 Я счастлив, чтобы быть здесь и прожить Тибон. Я был рад, чтобы ты с Тибоном collaborировал уже 15 лет назад. И, в принципе, я бы хотел спать 2 минуты и поговорить о этой старой папе. Не для того, чтобы сделать драматику, но мне это нужно. Игра была очень простая. Повторим статик Блэхол-метрик. Повторим ТТ компонент. Повторим маленький диформатор и маленький параметр для ГТТ. И потом геометрия Блэхола сразу меняют. И вместо Блэхола, у нас есть геометрия. Их есть интересные феочи. Как я сказал, у нас нет хорайзона. Хорайзон был располаган из-за геометрия Блэхол. И здесь появится новая таймска. Новая таймска, которая зависит от Апселона. Это время, что нужно, для того, чтобы влезть из-за геометрия. Для того, чтобы влезть из-за геометрия и вернуть. И, assuming that the Apselon is exponentially smaller, эта новая таймска может быть очень маленькая. И, смотря на эту таймску, для того, чтобы влезть из-за геометрия, мы не увидим никакого различия. И эта таймска, которая зависит от геометрия Блэхол, и эта таймска, которая зависит от геометрия Блэхол. Мы считаем, что статикный и нератитный случай был считан над этим людям. Так что, в этой статике, в этой статике, эта метрика была просто какая-то абстрактная метрика. И она не satisfactorяет любые гератитационные вопросы. Но сегодня, в моем статике, я попробую демонстрировать, что геометрия Блэхол в этой статике очень естественнее в статике геометрии для больших классов квантомстейц. Итак, я говорю о геометрии Блэхол, которая, как и в этот workshop, это типичные решения, которые в статике геометрии. Speaking about quantum theory, the first step would be quantize metafills on the background of this classical black hole. And then what appears immediately, let's list certain freedom, when we do it, freedom, which is usually phrased as we can choose different quantum states. Speaking about renormalized test in the tensor, there is also freedom to add some certain piece to the T-menu, which is traceless and covertly conserved. And let freedom bring us possibility to define different quantum states. The quantum states known in the literature are the following. So Hachtel Hawking state. In this case, stress in the tensor is a regular at the whole horizon and at infinity it describes a quantum thermal Hawking radiation. Another state is Boulvard state. It is singular at horizon, but it is vanishing at infinity. So there is no radiation at infinity. And more realistic state is Unruh state, which describes states, quantum state for collapsing star. So black hole is formed as a result of gravitational collapse. That would be most appropriate quantum state to describe such a process. Unruh state is more realistic, but it is non-static, non-stationary. And I will not discuss it in my talk, and will focus on Hachtel Hawking and Boulvard state. So let two pieces of picture, right? Classical black hole, quantum field on the black hole. But we would like to have some unified picture. We want to put these two ingredients in one, let's call it quantum gravitational theory. And that would allow us to talk about quantum black holes. And what kind of questions you would like to address? Here is a list of my questions. So first of all, I would like to know what happens to black hole horizon when I take, well, in this quantum gravitational theory. And are there still solutions to horizons? And what are the new features which possibly appear due to non-locality of this quantum gravitational theory? And the second class of quick questions, what are the answers to this question, first question is related or depends on the choice of quantum state. And most importantly, what are the back reacted geometries of the quantum states, which I discussed. I should mention previous work, these kind of questions we discussed for many years in the literature. So my theory would be, my quantum theory would be semi-classical gravity. So in this picture metric is not quantized, but meta fields propagating this classical metric are quantized. And if I integrate over such quantum fields, I end up with some contribution to gravitational action, which I call quantum action. And in general, that would be my new gravitational theory. So it's classical theory plus new terms due to quantum fields. It is functional metric. And new quantum term is generally very complicated. It is non-local. There is an approach by Barwinsky, Volkovsky, Gusev and Zhitnikov, developed for almost 10 years, how to compute this quantum action. These are suggested expanded in powers of curvature. And this is very promising approach, but it's rather complicated and difficult to use. For instance, the cubic terms, which are cubic in curvature, they have 29 invariants. 29 invariants. And in quadratic they have 6 invariants. So it's too many and it's quite complicated to use this approach in a real problem. But we can see here that gamma, some function of Laplace operator, and here for cubic terms, they are non-local objects. So gamma, this little gamma, is logarithmic in Laplacian. Big gamma, because like one of a box. So they are very, very non-local objects. In two dimensions, things are simplified. If I had a choice, I would probably lift in two dimensions. There is only left and right. It's easy to make choices. In two dimensions, if I speak about quantum, conventional conformal field theory, then it's integrated, after integrating out the quantum fields. The action can be easily computed. This is so-called Palikov action, famous Palikov action. And we see that it is non-local. So it has one of a box. So in reality, one of a box should be replaced by Green's function. So what we see here is two reaches colors and Green's function in between. But it's quite simple, and that makes two-dimensional models very attractive. It's toy models to address quantum black holes and information problem. And since 1992, this subject was very, very popular. And discussed in the literature. So in my talk, I will first discuss two-dimensional dilaton gravity model and address these questions in two dimensions. This part is based on ongoing research with Yuan and Depp. Depp is a professor in India. Yuan is a student, PhD student in Turb. And second part is four-dimensional story. It's based on early work with Cleman and Depp. Cleman used to be our student in Turb. Now he's a postdoc in Canada. So I understand that time in this nice lecture hall is faster than outside. So my program is minimum to speak about two-dimensional gravity. And if I have time, I will discuss four-dimensional story. So in two-dimensions, as we all know, Ritch's color doesn't define any grittational equations. And we have to introduce some extra field to define grittational dynamics. And usually that would be one of possible ways to introduce color field, so-called dilaton. And this action, which is present here, is the most popular action. Dilaton action, so-called trinospite action. And the solution to this, well, we vary with respect to metric, we vary with respect to dilaton. We get some equations. These equations can be exactly integrated. A nice thing about these equations is that a trace-free grittational equation takes this form. And that means that I can define... So it implies existence of killing vector. So epsilon-minu contracted with gradient of phi is an automatically killing vector on these equations. That means my general solution to these equations is static. And it can be brought to shwasher-like form. And it describes generically... Well, it describes it to dimensional black hole with a temperature proportional to lambda. So it doesn't depend on the mass. That's a peculiarity of two dimensions. It has also certain entropy proportional to exponent of minus 2 phi. And phi is the value of dilaton at the horizon. Speaking about quantum field. So, as I said, I will integrate over quantum fields and the entropy of political action. I prefer to write this action in local form. This is local form. So if I vary with respect to psi, let me give you this equation. And technically psi is one of box A. And I come back to usual non-local form. On dilaton... Dilaton gravity reaches color is minus 2 box phi. And it means that there is a relation between psi and phi. That basically is relation. Psi is minus 2 phi plus w. And w is the solution to homogenous Laplace equation. There is also always a freedom to choose this w. For static metric, this equation can be easily solved. Basically, it takes this form. So it's c over g. And c is a constant. And you will see in a moment that this w or constant c contains information about the quantum state. So, computing stress in the tensor, which is just a variation of political action with respect to metric. This is the tensor of quantum field. We find for energy density, this expression. And we see that it generally contains a divergent term. So it is divergent at the horizon, where g of x is 0. At infinity, it's positive, proportional to c plus 2 λ squared. So now I can define my Hartley Hawking state. Hartley Hawking state, by definition, the state, which is regular at the horizon. So this guy should vanish. And that's our first surprise. We have two solutions for c. c equals 0, c minus 4 λ. At infinity, as it should describe the thermal Hawking radiation, it horizons a little small negative value. Now, Boulvard state, by definition, is the state, which doesn't contain any radiation at infinity. And there is only one state, value c minus 2 λ. So at infinity, it is 0. But, of course, it is singular at the horizon. But in general, we have a family of states parameterized by c. There is an interesting symmetry. So if I take a difference of energy density for two values of c, it is 0 if not only when c1 equals c2, but also when c2 and c1 are related by this relation. Let's explain why I have two values of c for Hartley Hawking. So this kind of symmetry exists in only space of quantum states. And Boulvard state is symmetric under such a map. Now I want to bring all pieces together and consider a consistent picture. So my two-dimensional gravitational theory consists of classical gelatin gravity plus поляков-экшн, plus a new term, which was added by Rousseau-Saskind and Tarlatsou. This is local quantum, local term. And nice thing about this section, that equations are still exactly integrable. So a combination of dilaton and trace of T-menu. T-menu now has these three pieces. It leads to this simple equation. And here there are two possibilities. I, the phi, is constant. Then from other equations follows that a richer scalar should be constant. Minus 2 lambda squared. This is the center space time. It might be interesting by itself to discuss this solution, but I will not do it here. The other interesting solution when R equals minus 2 box phi, and that is precisely what we had in classical case. So again, if I introduce psi, there is a relation between psi and phi, and there is a freedom to choose W and W is solution to Laplace equation. Then a quantum corrected black hole or Hartley Hawking state corresponds, as I already discussed, to omega to the case, when W is zero, so psi is just minus 2 phi. Still equations, trace-free grittational equations obtained by varying this complete action with respect to the menu, takes this nice form, and this form tells me immediately that there is still a killing vector. General solution is static, and that is precisely the form for this metric. I should say that, of course, black hole solution in ICT model was obtained by ICT people. I here just rewrite this solution in a nice charge-like form, which is more appropriate for what I am doing. If you look at this solution, we find immediately that first of all Minkowski space is not a solution. And let us clear why, because at infinity of Hartley Hawking state I have a thermal gas. So thermal gas curves the spacetime, and it's not anymore Minkowski. What it is, is clear from this metric. So if I put 8 to zero, let still geofi is given by some function. So this term is actually produced by... It's a back-reaction of thermal radiation. So this metric still describes a black hole, generally a black hole, if A larger than A critical, it still is a black hole, will the horizon, and temperature of such a black hole is still lambda over 2 pi, as in classical case. General C. A general C state, so I remind you that in this case psi is related to minus 2 phi plus W, a W is C minus over G. And equation still can be integrated, and this is general solution. This is our new result. And it might be difficult to digest this form in few seconds, but let's discuss it. So metric here, metric function G, it is related proportion to Z, function of phi, and relation between Z and phi is given by this equation. So GHH of phi is what I had in Hartley-Hocking state. So GHH has a horizon, but G for general constant C or constant A, A and C are related like that. Z, well, I have a horizon in this metric, so G vanishes, if Z vanishes, but Z never vanishes at any finite phi. And it means that for any value of A different from zero, this metric doesn't have a horizon. A more careful analysis shows that if A of C is negative, this metric describes a wormhole, and if A of C is positive, then this metric describes a naked singularity. So let's look a bit more carefully at the back-reacted geometry for Boulvard state. There are the same equations, which I had before. So A here is, I take exact value for A. So G, there is a minimal value for this function, and this minimal value is a couple, well, I can express it in terms of classical entropy. It's basically related to value of a dilaton at classical horizon. And this metric function never vanishes, and the whole space-time is actually two copies of the same asymptotically flat space-time glued together at some point, which are called phi-minimum, which is a throat wormhole. So this is a completely regular space-time. It doesn't have any singularity. And interestingly, now flat space, Minkowski space or more precisely linear dilaton vacuum is a solution to field equations. It belongs to this class of metrics for certain value of A. 7 7 7 means is okay. Sorry. So in four-dimensional story as a set is more complicated, much more complicated and one has to use, well, we cannot use that expansion in curvature and because it's well, it's difficult. And we need to use some smart tricks. We'll describe what tricks we can use. So let me just say remind you about certain universality at the horizon. So let's take any spherically symmetric metric. Suppose that there is a horizon, means that there is a zero for function g. Finite temperature means that there is a simple zero. And if I factorize g, so now what remains is so-called optical metric when gtt is 1 or minus 1, it depends on the signature. Then near horizon this g of z because exponentially it's universal behavior, beta contains information about temperature. And what remains here is optical metric and optical metric is a product of one-dimensional space, which corresponds to time. And 3d, usually it's 3, here it's written for any d. So if this 2 is 3-dimensional space which I call here m3 and there is again universality if I approach horizon, so z goes to infinity this space becomes approaches hyperbolic space we will use beta over 2pi and I remind that beta is related to hock and temperature. There is another interesting property of horizon, it is a minimal surface. In fact, we can universal logic assume that there is a surface in spacetime, in static spacetime which is minimal and it will immediately follow from Einstein equations that it is the surface of the horizon. We will use that and let's class of metrics which I will consider. This is a general static metric with 3 functions omega n and a. So if there is a horizon omega is I didn't write it here. So omega is exponent of sigma because like that and the optical metric approaches this universal form. So our idea is to use space. We cannot unfortunately in 4 dimensions we cannot integrate the conformal anomaly but what is known is relation between quantum action computed for conformally rescaled metric Well, there are 2 metrics metric G and rescaled metric Sigma is conformal factor and the difference is given by a local function. A and B are some constants which are called conformal charges. But so we use this actually we use this information assuming that assuming that exponential of 2 sigma is my omega. So I factorized GTT what is left is optical metric but we still need we still need this term here computed on optical metric and here is my optical metric it's basically in general it again can be expanded in some power in series with respect to curvature all this 3-dimensional space and we draw the high curvature tomes general structure can be found in the paper of Gussiev and Zelnikov and this result is actually exact if M3 is a 3-dimensional hyperboloid. So now migratational action has 3 functions sigma or omega n and n and variation respect to n gives me certain term which is divergent. It is divergent and that means that what we actually consider is a boulevard vacuum accidentally for certain choice of n, 0 and scalar fermions and vector fields this leading singularity disappears and curiously it disappears exactly for n equal 4 super Young Mills but that might be just an accident and we have to look at some leading tomes. So generally we don't have a solution to these equations but we have horizon which is natural because we have this corresponds to boulevard state. Well, instead of looking at horizon we can look at minimal sphere let bring us some restrictions on conditions on alpha and omega. Well, let path is very technical and bring it to the final formula. Let's restriction which comes from equations on the minimal value of omega. I remind you that omega is GTT component of the metric and from this equation we see that it cannot be 0 because the right-hand side is positive and omega 0 is exponential so omega is between 0 and omega 0 and this omega 0 is actually exponential of minus black hole black hole entropy if you compute it in classical case. And if A goes to 0 this parameter A omega 0 becomes 0 and we and the throat becomes a horizon so it's limiting procedure to get classical black hole. So, here's my conclusion back reacted geometry of boulevard state in semi-classical gravity is a warm hole we demonstrated this in two dimensional case and in four dimensional case. Minimal non-zero value GTT component of the metric is bounded by classical black hole entropy so it becomes one of entropy in two dimensional case an exponential of entropy in four dimensions and the two dimension examples show that there is an infinite actually family of possible quantum states which are represented by horizon free back reacted geometries and it looks like Hawking state is the only state whose back reacted geometry has a horizon we see it in explicitly in two dimensions and it's very likely to be the same in four dimensions but always deviations of the warm hole geometries and very hard to measure in real life so here's my question I first wrote like five questions but when I raised all of them left this one so if you look around us and we find that there are many black holes detected directly or undirectly so I'll be sure that all these black holes are really not warm holes and let me leave you with this question thank you very much and happy building