 O ką, patičiau, kad būdome mėgai žaidimus, priežbūrėme, kad mėgai, kad mėgai žaidimų žaidimų, ir taip, dar, taip žaidimų pirmos, o šiek ir negračiu, ir sėgau, ir ką, jis žaidimus, jis jums ir jis su Vosaks, 2011. ir laukėjome nebūtų horizinojų, ir tos būtų ir nebūtų, tai būtų ir būtų, jie mūsų priešsės, ir mūsų priešsės, ir jie mūsų priešsės, ir jie mūsų priešsės, ir jie mūsų priešsės, Lien, desiter, Minkolskyi. Aš laimėjome, žaidėjome, kad mūsų žaidėjome, žaidėjome, žaidėjome, žaidėjome, žaidėjome. So, IDS-CFT, well, is a gold standard example of holographic description. Well, in general, in holographic description well, idea is, as you will know, let we can describe physics in a bulk by some data specified on a boundary. Let boundary is usually called a holographic screen. And in known examples of holographic description, this data is agonized in a form of conformal field theory. So, the first, the main example of this holographic description is IDS-CFT. And we can see the symptotic IDS metric. So, y is sort of a radial direction. L is a radius, this L is the radius of anti-dissitter. So, when y goes to infinity, let's say leading behaviorally metric, and what stands here is metric on the dimensional boundary. So, our safety is supposed to live here, has a dual description in terms of gravity on anti-dissitter. So, scalar field. So, in excitation on anti-dissitter, for instance, scalar field. I apologize, I will spend some time telling you very known facts, but I will use them later. So, scalar field will mass m, assuming that phi becomes exponential of h y l. We find equation for h, which is that form. And from here, we find two solutions, h plus and h minus. So, one value h plus is positive, h minus is negative. So, one mode is growing, sorry, it's minus. The other mode is decaying. And, well, for large y, we have an expansion. In ADSFT dictionary, this mode describes a source on the boundary, and that field would be a technician value of a dual operator. So, in order to reconstruct, well, this field in the bulk, we have to specify both the source and the expectation value. I have minus on both. Simply h minus is negative, h plus is positive. So, similar story exists for gravity, assuming that metric in the bulk satisfies and stand equations with positive, sorry, negative cosmological constant. We have well-known Phaferman Gram expansion. Well, let metric, well, let's explain it y. Well, there are some sublidian terms plus. In this expansion, two terms are not determined. The boundary metric and this term. This second term has interpretation as expectation value of boundary stress engine tensor. It can be also thought as a proportion to extrinsic curvature of the boundary, of regularized boundary. And in order to reconstruct metric in the bulk, we have to specify both the metric and the expectation value of stress engine tensor. Local conformal symmetry on the boundary is a part of conformal symmetry. On the boundary is generated by bulk difumorphism. I use coordinate rho instead of y here. Xa rho is alpha x. And xa i will satisfy certain equation, which can be determined in powers of rho. And under this difumorphism, the boundary metric transforms conformally. So, let's list under story in ADCFT. If you want to do similar analysis in the sitar spacetime, well, one way is to analytic continuation for L squared. So, L to iL and also y to it. And the sitar spacetime is a, well, it looks like that. That's global time tau. And in this case, we have two boundaries at tau minus infinity and tau plus infinity. So each section of the sitar space is a sphere. And, well, in this picture, with Strominger in 2001, we can associate to each with surfaces conformal operators, which satisfy the usual conformal correlation functions. And, well, in this case, if you look here, if you do analytic continuation L squared to minus L squared, there is minus here. And H plus has interpretation in the dual series conformal dimension. It may become imaginary if mass is sufficiently large. So that is one problem. The second problem is that, well, if you want to describe quantum evolution on the sitar spacetime, in principle, we could describe evolution in terms of correlation functions between operator here and operator there. But any single observer will not have access to all data which is specified on each sphere. So that sort of evolution would not be observable. Any observer will not have access to the whole information on the screens. Well, similar story exists for fifth-ramangrium expansion. Well, in this case, it was for the sitar spacetime. It was discovered by Starobinsky completely independently in 1983. So in these two examples, anti-desittery desitter, conformal theory lives on the co-dimension one. Hyper-surface. In my next example, conformal theory will live on co-dimension two. Hyper-surface. Well, this picture was suggested in our paper William de Bourre in 2003. So now we are in Minkowski spacetime. Aksmological constant is zero. And we look at light cone. Take any light cone. So then inside light cone we can slice spacetime with hyperbolic slices. So here spacetime looks like that. Spacetime metric minus dt squared plus t squared. And here metric of hyperbolic space. So t goes that way. And outside the light cone we can slice it with desitter slices. So here metric, flat metric looks like du squared plus a squared metric of desitter. Yes. And all these slices have the same boundary. Well, actually two boundaries. That's sphere and the second sphere. That would be s minus d s plus d. So here is to relate, well, considerably, du layer standard holography on each slice here on layer. And these two spheres would be two holographic screens where all data, conformal data are defined. So if we start with, as an example, scalar field, mass m. Then suppose I look at here outside, look at solution to this equation outside light cone. I assume this behavior, sorry, theta. Then this field phi, okay, phi lambda will satisfy massive, well, scalar field equation will set a mass on the anti desitter slice. And mu is related to lambda. And what is interesting, if you look at propagating nodes, propagating nodes are plane waves. Then these nodes are described by complex lambda. And respectively, if you now compute conformal waves for that mass using that expression, you'll find that, correspondingly, conformal operators have a complex. So in this description, in this description, the propagating waves, plane waves are described by conformal operators defined at the here layer with complex conformal weights. Well, there is ongoing research. What does the holographic screen see in the Penrose diagram? Well, on Penrose diagram, I guess that would be my light cone, and that would be this sphere. So you take special points on sky plus and sky nine. Which depends on the origin. It depends on the origin where I put this point. So I can, of course, move this origin, and that will generate certain symmetry on my sphere. In particular, you'll have generators of BMS group generated on that sphere. You don't have to consider this the whole co-dimension one. I could, but all these slices have the same boundary, and it's more economic defined data here. I would have thought that this holography has to play with the full sky plus and sky nine. If it exists. Well, in these pictures, like let's, we pick one cone. In the other example, you didn't pick any special observance. Here you are showing the past and future life forms of the hidden observed. It is, it is, yes. It is attached to this particular observer, which goes that way. Well, there is some ongoing research on different of two groups. One group by Strominger and Lala by Sand Room, which are building on this picture. Well, in particular in the paper by Sand Room, they have, so in principle, we could decode S matrix in terms of correlation functions, conform correlation functions defined on these spheres. And they have a one-to-one correspondence, which was already anticipated in our paper, between endpoint correlation function and scattering amplitude. So if you have any, and vice versa, it's invertible. Scattering of digital massless fields? Massless fields, and also massive. Massive are more difficult to describe. Let is true. Well, we can see that massless fields, but there is also, there should be a description for massive fields. There is similar, similar construction for massive fields. Okay. And now... What are the new force degrees to safety degrees for new fields? The statement... Quantum gravity on a constant space? Well, we didn't consider quantum gravity. We just look at simplest scalar field. So for scalar field, the statement is that quantum evolution, or if you like scattering in Minkowski spacetime, can be represented as sort of collision functions between conformal operators defined on these two spheres. Sir, do you think of a very particular sphere on square minus? Let is true. So suppose I send a wave packet from here and detect wave packet there. And always Larisa, well, it's not necessarily picked at this point, but there will be a tail, which can be detected at that point. I mean, why doesn't it depend... What's up on the scribe plus at that sphere? Why doesn't it depend on the whole of scribe minus, or big part of scribe minus, and not just that? I mean, I don't see how you get a unique mapping from this one sphere on scribe minus to this one sphere on scribe plus. It doesn't know what happens at scribe plus depend on other places on scribe minus rather than just that one and two sphere. You mean it may depend on the point, that point? I mean, suppose you have the field that's on this two surface on scribe plus, that depend on the field, not just on that particular two sphere on scribe minus, doesn't it depend on the whole distribution of the scalar field on scribe minus? If you send in two particles, the result will be the same as the impact parameter. There would be... Yes, it would be the same result. The same point on scribe plus scribe minus is just the sphere on the conformal sphere. So, I mean, this picture will wave packets. You don't like it? If you send a single particle, then the picture is correct. You have interaction with the ball. If it doesn't interact, then it will just go straight across. But let me continue near horizon optical metric. Let's take any static d plus two dimensional metric. I use function f, which describes a killing horizon at some point. It is conformal to what is called optical metric. And f of r has a simple root, and beta h is related to inverse temperature. And that metric is called optical metric. So now, because... Well, I introduce here y, and y is related to coordinate r, at least. So near horizon r minus rh is proportional to exponential minus 2y over beta h. And f of r is proportional to minus 2y over beta h. And this factor, this factor, so r squared over f of r, is proportional to some constant c, exponential to y over beta h. And we see that asymptotically, if y goes to infinity, so y goes to infinity, r goes to rh, y approaches horizon. This metric is actually hyperbolic d plus one dimensional metric. So universally near horizon, in the optical metric, we have the product of r1 and hd plus one. So it is universally, but only to leading order y goes to infinity. Horizon, let me... metric horizon is d sphere. In optical metric, it becomes boundary of... Horizon becomes boundary hyperbolic space. And beta h, with inverse hocking temperature, is radius of hyperbolic space. So the statement that this behavior is only to leading order in y, but there is one example when it is so to all orders in y, to all orders in y. And that's exactly when our spacetime is a desitter. So when spacetime is desitter, r squared over f is let. So l desitter radius is beta h. And, well, optical metric is minus dT squared plus metric of anti desitter. Minči vašyt desitter? Well, if no, pure desitter, pure desitter, pure desitter. Consider scalar field. Consider scalar field on this original initial metric, which describes a black hole. We redefine phi, okay, phi optical. And then this phi optical will satisfy, well, sort of massive equation in optical metric, phi optical, where m squared, well, mass effectively becomes bi-dependent. And there are several terms. So mass is just multiplied by f. And there are also terms which don't depend on m. They simply come due to this multiplication. The leading term, first term is d over 4 squared plus other terms which vanish if f is 0. So now when y goes to infinity, that becomes 0. And that becomes minus d over 2 beta h squared. We are looking at solution with particular frequency phi optical omega t. Define that this phi omega satisfies equation on anti desitter phi omega, where this effective mass in linear horizon region is a combination of two tones minus omega squared minus d over 2 beta h squared. So now again we look at this formula and we see that if you compute conformal weights, they become complex d over 2. And phi omega has two modes respectively with h plus and h minus. In fact, I can play similar game for interacting fields. So I add some self-interaction phi n minus 1. So let in long range that would be phi n. So I do exactly the same trick. I rewrite everything in an optical metric. Then phi optical will satisfy, well, similar equation, but now the coupling constant becomes y-dependent and it's multiplied by certain power of function f. So for certain values of n, this n, lambda y is decreasing. For other values it is increasing. So if n is less than 2 plus 4 over d, lambda y goes to 0. So n equals 2 plus 4 over d, lambda y is y-independent, and n larger than 2 plus 4 over d, lambda y goes to infinity. Well, I prepared this talk. I discussed this point with Gary and he suggested that maybe there is a, well, this condition singles out renormalizable theory. And this seems to be the right intuition. So in this case lambda has a positive mass dimension that would be 0 and negative. So renormalizable scalar field near horizon is effectively massless, so mass will disappear, and it's also free. So lambda switches off asymptotically when we approach horizon. So now we can, well, continue the same path and identify conformal operators. So when y goes to infinity, y-optical can be represented as by this integral. Well, let's basically left and right moving modes near horizon. So that would be left moving and would be right moving modes. And respectively, Green's function. So Green's function on optical metric when both y and y-prim prime go to infinity, can be decomposed on Green's function. So that would be Green's function on anti-decider, satisfying, well, let equation. So we fix, we consider let behavior, well, form all the scalar field as boundary condition near horizon, and as usual, we compute on-shell action on a solution to wave equation, which is functional of this boundary data. Take two variations and we get correlation functions for dual conformal operators. So gamma is the geodesic distance on the sphere. So we have two points. That would be gamma geodesic distance. And for these two operators related to left moving and right moving modes, but on the creation function same type of operators produce data function. And what is interesting, well, there is dependence on the frequency here in the correlation function. And this factor, this function contains conformal, sorry, thermal factor. Well, I should probably say also that conformal symmetry, again we can play similar game as in anti-decider, conformal symmetry generated by bulk difumorfism. So we take, we generalize this metric. We put here conformal factor. So asymptotically when rho goes to zero, now horizon at rho equals zero, this metric should approach metric of anti-decider. So we have a decomposition for sigma in jai jai plus some corrections plus some rho corrections proportional to rho. And induced metric on the horizon is multiplied by sigma over rho g0. And again, we look at, as in the case of anti-decider, we look at difumorfism, which preserve the formula metric. And that would be, well, generated by vector ksa. So ksatį is just constant. Ksaro is alpha of x. And ksai, well, in the similar way as a anti-decider case, can be determined by rho expansion and rho. Under this difumorfism, sigma is changing by alpha, alpha x, respectively g0 is changing. And induced metric is, of course invariant. There is a generalization of this difumorfism when CT is not constant, but linearity. And let difumorfism will effectively rescale beta H hocking temperature. You get some interesting algebra? You get some interesting algebra? Well, to lead in order is just usual conformal transformation. You mean... You don't get functions. No, no, we do get functions. So this ksai is expansion in rho. And we get this expansion. How many parameters? No, no, just one. So it's this function alpha of x, infinite. So it's like an anti-decider. So you get like virasora? Well, in this case, this is conformal rescaling on d-dimensional sphere. 4D is 2, we get virasora. So like in 4D sphere would be 2-dimensional, it will get virasora. Let me just... Well, can we reconstruct metric? Well, I already raised the anti-decider case. So in anti-decider we could fix the metric on the boundary, conformal boundary, stress and jatensor on the boundary. And that would give us... Using this data, we could reconstruct the whole metric in the bulk. So... Can we play a similar game here? So in fact, we use same metric. So we use this metric. All this form, we fix sigma 0. We fix j0. And suppose the whole metric satisfies Einstein equations in the bulk. We'll possibly non-trivial cosmological constant. And we look at static solutions. So described by this metric. And then we'll find... So we fix sigma 0 jaj 0 as our data, boundary data. And then all these terms in this expansion for sigma and for jaj jaj are completely determined. Sigma 1, sigma 2 and so on, respectively g1, g2 and so on are completely determined by this data. So... In this game jaj jaj 0 should not be necessarily a metric around sphere. It can be more or less any metric. But of course this solution, if you continue it far from horizon, will not be asymptotically flat, asymptotically Mankovsky. And that means that this metric will not violate this uniqueness theorem by Bernie Israel. Did you have some gauge condition to fix the problem? Away from the... Gauge condition, you mean? Like for for mangan coordinate or something to fix... I let my gauge condition. I use this form for the metric. Yeah, it's everywhere. I assume this metric takes this form. I put it in bulk Einstein equations, solve it iteratively as expansion in row and find that all terms in this expansion are uniquely determined by these two data. So let me finish with... No. That's a very good point. I don't get any analog of stress engine tensor. Well. Let me say why. Because... Well, potentially stress engine tensor would be extinction curvature. Extinction curvature. So like first term in this expansion. So it start from a row squared. First term in this expansion would be extinction curvature and extinction curvature vanishes on the horizon. It's sort of property of horizons. And that's why we don't have the second tensor or stress engine tensor. So the CFT will have continuous spectrum of operators complex dimensions and volume and tension. Yeah. So it's very exotic. Well, it is exotic. But let me finish with some idea or proposal. So how we could define, I don't know, S matrix on this iter space. So let's take a causal patch of this iter. So it's d plus 2. So that would be a word line of inertial observer. It has past horizon. It has future horizon. And that causal diamond is all this observer can see. So it has access to the whole data in principle. When everything what comes in this region comes from here. What this goes there. Respectively, these are in going modes described by some conformal operators. These modes by other conformal operators. Okay, out. And quantum evolution in this patch could be while using this technique can be encoded in collision functions. Conformal collision functions. And similar similar structure in asymptotically flat spacetime wheel horizon. So we have well past horizon here. Future horizon there. So these modes incoming outgoing modes near horizon can be described by this effectively by conformal operators. And well. That would presumably give us some idea how to give quantum evolution in this case. Let me stop here. Your calculation seems to show that for a non-renormalizable theory things are very different. We do a nice formula here for C of omega. Yes, well for non-renormalizable theories and is larger than that number lambda is lambda is increasing function y. And it blows up near horizon. That means we cannot do perturbation theory in this case. So in this case, in this sense normalizable theories are nice because well here in there theory is free and we will define asymptotic states here and there which is not the case for non-renormalizable theories.