeah. But this phi X, you can just imagine it as some complicated operator in the CFTR. Okay. Yeah. But just because of the holography, we can indulge some kind of geometric meaning to this phi X. Yeah. But in principle, we can just view it as some complicated operator in the CFTR. And the holographic coordinate, would you interpret it as just some scale at which the operator is defined? That's right. Just say because for the region outside the horizon, we have the standard correspondence based on generalized, say the bulk freefield corresponding to generalized freefield in the boundary theory. So, yeah. So, yeah, as I mentioned, we use the, we use the bulk reconstruction outside the horizon as the input. So we assume that the, you can actually use in the standards wash your time to already reconstruct the bulk. Yeah. And the radial direction to reconstruct the region outside. Yeah. We want to see, with that as the input, how does the interior region arises? But isn't this very generic? Like, couldn't I take this phi X to be any, like any two systems, right and left and take phi to be purely in the right system and then evolve it with a generic U, wouldn't it, at some point, start mixing? No, no. If you just do it generically, then you will not say, if you do it generically, you either have the following two situations. Either they always mixing, depend on your choice of U, they always mixing, or the level mixing, okay, the level mix. And the non-trivial thing here is that there exists a sharp value S0, they don't mix before S0, but mix after S0. Somehow, we are saying if I start from something that is not mixed, so is this my, my phi of X, the thing that I start from is just made of the right degrees of freedom. Then I apply a generic U that involves both right and left. Are you saying that generically it will not mix with the left? No, no, no, no, no. I'm not saying that. No, I'm saying if I just take a right operator only in the right, and then I evolve it with some U S involving both left and the right, then generically, we'll just always mix left and the right. Just even S, any infinite S will already evolve. Oh, I see, one finite is not to, okay. Yeah, so the non-trivial thing is that it exists a finite S0 greater than zero, somehow before that does not mix, and after that mix. Yeah, here is the most important one. So does that answer your question? Yes, thank you. Okay, good. Okay, good. So now let me explain a little bit the value of S0. So this value S0 actually depends on your choice of initial operator, and also will depend on the U S, excited choice of U S. So we can construct a particular U S and so let me just show the Kruska diagram just here like this, and then the U and the V axis corresponding to the horizon. So here I'm just surprised that the boundary and the singularity. Then I imagine we have a point, initial point is in the right region with initial value U 0 and B 0. So I want to emphasize in the boundary construction we don't use the Kruska coordinate or we just evolve it, okay, and then somehow this Kruska coordinate emerges. Okay, then it turns out if it happens choose the say X to have this U 0 and B 0 as the initial point, and then you find that this S0 is actually given precisely by the by the law of distance along the U direction from this point to close to the horizon. Okay, so this is S0 just give you the minus U 0. Okay, so there is such a U S which somehow just precisely give you this minus U 0. And furthermore if you take this point to be close to the horizon and then you find that the transformation is actually local. So actually just give you a Kruska law translation. Okay, so some for this particular choice of U S. And so in general if you take X some generic point in the box in the right region actually the transformation is not local. Okay, the transformation is not local. And but not local in the way which preserves the across the structure in the sense you still even though has a finite spread but still it takes only the value of S0 for it to cross the horizon. So the cross in the horizon is still sharp. Okay, so even though the the transformation is not local. And you can actually explore the the causal structure a little bit more sharply. So let's consider again this is a Kruska diagram. So the so the U and the V axis corresponding to the horizon. And now we can consider you evolve this operator in the right region. Okay, start as X1. And then you try to look at the commutator with the operator in the left region. Okay, so so you look at this commutator. So so in order for the commutator to be non vanishing. So this evolution have to take you inside the light cone of this X2. Okay, so so that's the only way the commutator can be not there. And then you see indeed the similar thing happens. So so there's a law of distance to go from this X1 to cross the the the light cone of X2. So that's essentially the law of distance is this U1 plus U2. So remember the U1 is negative for the point in the right region. Anyway, so even though this evolution is not local. So here is just a cartoon. I don't take this yeah so this shaded region just means this operator is spread. And but but don't take this spread to literally just say it really is a cartoon. So so so it turns out for smaller than this value again, you get zero. But then there's a sharp S value. And then indicate you have crossed the right across the light cone of this X2. And then now you suddenly find that this is not there. Okay, again, it's like some kind of sharp transition. Okay, a sharp transition. So this tells you that even though this evolution is not local, but somehow this actually presents somehow exhibits still exhibit the sharp course of still exhibit the sharp course of structure which you would like to see in the book. And but actually, there's a regime actually this transformation actually is local. So this for the BTZ case, this is local. When you take the dimension of this boundary operator dimension to be large. Okay, and we know that the when the operator dimension become large, there's some simplification say in the two point functions. Yeah, yeah, just there's some new structure in the in the behavior of these operators tend to in this regime. So if we also say average over the boundary spatial direction just for simplicity. And then you find actually this transformation become local in the sense if you start with the point in the right region. And then this takes you to a lot of points to the box get a feel that a lot of points are evaluated a lot of points on the gravity you can match it. Okay, so so we just evolve it using the boundary operator. But then you find that the result of the evolution can be matched exactly with the with the local operator in the book. And the and the intense out of the transformation is like this. So, so the transformation expressed in terms of this cross coordinate. Yeah, I'm just telling you one example, corresponding to you zero plus s. And this vs transforming this way. Okay, so the so the cross coordinate transform this way. And for you, we just the law shift. So we can geometrically we can draw the diagram. So if I start with a point some point here. And then you find that the evolution for each point is just like that. It just evolves. You can just draw this flow flow line based on these formulas. Okay. And and this is the this is the constant s slice. So if you start with s, you start with the t equal to zero slice for the black hole. And then let's see, then, then let's look at the finite some constant value of s, then you find it's actually evolves this way. Okay, so it's closer to the horizon, etc. And so, so, so for this evolution, the, the, the, the near the horizon is actually just a lot of translation. Yeah, it's a lot of translation in, in you. And we, we also construct similar transformations, which is the law, which corresponding to the law translation in v, but then the u transformation will be similar like this. Okay. So when you say large mass it is with compared to the strings key compared to. Yeah, this is just the large mass just means that so the mass of the bulk field or the dimension of the boundary operate will always take to be large or the one radius in the radius. Yeah, that's right. That's independent of n. It's always independent of n, but it's parametrically large. Yeah. Okay. Okay. So, so the bottom line is that we want to start with the bulk reconstruction of our standard safety dictionary, say, for the right region and for the left region. So if you just using the standard bulk reconstruction, you don't really know whether this L and the right region are connected, whether they exist future and the right path region. So by construction, this us essentially, you build this whole space time diagram. Okay. In principle, you can build this whole space time diagram. And so this us lets your new, say, emergent informing time. Okay. Yes, I showed you can take you from one slice to another slice, etc. And in particular, actually, there's an infinite number of choice of such informing times. So this is, so this is consistent with our intuition actually in the somehow, yeah, on the gravity side, there, there, of course, there's an infinite choice of times, but we're not sure whether those times are different. Yeah, whether we're, whether, but normally, we're not sure whether all those times corresponding to some kind of different morphism environment, the notion of time evolution. But here actually, we actually see actually there. Yeah. So this construction is by definition, a different morphism environment. So this gives you actually infinite number of choices of different morphism environment time to describe in the book. Hi, I have a question. So do you start with an unentangled state? Or do you start with some sort of thermo field double? Because if there's always some level, yeah, it's always some of your double. Yeah. So here, I'm only talking about some of your double. Okay. No, in the sense that I mean, if you start with some entangled state, then I mean, it's, isn't it obvious that the two CFTs, I mean, they're at least in the bulk looks like the eternal radius, what's it? No, it's not obvious, right? It's, it's, it's certainly far from obvious. If you start, if you start with the, the standard dictionary, say, for the, for the finite temperature is two or two, the region outside the horizon. And, and if you even think about some of your double picture, say they entangled, it's not clear that actually they're connected in the bulk is connected. So there are various conjectures based motivated by this internal blackout geometry, say they should be connected in the bulk and say, yeah, you can do EPI, et cetera. Yeah, but this, these are all motivated from, from already looking at the internal blackout geometry. Yeah. So here we want to just say, let's just start from this picture. Okay, let's just start from this picture. I know that finite temperature CFT, I can really describe it using the region outside the horizon and the same thing for here, but the connect actually really constructed this whole, whole connected picture. Yeah. Okay, good. So, so now let me talk about the yeah, remind you, yeah, just talk about some, some features of this entanglement in the right V6 CFT, which we'll use as a motivation to explain how we do the construction on the gravity side. So, so first, just very quickly remind you, say normally when we talk about entanglement of quantum systems, we say they separate into two subsystems. And we assume that here was space factorizing to them. And then you can construct reduce density matrix in some general states, we can construct reduce density matrix to trace out one subsystem. And so, so this will reduce density matrix we contain all kinds of information we are interested in. Okay, for example, you can construct the von Neumann entropy, et cetera. Okay. So, so in this context, if you just know the row one, you already know essentially all the entanglement information about one or two, yeah, contain much more than you want to know. Yeah. Anyway, but, but, but there are situations, which is row one, the row two, so you can also define row two, are both full rank. Okay, full rank means the two systems actually are highly entangled. Okay. So if they're not entangled, then row, row will be rank one will be a pure state, that will be rank one. So, so if it's a full rank, and then that means these two system are highly entangled. But then they are full, but when they are both full rank, actually, there's additional algebraic structure. So this is now you can define an operator like this. So called a modular operator, we take a row two times row one minus one. Okay. So this operator has an inverse because row two is also invertible. And so in this case, so this operator have the last feature that when you act the operator in subsystem one, it remains in the subsystem one. When you act, say if you explain it, yeah, by some imaginary amount, so act on the operator, yeah, it can generate the unitary flows, which still take the operator in region one, still remain region one, and the operating region two still in region two. Okay. So, so this module flow generate automorphism, say of each subsystems. Okay. The operator algebra of each subsystem. So the existence of the module flow also is an indication that these two systems are highly entangled. Okay. Well, this is only possible when these two are full up. Sorry, in the definition of the modular operator, is there a direct product or it's just a matrix product? Oh, this is a direct product. Yeah. Yeah. Thanks. Yeah. Yeah, I'm just using the simple notation. Yeah. So now let's go to entanglement in the quantum field series. So let's consider QFT in the Minkowski space time. So the simplest case, you just can see the entanglement between two half space. So that's corresponding to, you can separate the space time into window regions. And so the causal diamond of the right half space is just the right window region. And then that one for the left window region, then you also have future and past regions. So it's often said that Minkowski vacuum state can be interpreted as some of your double state for the right and L window patches. So this is a famous on-route story. So yeah, but this statement is actually not true. Strictly speaking, okay. So the strict speaking, the statement is only correct when you actually discretize the theory. Okay. So if you put the system on the lattice, then, and when L and R don't share any lattice points, and then this statement is rigorously true. Okay. And the, but in the continuum limit, actually this statement is not correct. Okay. And the, but, but normally this subtlety is not that important for, for many physical purposes. Okay. But this subtlety, the difference between the continuum and the discrete limit actually is important, very important for our current purpose. So let me just emphasize that. Okay. So I want to emphasize there's some fundamental differences. Between the discrete and the continuum cases. Okay. So let me just contrast you the discrete and continuum limits or the differences. So in the discrete case, there's a local Hilbert space for L and R. So you can always factorize them. So essentially by definition, but in the continuum case, there's no such factorization. Okay. So, so none of any states which can be factorized into HR and HL actually have infinite energy in the continuum limit. And so, so in the continuum limit, those states just drop out of the Hilbert space. And also here, you can just trivially define reduced density matrix because you have a tensor product structure. But here, you cannot because you no longer have the tensor product structure. And here you can have finite entanglement entropy, of course, this entanglement will depend on the cutoff, net spacing. But when you take the continuum limit and this entanglement will become infinite. Okay. It's not very fine. So the only thing remain, which is common, when you take the continuum limits, is that actually the module operate and the module flow exists. Okay. So all are the standard description of entanglement using reduced density matrix, they just don't apply. And by the module operate, the module flow still exists. Okay. And that becomes the only way. Only way we can simply talk about say entanglement in the intrinsically continuum limits. But still there's a fundamental difference between the module operate and module flow between the discrete and continuum case. So in the discrete case, as I mentioned before, this module operator can be factorized, you can write it as a row two times the row one minus one. But in continuum limit, actually, there's no such kind of factorization. You cannot factorize this module operator, say in terms of some product of the operator in the left region or in the right region, just cannot be done. Okay. So you involve both L and the right operating is somewhat complicated. Yeah, it's somewhat more intricate. So in particular, in this case, there's no sharp right cone. Okay. So when you put the system on the lattice, so whenever you evolve the system a little bit, and that always have some kind of small tail, especially small tail outside the right cone, so there's no sharp right cone. But of course, in the continuum limit, there is sharp right cone. So it turns out that all these differences between the discrete and the continuum case, which normally we treat as some kind of technicality, okay, say when you evolve UV divergence, et cetera, some kind of technicality, it turns out that all these differences, they can be attributed at more fundamental level in terms of the difference of the operator algebra structure in the two regions. Okay. It turns out, the operator algebra in this right window region have a very different structure when you're in the discrete or in the continuum case. So in the discrete case, you have type one monomer algebra, and in the continuum limit, you have type three monomer algebra. So we're not going to detail how to define monomer algebra under the classifications, et cetera. So let me just, the type one monomer algebra is the one which you can always define it on the Hilbert space. Okay. So because here, it's always factorized, so you can, yeah, so you can always define on the Hilbert space. In particular, here, you just have the standard projection operator associated with a Hilbert space. But for type three one monomer algebra, actually in here, all the projectors in type three one monomer algebra, they are infinite ones. Okay. Under the equivalent to identity. So, yeah, so this is highly non-intuitive kind of algebra. And, but you can view all those features as some kind of physical manifestation, say, of this type three one monomer algebra. Okay. Good. So the story is very general. So you take any local region, so emphasize any local region in the, in the relativistic QFT, and the local operator algebra cannot be associated with the type three one monomer algebra. So it's always the type three one monomer algebra. And so this is a universal statement. And, and in fact, this is the only way we can actually talk about the modular flow associated with this one monomer algebra is the only way we can talk about the, the entanglement between this region and the complement region, say, in the intrinsic continuum limit, okay, in the intrinsic continuum theory. Yeah. Good. So this type three one monomer algebra turns out, give you some additional structure, which is, which is not present in any other kind of monomer algebra. Okay. And so this new structure associated with this type three one is called half sided modular translation and inclusion. So let me just first state the mathematical language. So suppose M is a monomer algebra, and you consider some vector, which is cyclic and separating for M. Okay. So translates into physical language is that let's just take this M to be the render operator in the right region. And this omega states can be considered just as a vacuum state of the your standard the minkowski vacuum states. And the saying that this omega is cyclic and separating for M for this region in the right for the operator algebra in the right window region is the standard statement, say, of the rich rider. Okay. Should I that means that if you act the operator algebra in the right region on the minkowski vacuum, the resulting states are dense in your Hilbert space. Okay. And the same thing if you use an operator algebra in the left region to act on the minkowski vacuum, again, again, the writing Hilbert space is dense in your in the full Hilbert space. So and this cyclic separating can also be viewed as the continuum limit. The language appropriate for the continuum limits, what we said in the discrete case, that both row one and the row two are full rank. So this is like the continuum generalization of that kind of statement that the row one, row two are full rank. Anyway, so suppose you have such a volume of algebra, which is we have, say, for example, for the window space. And now let's ask whether there exists a lot of volume of sub algebra of m with the following property. Say this omega is cyclic for m means that if you act n on omega still, the resulting space is dense in your Hilbert space. And then a lot of property is said whether when you act the modular flow of m on this n, whether it takes you outside the n or not. Okay, so if so we define this thing by modular translation, yeah, so suppose there exists some algebra, which actually does not take you outside n for for all the modular flow with this t parameter smaller than zero. Okay, so in this case, we say that this n and m form a pair of this volume of algebra, which they have this half sided modular inclusion structure. Okay, so suppose such a kind of n and m exists. Okay, and then there's a powerful theorem from in the 90s by Borchus and Westbrook, there's a number of powerful theorems. Then they showed that for type three one algebra, then they exist a unitary group U.S. with the following property. So this U.S. can be constructed using some Hermitian operator, they exist some Hermitian operator, which is bounded from below. Okay, and then you can then construct second U.S. Okay, so so this is a existing statement. So whenever such structure exists, such kind of operator must exist. Okay, so as I have said before, whenever you see an operator be spawned from below a spectrum boundary flow, then this is a candidate for Hamiltonian, and this is candidate for emergent time. Okay, and yeah, they also prove that this evolution actually leaves your vacuum environment. So this actually can be really considered as a time translation symmetry. Yeah, not this, not symmetry, this is a time translation, which actually leaves your vacuum environment. So then this can be used to generate the new times. Okay, this can be generated new times. And so this is the structure we will use to generate the new time to, or corresponding to Kruskal time in the blackout geometry. Okay, so let me just give you an example to show such kind of example. Yeah, let me just give you a triple example, which is corresponding to this Rindler story. So let's just take the m to be the operator algebra in the right region. So this is a von Neumann algebra. And now let's imagine, let's take n to be the sub algebra of m, which is associated with the operator algebra in the, in this smaller wedge region of n. So n is still associated with a lot of infrared Rindler wedge, but this is the, but this inside m. Okay, so n is the sub algebra of m. And again, just from the Riesch and Eider theorem, n is cyclic with respect to the Minkowski vacuum. So the modular evolution of m here, just the boost. So clearly, when you do the boost in wind racking, this n, you take n inside itself, you don't take n outside itself. Okay. So, and then you then here this m and n satisfy this structure I mentioned earlier. And then in this case means there exists an operator which is bounded from below. Okay, then you can generate a evolution. So then of course, in this case, G is where long because yeah, it's Minkowski below everything about Minkowski series. And here it turns out that G just generated the translation along the X minus section. So it turns out G here just a long homotonic of your QFT. Okay. And, and so, so, so you can also, so now you can also act to this U on M. So, so the U on M, so the U is just a large translation along this X minus direction. So if you act by as smaller than zero, you just translate along this direction, of course, take you still within M. And also in particular, you take you to N, there'll be some choice of parameters anyway. But when you take S greater than zero, then you can take M to flow in the outside the original M. Okay. In fact, you just by taking all value of S greater than zero, you can just take M then you can translate to generate the full F region. Okay. So, so by use this kind of half-sided modular structure, if you start, if you give me the Rindler, if you only give me the Rindler right and L region, then I can actually construct the full Minkowski spacetime. Okay. I can actually construct the full Minkowski spacetime using this structure. So, so this is the way we are going to construct the full black hole spacetime using the, using the right region and the left region of the black hole. Okay. So, I have a question, sorry. Yeah. The argument of U when you define the operator N to be only negative one or any negative number. Sorry. The argument of time evolution operator for N, it is only negative one or any negative number. Well, can be any number, can be any number. It's from minus infinity to plus infinity. You can be any number. Yeah. Just is for S smaller than zero is take M to itself. It does not, but when S greater than zero, when you take S from zero to infinity, then actually that can be, this now a translation can be used to cover the full F region. And so when S take, you will take S to be minus one just based on our original choice of S, and then actually just take your list to N. Yeah, N is the sub edge. Yeah, so this is the, in this well understood example, so this structure can use to generate Minkowski spacetime from the renderer. Okay. So let me make a comment here. You say, oh, maybe if I look at quantum field theory to use the renderer region to generate the full spacetime, this might be, yeah, maybe some triviality, but this is actually not so trivial, right? Because I actually, here I only need to assume certain edge break structure. I don't have to assume you actually have a Minkowski spacetime. I only need to have this right and the left wedge. Okay. And somehow this Minkowski time will emerge out of here. Okay. So now the key is now you need type 3, 1, von Neumann algebra and some appropriate chosen sub edge bus, then that leads to emerging time. Okay. Yeah. Sorry. I'm running out of time. So let me now quickly tell you how we use this structure to actually do the, do the boundary construction of the bulk emerging time. So you don't need to hurry. Okay. Okay. Good. Okay. Thanks. Yeah. So the basic idea is that this black hole is described by this CFTR times CFTL in the sum of your double states. Okay. So let's go back to this description. And so in the finite N, the boundary-operated algebra of CFTR or CFTL is the so-called type 1 edge. Okay. Because of the, because of the, because of the operate the algebra of the CFTR, just act on the right hubed space of CFTR. And yeah. And by definition, that's type 1, von Neumann algebra. They just standard, yeah, just standard operate the edge by acting on the hubed space. That's just type 1. But the things become subtle when you take angle to infinity limits. In particular, when you take angle to infinity limit, in particular, if you want to include the states like black hole kind of states. Okay. So the black hole kind of state corresponding to energy, say proportional to N square, say, if we, say if we take input it was some way a mill theory. And the, and the, and the, those states have energy proportional to N square and dimensional hubed space proportional to exponential N square. So it's become highly tricky to define that cube space and operate algebra when you take angle to infinity limit. Okay. Actually, strictly speaking, the angle to infinity limit don't exist. Okay. If you look at the sector in moving black holes, but we will argue actually there's an emergent type 3, 1 von Neumann algebra in the larger limits. Okay. Which can be well defined. And it is this emergent type 3, 1 von Neumann algebra, which actually leads to the emergence of the sharp horizon and the interior. Okay. So, so, so this to this informing time. Okay. Yeah, actually infinitely number in front. So, so the key is to look at the edge by generated by single trace operators. So, so in the larger limits, the algebra generated by single trace operators is still well defined. Okay. So, so when you take angle to infinity, so the single trace, you take any final, take any products and some of single trace operators. And this is a well defined edge. And in the, in the larger limit, as I mentioned, the original hubed space, including the, including the black hole states become highly complicated and not well defined. But in the larger limit, actually, there is a well defined hubed space based on a sum of your double states. So this can be constructed as following. Okay. Heuristically, this is very simple. So, so let's imagine we, we can just act all the single trace, all possible single trace, we can just act that this edge, all possible single trace operator or product of them on the sum of your double states. Okay. And, and it turns out the space you generated by doing this. Okay. Actually have a hubed space structure using so called the GNS construction. Okay, GNS is the name of three people. So, so you can actually so construct the hubed space of excitation around sum of your double states. And this hubed space actually have a well defined larger limit. And you can define the well defined operator algebra in this hubed space. And then turns out that, and we will denote the MR, the representation or the action of the the single trace operator algebra in this GNS hubed space. Okay. So, and so this GNS hub is just the small excitation, just the physics around sum of your double states. And this representation of the single trace operator algebra, let me just give you a different name, I call it MR. So MR, so AR is the is the single trace operator of the original series. Okay. And MR is its representation in this GNS hubed space. So, so this distinction is very important. So, now the conjecture. Sorry. AR is at finite and AR is not closed, right? Yeah, that's right. So, at finite and AR cannot be strictly defined. Yes. Cannot be strictly defined. But in the infinite n limit, essentially AR is the only sensible algebra you can define. Yeah. And then, and then, then we can also define in the infinite n limit this GNS hubed space around sum of your double states. So, now, now the claim, so you can also do similar thing with ML, AL and ML, okay, which is, yeah. So, now the claim is that this MR and ML are type 3 volume magic. Okay. So, so even though at finite n, the operator algebra in the CFG are type 1, but, but then in the infinite n limit, then they become type 3 volume magic. So, there's worries, support we can give, or we can say partial proof, we can say to support this statement. Say the sum was, you can argue from the perspective of the sum of the behavior of the spectral function of single-trace operators that we're not going to there. You can also show that actually this algebra have this half-sided inclusion and translation structure, which is only applies to type 3 1 theory. And so, in fact, from this, I may be able to already give a construct rigorous proof just based on this one. Okay. We do show actually they have this half-sided inclusion translation structure. And then the finally, you can also motivate this by duality with the bulk. So, this last aspect is the simplest one. Let me just talk about that. So, so on the gravity side, so in the large n limit, on the gravity side, just corresponding to, by large n limit, we just means perturbation being 1 over n. Okay. So, in the gravity side, corresponding to the perturbation in genutin. So, perturbation in genutin, essentially, you just have a quantum field serve in the curve space time. Okay. It means the genutin as the perturbative parameter. And so there, we can define the following structure. We can introduce heart-hawking vacuum. And then we can construct the fork space of the all excitations on the, on the heart, based on the heart-hawking vacuum. Okay. And so, this is the standard, the fork space you can define for the bulk fields, which can be defined perturbatively in genutin expansion. And we also call the opiate algebra, the bulk opiate algebra in the right region m tilde r, and the opiate algebra in the left region m tilde l. And then the standard duality between the black hole and the CFG outside the horizon. So, restricted outside the horizon can be formulated using the following algebraic statements. You said the GNS Hilbert space around the sum of your double states should be identified with this fork space in the bulk. So, this is our standard identification of the Hilbert space. Okay. But phrase it more precisely in this context. And then they, then we identify this, the vacuum in the GNS Hilbert space, which essentially corresponding to the sum of your double state to the heart-hawking vacuum. And then the statement of the bulk reconstruction, say for the operator in the right region and the operator in the left region can be just written as identification of the two algebras. Okay. And so, this is, so from the algebraic statement of the standards, ADS CFT for regions outside horizon just can be written in terms of those statements. Okay. In the larger limits. And now this f m l tilde, m r tilde, they are just sub algebras, some operator algebras in some sub region in the bulk quantum field theory. And so, so, so just from the quantum field theory in the curve space time, which we do believe these types rewind structure still exist. And so, so there must be, from this perspective, there must be types rewind. Okay. So, so at least self-consistency with the duality with the black hole geometry also implies that somehow this m and l, they must be types rewind. Okay. But, but the first two, which I think both of them can in principle be made into rigorous proofs. And they, they don't rely on this kind of bulk duality. They can be intrinsically formulated from the boundary theory perspective. Okay. Good. So, so now in order to find the emerging time, we have identified this monomer algebra of these types rewind structure. And then, then to find the emerging time, we just need to find some more appropriate subject. Okay. All of this, this emerging the volume. And so before doing this, let me just emphasize one simple point. But it's, oh, yeah, before doing that, let me just make a couple remark. Each GNS is an eight GNS is Sorry? Eight GNS is a subspace of the product space, or is it for the right or left separately? It's a, so HGS is the full space. So the bulk, so the bulk fork space cannot be factorized between the L and the rights. So this is just the full, full, full cube of space, or a perturbative engineering Newton. And that space, that hub of space cannot be factorized between L and R. So each GNS is a subspace of each right direct product with each left. So, so, so that's a little bit tricky to describe. Because when you take the angle to infinity limit, when you include the black hole states, the hl, hr times hl can no longer be precisely defined. Yeah, yeah. Yeah, but, but he basically, we can say the following. We can say, let's look at large, but finite n, okay, we can look at just finite them, but n very large, say n to the 10 to the world 500 say we can take a large and finite n. And in that case, then, then in principle, we can still talk about hr times hl because we still have a very defined hub of space. So in that case, we can say, then we can talk about this AR in the, in the approximation, which we do one way expansion. Okay, then in that case, the one way expansion just can be considered as approximation to this finite, but the large n. And then this GNS hub of space is emergent in that kind of approximation. And then you may heuristically think that this GNS hub of space may be considered as a subspace of hr times hl. Yeah, in that perspective, yeah. But mathematically, I don't think the, you can say, rigorously. I'm going to define it with respect to an algebra, so I say, okay. That's right. Yeah, just rigorously, mathematically enough, say this is the subspace. It's defined using an algebra and not the Hilbert spaces. Okay. Yeah, that's right. Good. Good. So, so yeah, let me just make some remarks where the theorems of half-sided modular translation ensures the existence of this G and the US. And finally, explicitly in general, it's very, very difficult. Okay. We only know some small number of examples, say, Rindler, yeah, et cetera. Yeah, actually, Rindler is essentially the only non-trivial, yeah, the only, yeah, they are isolated example, but, but the other examples other than Rindler, they are all a little bit special. So, but in the larger limits, the algebra of the single trace operators in GNS hub of space actually can be described by generalized free theory. So in this limit, I have an elementary question. So, when you identified this vacuum state of CFP, omega with the heart locking state, I mean, doesn't it already assume the validity of ER equals CPR? No, no, no, that does not. No. So, so, so, yeah, so when you do the GNS construction, you start with a somersfield double state, and then you just act the operator, a single trace operator on the somersfield double state, and then you construct a new hub of space. So this new hub of space, yeah, I'm not describing the detail of the construction here. So this new hub of space has, yeah, so essentially just each operator in this operator algebra, corresponding to a state in this table space, okay. And the vacuum is the one corresponding to the identity operator. And so in that sense that the vacuum in the GNS hub of space corresponding to the original somersfield double state, it should be viewed as a mapping rather than as identification. Yeah, mathematically should be viewed as a mapping. Okay, so, so, so, so in this case, but actually there's a generalized refilled. So in this case, it turns out it simplifies a lot. But it's still in this case, it's not easy to find the g, but turns out it's much easier to find the us. Okay, to find the g is still difficult, but finding us is much easier. So for worries constrained, etc. So, so, some will explain again in the second talk. So, so in that case, like, you can actually constrain this us for the generalized refilled to, to, to have a universal form up to a face factor, which depends on specific edge. So, so in the general as refilled case, even though the g is still complicated to do, but actually this us actually turned out to be much simpler. And you can actually constraint to some universal form. And, and the universal form up to a face fact. Okay, so, so the expression of this can be essentially be fully determined up to a face. And then the only non trivial thing when you find the specific sub algebra is you find that face. Okay, so, so now before talking about sub algebra, let me just mention one quick thing, which is a simple statement, but it's actually highly non trivial, leads to highly non trivial consequences. So let's just look at the causal diamond, say, of some courses, some slice, okay, some, some region, okay, some, some spatial region, some spatial region, look at the causal diamond associated with it. And so let's look at one slice. It's all played the edge of such if it's nine slides, we call a one and open the edge of social with another slide, we call a two. So in quantum field theory in the center of quantum field theory, a one just the same message is because you can always evolve because a two can be evolved from from a one through Heisenberg evolution. Okay, so, so a two, anything a two can be expressed as what position of operators in a one, and anything a one can be expressed in terms of position of operators in a two. Okay, so, so they just equivalent. But this is not true in the infinite and limits, when you look at the edge of a single trace operators. So, so this is easier to understand from the generalized free field perspective. So when you have a generalized free field defined on the causal diamond, because generalized free field, they don't obey any equation motion. So there's no way to relate one course slice to a lot of course slice. So, so the a one a two, they just in equipment. So you can also form describe more generally from the, from the perspective of the single trace operator as a follows. So when you try to evolve single trace operator using standard Heisenberg evolution, then then actually that take you outside the single trace operator. Okay, the standard Heisenberg evolution, say, say, let's take a super Yang-Mills theory, and then let's look at the single trace operator. The standard Heisenberg evolution take you outside the edge. And so, so, so again, of course, this is just another perspective states that this is generalized free field. So a one is not equal to h. And then this just give you tremendous new possibilities to generate new sub edge because of this in equivalence. So, so now we can now talk about so emerging the boundary times. So now it's, so now let's just use the example of this one plus one dimension boundary. So let's imagine the boundaries on the circle. So this is the time direction. So this is a boundary manifold. Okay, it's cartoon or boundary manifold. So this is a vertical direction is the time direction, and the horizontal direction is a circle, which we identify. So as we said, the single trace operator but defined for the full space time on the boundary is the type three one only my edge. But now let's look at a sub region, sub space time region associated with say t smaller than zero. Look at all the single trace open edge of associate with t smaller than zero. So, so this open edge of code and okay. So now, so, so the key thing is that at the level of in the infinite and yeah, or particularly one of them. The single trace open edge for m and n they're in equivalent. Okay, precisely due to the reason I said here. Okay, they corresponding to in equivalent algebra. If I define at n, you look at the operator algebra for the full space time and the operator algebra for this sub region, they're just trivially equivalent, because they're all equivalent to the operator algebra on the single course is less. Okay, and the box in the infinite n, we look at single trace open edge bro, and now they're in equivalent. Okay, so, so, so now there's an emergent sub edge bro, which is not present in the in the in the five n case. So you can also consider many other examples. For example, take the n to be the time in the above. Also, you can trivially show. Okay, the the the satisfy this half sided module inclusion structure. So here the modular time just time evolution. And so this trivially satisfied this half sided module inclusion structure. You can also take the case which are inhomogeneous in spatial direction. This sub edge bro defined in the region which which say defined by the inhomogeneous in the spatial direction. Yeah, this way, etc. So, so you can have infinite number of such sub edge brush. And then they give rise to infinite number of emerging times. So now let me take you just tell you this specific example. Okay, this specific example. So, so as we said, so, so let's just take n to be given by this region. And now we just need to find the US associated with this sub edge. And now this is actually strongly, but actually, as we said, there's a generalized refilled never we can reduce this US to a single phase factor. But finding that phase factor still on trivial. Okay, actually, it's still a strongly coupled problem because because this generalized refilled the correlation functions and the example of this generalized refilled actually are controlled by the strongly coupled physics. Okay. But actually, there's a trick, we can actually find that this phase factor. Okay, so the trick is the following. So we actually use the bulk duality to find that the this phase factor. So, so here we need to make a lot of proposal. So we propose that the operator algebra in this sub region n, okay, so corresponding to the boundary space time, say for T smaller than zero. It's actually can be identified with the operator algebra in the bulk in the black hole region in this region in this causal wedge of n, okay, in the black hole geometry. So in this n tilde region. So if you just draw the causal wedge of n in the bulk, and then you look at the bulk operator algebra in that region. So so our proposal is that these two operator algebra can be identified. So we're not going to detail here, there's a worries, a consistency check you can do, okay, and the which we give in the paper. So now since now this operator algebra become identified, and this n tilde actually is a free free algebra in the bulk. Okay, now this just become generally free algebra in the graph design. And now we can actually try to find that this phase factor using this n tilde. And now that can be found. Okay. And it turns out this phase factor is precisely the phase shift. So now suppose we act on the scalar field, okay, it's supposed to act on scalar field. And then this turns out this phase, this phase factor is the phase shift for the scalar field at the horizon. Okay, so the so this phase shift come as follows, okay. So as we know that any bulk fields, if you look at the behavior here the horizon is given by two plane waves, one corresponding to a plane wave which going outside the horizon, and the one corresponding to a plane wave, which go inside the horizon. Okay. And yeah, just for any field near the horizon, just like a plane wave. And now if we impose that this wave function is normalized by infinity, and then that determines a specific wave phase shift between these two are infalling wave and outgoing wave. Okay, so you can imagine this as a scattering problem. You have a wave coming out of that horizon scatter on the boundary, and then fall into the horizon. Okay, so this scattering problem has a phase shift. And this phase shift turned out precisely the phase shift corresponding to this how corresponding to this us. Okay, so actually personally have have wondered about this shift for many, many years. So one of the first days you do the ADSF key you write down the wave function in the back, and you see this phase shift. But that hasn't appeared in any boundary quantities. And actually now it actually appears in this in this us. Anyway, so now you can just construct that phase shift doesn't doesn't depend on the field. Yeah, so so the us does not depend on the fields. But the action of you on the specific operator that depends on that field. Okay, yeah, yeah, because we cannot really write down the general us, we because we cannot write down that general g that is very complicated. What we can do is you can, you can turn the action of us on specific operators up to your face. Yeah. Yeah, and that phase then depend on your specific type of operators. Yeah. Good. And then, and then this way you can just construct this us explicitly. Okay. And then and then this is a flow pattern we see, which I showed you earlier. So this is like, what we call the you time kind of shift, which corresponding to you take this and given by this way. And then, then that give you this kind of shift. And if you take the end given by this way. And then that can be argued to be due to the bulk region like this. And then that give rise to the cross collect beef flow in the gravity side, which, which the, which the horizon translate along the view of the direction. Okay. And again, that generates some kind of times nice. And then you can consider others. You can also consider compositions of such groups are you and the V type flows. And yeah. Anyway, yeah, so, so I'm way out of time. So, so I can just stop here because I have already talked about the the, the main thing. Yeah, I wanted to talk about. So the rest is just some, some general remarks on generalizations and, and, and discussions. Yeah, I can just stop here. And then I would propose that we stop the formal part of the talk here. And we thank Hong and the people who want to hear more can stay afterwards. So thank you, Hong. So you want to say a few more things about the right. Okay. Yeah. Okay. Yeah. Yeah. First, do people have questions? Okay, I have a couple of questions. Okay. So is it easy to understand why this type three one structure is really essential. I mean, in what way it is essential for this haps, half sided modular flows. I mean, yeah, what, what is the basic idea why the type three is so crucial. And if it is type one, you cannot do it. Yeah. So, so I haven't really gone through very carefully there, the mathematical proof. It's a theorem, which they can prove. I have it is what's the physical I sort of what's the basic idea. So, so, so I think the one special thing about this type three one structure. Yeah, I think that the following intuition might help. Is that the this type three only for type three, only for type three, the modular flow is external automorphism. And for type one type two, they are all in the automorphism in the sense that they can be constructed using the operator within the edge by itself. And, but, but, but this type three actually is the outer automorphism. And yes, I think it's this structure. Yeah, it's important there. Yeah. Yeah, but I cannot say too much more because I don't really understand the proof very well, a mathematical in myself. Yeah. Sorry, but if you take your U of s that you compute at infinite n using your techniques, and then you just take the same U and you apply it to finite, but very large n. Yeah. Probably you will get correlators that seem to be reasonable as, you know, correlators inside the black hole. Yeah, yeah, yeah. So the construction has some robustness under taking finite, but very large n. That's right. That's right. Yeah. Yeah. Yeah, just it's a similar. It's similar to to when you can see that the window of space. When you put it on the lattice. So if the lattice spacing is very small, even though strictly speaking, you don't have a sharp light cone. So you actually lose the sharp light come because you always have this kind of small experiential tail when you're in the lattice. And but still, whether you have a tennis, 90 spacing or whether you have continuum limits, for most purpose, they're not that different. For most physical question, they're not different, even though certain sharp feature has disappeared. And I would think of the similar thing between the large n and the finite, but very, yeah, including infinity and the finite by large n. Just both of those structure, they're there, but they always, but now they have all kind of small tails. And then they're no longer exactly sharp. Yeah. Can I ask a question? Yeah. Yeah. So you mentioned that one of the evidences in favor of the fact that it's a type three one algebra is something to have something to do with the spectral functions. I was wondering what information the spectral function give you vis-a-vis the type of algebra. I mean, why the spectral functions you'll see with type three one as opposed to type two infinity. Right. Right. Yeah. The reason the spectral function is rated is the following. The way you construct. Yeah. So this algebra is the algebra on the gene S-Hilbert space. And the way you construct the gene S-Hilbert space is you associate the action of the operator on the, yeah, you associate the action of the operator on the sum of your double states as the state of your Hilbert space. Right. And in particular, the inner product of this Hilbert space are given by the two point function of the single trace operators. And so that inner product structure actually depends on the two points, the structure of the two point functions. So that's why the behavior of the two point functions actually important for the algebraic structure. Thanks. I have a question about the, does somebody have a question? Yeah, but you go ahead. Thanks. Hi, Hong. Thanks for my stock. Yeah. I just wanted to ask that you are showing these plots where like how these U of S can kind of move a Cauchy slice, let's say it equals zero forward. Yeah. But also at some point you were mentioning that the property of this U of S is that it fixes the state that you define it from. So in the case of the Rindler example, you were saying that this is because, you know, like the light can only fix the vacuum. Yeah. But if this is a general property, that this seems a bit surprising in the term of the double where if you have sort of like because there's no isometry that you would have for both forward evolution and you expect the state to be like a non-equilibrium state. Yeah. It seems like you act on the boundary as well. Right. Yeah. Sorry. Yeah. Let me just understand your question. So you're asking, say the fact that this U of S seems to leave the GNS, seems to leave the some of your double state environment. You said what you're asking, you say this is a little bit unintuitive. Yes. Yes. Yeah. Yeah. So this is indeed a very interesting feature as a consequence of that theorem. So this is in the sense, Nitro from the Minkowski space point of view, in the sense that somehow your time translation somehow should not change your vacuum structure. And but it's indeed, I think it's surprising such kind of thing can be this in this kind of curve space time. And somehow you can have this kind of time. Yeah. So there are several surprising things here. So one surprising thing is that you can have this kind of evolution. Yeah. Normally when we think about the Kruskal time evolution in the gravity side, we say, oh, if you want to write down evolution over the corresponding to Kruskal time evolution, then the corresponding Hamiltonian must be time dependent. You cannot, must depend on some kind of time, because there's no global killing vector. Okay. But the fact that you can, they exist, yeah, then when you write the evolution operator, they will be corresponding to some complicated past orders, say, say evolution operator. Yeah, because your Hamiltonian should be time dependent. And by the way, surprising thing is that they exist this all infinite number of G, which you can just explain to generate this kind of time flow. Indeed. So this is one surprising thing. And another surprising thing is that this G turns out to allow your GNS, we allow the sum of your double states. So in the sense that this is really some kind of global time evolution of your theory. Yeah. Yeah. Despite that you have this kind of curve space time structure. Yeah. I have a question. I didn't understand exactly where in the construction you need that there is a left and the right. So suppose that you only have the right CFT. Can't you equally well define this sub algebra and construct the U of s and define everything just using the right side? Right. So, yeah, indeed. So it depends on, so if you just start with the right algebra, then if you just act on say, say we just, yeah, you can do it in several ways. You can, yeah, let's just start with the right algebra. You can either act on the thermal density operator, or you can act on the sum of your double states. In either way, you will get the same GNS curve space. And in the same way, there's always a left algebra will be generated from this procedure. So even if you don't start with the left, the left will be automatically generated. I mean, are you saying that you will not have a cyclic separating vector? I mean, if you did not have the left, what vacuum will it choose? So, for example, you, yeah, so you don't have to, so in order to generate this GNS curve space, you don't have to say, yeah, say, let's do the following thing. Let's just take the right algebra. Let's just purely take the right CFT that act on the thermal density operator. And they don't have to talk about the left at all. Just act on the thermal density operator. It turns out that it's a feature of this GNS construction. Such kind of construction automatically purifies the states, okay? So this GNS, so this density operator under this GNS construction is mapped to a pure state in the GNS curve space. And the left algebra is automatically generated in this GNS curve space as a commutant of your right algebra. So you actually don't have to start with the left theory at all. In the sense that this can be used as a way to predict, okay? That somehow if you want to describe, describe small excitations, yeah, this can be used as a predict. If you want to describe small excitations around the thermal states, even for the right theory, you actually have to have connected internal black hole geometry. And this can be used as a derivation. So you don't have to start, so the standard thermal field double state is a gas based on the geometry on the gravity side. But using this GNS construction, you can actually, I would say you can derive it, okay? Then the left is actually generated. I had a similar question that it should be possible to formulate everything in terms of a thermal quantum field theory, right? Because- Yeah, exactly. Yeah, yeah. Yeah, yeah. Yeah, somehow this magic of this GNS construction automatically purifies for you. Yeah, yeah. Yes, sorry, there was- No, I think I want to say that here there's a slightly related, but maybe different question, which is what happens in the case where we have one CFT in a pure state? So far you were talking about a thermal dysthmetics, I guess. But we could- I think the same thing could probably be done for a typical pure state. One would have to be more careful how to take the larger limit and how to select the typical pure state. But I think probably can be done. And for example, your criterion that to show that the algebra is type three based on the spectral properties of correlators. Yeah. I think you can argue that on a typical pure state at large end, the correlators or single-trace operators will be very close or the same as the thermal ones. So I would imagine that some sort of emergent type three property could actually be defined for even for pure states of a single state. Yeah, absolutely. So for pure states, you should be able to do this. But for pure states, in the sense that for pure states, to do the- yeah, so for pure states, physically we expect everything to be there. But the tricky thing about pure state, if you if you just do the GNS construction on the pure states, it's not- you don't see such structure. It's because the pure states don't give you- it's already purified. But I think it's maybe a question of order of limits. For example- Exactly, exactly. So the key for that case is that you have to understand that this large end limits much more carefully in the pure state case. It's because for the sum of your double states, the large end limit is more or less straightforward in the sense we understand much better. But for the pure state, the large end limit is much more trickier. Yeah. And so that's just related to that issue. Yeah. I mean what you're saying is that you're saying this because a typical pure state is close to thermal state. Is that the idea that you- Yeah. Yeah. So I would imagine that- I mean I guess one- if you take a typical pure state and you act on it with a very large number of single trace operators, large but not scaling with them. So you have to be careful how we take this large end limit. You can define something that is usually called the cold subspace around a particular pure state. And I would imagine that in the larger limit, this will be identified with the dreamless Hilbert space that you're talking about. Right. Yeah. But the key question is just how you really define that limit. Yeah. Yeah. Yeah. Just defining that limit carefully should be very tricky. Yeah. And but we do know somehow that something like that must exist. Yeah. So maybe can I ask a question about- since we're talking about the larger limit now, if you take the infinite end limit, then there's a clear way of defining the subalgebra end that you were talking about. But when we start considering one of our end corrections, then there are issues with the fact that the Hamiltonian may have to be included in the algebra. Yeah. And then it's not so clear how to- I mean is it clear to you how to precisely define a subalgebra end time? Yeah. Yeah. Yeah. So I think the perturbatively order by order in one over n, this structure should persist. It's because when you include- Yeah. So the Hamiltonian, you see, so the tricky thing is the following. So the tricky thing is that if you talk about the Hamiltonian, yeah. So Hamiltonian as an evolution operator and Hamiltonian as a single trace operator, they're different. And they differ by fact of n. And that fact of n is crucial. So the Hamiltonian as an evolution operator have the structure is the n times the trace. Okay. n times the trace. But the Hamiltonian as the single trace operator, as we normally define single trace operator, you only have the trace. You don't have the fact of n. And so the Hamiltonian, which is in the single trace operator algebra, they only generate infinitesimal time translation. Yeah. Because just related to the standard OPE, that two single trace operator OPE is one over n. So the Hamiltonian operator in the operator algebra, in the single trace operator algebra, we include one over n correction. Still, they only generate the one over n kind of time steps. So that's why I think order by order in one over n expansion, you should still have a type 3, this type 3 structure. Yeah. Also from the gravity point of view is very intuitive. The gravity point of view is still have a quantum field theory in curve space time, but with the G Newton as an expansion parameter. And but then as Witten pointed out, but somehow if you have n, but if you introduce n in some different way, and then you have a type 2 because of this cross product structure. But this cross product structure would give you a type 2 infinity. But this type 2 infinity have to somehow mix different n. Yeah. I basically want to ask you if this discussion that Witten presented for the full algebra of single trace operators for all time, if you have thought how to apply it in the case where you have this algebra n. So if you start with n and you include the Hamiltonian in this perturbative sense that Witten is talking about, do you know how it works? No, I think I understand mathematically what he's talking, but physically how to interpret this type 2 infinity and this cross product, I don't fully understand. It's because you have to shuffle n. You cannot do it all by order in one way. All by order in one way, you will not have it. Yeah. I mean, I had a similar question, but you have to go from type, so Witten shows that you can sort of go from type 3 to type 2. Yeah. But you really want to go to type 1 in the end. In the end, you want to go to type 1. Yeah. When you go to the finite n, you have to go to type 1. Yeah. So what Witten is doing is that finite? I mean, how should one think about this type 2? Yeah. So that's why I don't fully understand. So mathematically, I more or less understand what he's doing, just physically really how to interpret the regime he's defining. I don't fully understand. Yeah. Yeah. Also, he has an entropy. And that entropy is not the blackout entropy. And that entropy somehow is also a little bit tricky. Yeah. So yeah. So how to interpret that entropy? Yeah. When you have a type 2 infinity, then you have entropy. And then how to interpret that entropy physically? Where is it finite? It can be finite, but the unit of that finiteness is not where it defines. It's more like the standard story. It's more like the standard story when we have generalized second law. So you have the area term, then plus the entropy for the matter. But then there's ambiguity. Yeah. Because the entropy of the matter is infinite. And the matter is UV divergence. And also, for the area term, you have the divergence corresponding to the remuneration of the Witten constant. And then you have the freedom to put some divergent piece, either to include it in the Witten or including. Yeah. Anyway, so this kind of ambiguity to shift both of them by finite amount. And what's finite amount you associate with them? Yeah. I think that ambiguity is ready to that. Yeah. So in his construction, there's no area term yet. I have a question. So the answer might be obvious to some members of the audience, but it's not to me. So morally, at least your construction seems very similar to the Papadodimus-Raju construction of meteor operators. So how does this construction differ from theirs? So I think that the so the meteor operator, so the meteor operator is automatic in the civil field double case. So the meteor operator is non-trivial in the single-sided black hole. And there is a highly non-trivial. I think it's a beautiful gas in the sense that somehow there must exist such kind of meteor operator in the single-sided black hole case. And yeah, this is related to the question Kiriako mentioned earlier for the single-sided black hole. Physically, you feel they must be there, but somehow to mathematically construct them is highly non-trivial. But for the summer field double case, such operator, for the summer field double case, this question somehow become trivial because summer field double case, such meteor operator, it just operate on the left side. And so they're automatically there. But just to make a small comment, one of the features of these meteor operators is that for a pure state is that they have to be state dependent because they are determined by the entanglement pattern. And I guess if we apply this half-sided modular inclusion story for a pure state or a typical pure state, I would imagine that this Z operator would depend on the microstate that we're starting with because it is defined by the modular Hamiltonian of M and of M. So it would reproduce this state dependent property, I would imagine, of the interior operators. So this G certainly will be state dependent, but the key question is we know there must be states independent aspect of this G operator. Because we know even though somehow each microstate is different, but the single-sided black hole geometry, they do have universal features which are independent of the microstate. And so in the sense that the, yeah.