 Ευχαριστώ very much for the invitation to this closing day and actually even though I was not a very systematic participant in this trimester, I really enjoyed it because it's true I hadn't spent much time in my life thinking about the quantum world, so I think I learned many things. Now I have to find how to change. Should I do it from here? So first a disclaimer, you know, I think I was asked, I'm not totally sure, the right button, okay. I think I was asked by these three gentlemen, the organizers, to talk about strength theory in a way that non-specialists will get at least something out of it, epsilon out of it. Even though I see some of my strength theory friends in the audience, I can tell them right away that they will not learn much from this talk, but hopefully I will convey a little bit to the others, to the people that haven't been working in the subject. And anyway, with this in mind, well, I'm not going to speak about my results or my recent results. I will just put in perspective a few, I think, well established facts by my standards of mathematical rigor and comment a little bit on the recent literature in relation with the issues that were raised in this quantum trimester. So it's some kind of poor, poor, poor man's version of a Burbaki seminar. And talking of Burbaki, you know, I was surfing one day, I didn't have anything better to do. I was surfing on the internet and I fell on this blog from a young American mathematician who happened. It was a she and she was in Paris one afternoon as a tourist and she heard about the Burbaki seminar. She dropped by but she didn't speak French. So she says something interesting happens when you are really lost. You notice things that otherwise might not register. For example, I noticed I was one of two women in the crowd and of about 40 plus people. From the picture on the left, you can also see that the median age is probably about 50. Also, it's apparently okay to fall asleep at the seminar Burbaki and that holds here to feel free. Now here are a very brief outline of what I will try to say. I will spend very little time, five minutes hopefully, just to say a few generalities about strength theory. Of course, there's no chance I can do any justice to the subject. I will just make some general remarks. Then I will talk a little bit about singularities. Again, the way in which singularities have been considered in strength theory. Here I'm talking about mostly singularities of space-time. Then in the last two parts, I will try to talk to you about horizons and entropy and about the emergent geometry. Again, some of my friends here, I see Yosif will probably have much more to add. We can see as things go to the end. Now, some generalities first of all. Let me just try to put the setting somewhere. No proofs in whatever I say trust me that things are usually proven to the satisfaction of a physicist, not of a mathematician or a mathematical physicist. But these statements usually are considered sufficiently solid so as not to pose questions. As you all know, here is the story. We have a relativistic quantum field theory. That's the standard model. It's pretty well defined. You can put it in some grand unified group. It's asymptotically free. Again, for a physicist, this means that it can be defined in principle at all distance scales, all the way to the deep intraviolet. Then this lives in a classical background. And the classical background is indeed described by the Einstein-Hilbert classical action. And we heard Klaus this morning talking to us about quantum field theory in classical geometry. And I think this doesn't run into any direct clash at all with present day observations, treat geometry classically, treat the quantum field theory in the background. Of course, there is a lot of rigor that can go into this. And this was explained this morning. But a priori from the point of view of a physicist, things seem to work perfectly fine. Now, of course, many things may hide in the sky. But again, there are three at least reasons why this story is not satisfactory. There is an optional problem, because some people think the solution is outside physics. It's in metaphysics, anthropic or other. Otherwise of dark energy. That's the one thing. No one has a good theory or beginning of a theory of understanding. Then there are obvious conceptual problems, conceptually incomplete issues. Should one think of geometries as coherent or mixed quantum states? And what about the information paradox of black holes? And finally, anyway, the theory is even mathematically incomplete. And there is another singularity of general relativity. So you cannot simply assume that Einstein's theory is the end of the story, because the theory has generic singularities. And that's where you have to do something, put boundary conditions, say something more, do something more. So I think everyone agrees that something should be done at the latest of the Planck scale. And actually, you know, people have tried all sorts of things for years. And the most timid ideas run very quickly into difficulties. You know, the most trivial idea would be to say, well, let's deform Einstein's theory to make it renormalizable. It's okay for a Euclidean gravity, but it has ghosts when you try to go to Laurentian signature. So this doesn't work. Then I think the idea goes back probably to Sahara first, who said, well, you know, maybe gravity is some kind of hydrodynamics. But in a sense, emerge from some regular quantum field theory, in the same way that hydrodynamics emerges from atomic or molecular physics. And actually, at least in a simple version, this was ruled out by the so-called Weinberg-Wittem theorem. This is a very simple kind of theorem. It's a theorem that, again, theorem by my standards, not by Yorkes or many of the people in the audience. It's a statement that says that there cannot be a massless spin-to-state in a theory that has a conserved energy momentum tensor and relativistic invariance. And the theorem is really sort of trivial. All you have to do is show that there is a matrix element of the energy momentum tensor between two states that has to be non-zero. It has to be non-zero because the energy is a conserved quantity. So if you integrate t00, you should get e. And furthermore, Lorentz invariance, you can check, quickly forces you to write down this as the answer. But this is obviously inconsistent. You have j spin bigger than one particles simply because this quantity on the left does not have consistent transformation properties. If you try to rotate by an angle phi around the momentum and the spin projection of the particles p and p prime, you know, it's a pretty trivial statement. Simply this is the vector representation of the Lorentz group. But if j is bigger than one, you cannot make this work. Are you taking questions during your lecture or after the lecture? No, I'm happy if they don't divert me too much. They do linearized gravity, which was done a long time ago. Then I have a spin as helicity to massless particles. Yes, but there is no conserved energy momentum tensor. Okay, it shouldn't. So this simply tells you that if you had a theory like QED with only spins... Sorry, I'm sorry. This tells me that if I have a theory like QED, a theory that has a local conserved energy momentum tensor, this theory cannot have states with spin bigger than one. And massless. Okay, it cannot have massless gravitons. You know, that's all it tells you. So if you take a regular quantum field theory, there is no way you will ever construct a massless bound state of spin 2. So of course, if you start with linearized gravity, you know, that's a different story. But there the answer is that the new is not defined. And actually, you know, basically I don't want to go into the details. You have to define gauge invariant or BRST invariant states. And that's the way you get out of this contradiction. Conserved means ordinary... Dimutimu nu is zero in Minkowski spacetime. So the covariant conservation is not... No, it's all Minkowski spacetime. No, tick. A relativistic one. I wanted to know if this theory applies only with the ordinary derivative in Minkowski spacetime. Yes, yes, yes. Because I'm using actually Lorentz covariance for the argument. You need both things. A local conserved energy momentum tensor and Lorentz covariance, okay. And indeed gauge theories get away from this because of gauge invariance. There's no conserved local current. Now, you know, people have tried all sorts of things. For instance, one way to evade the theorem is to break Lorentz symmetry either explicitly or spontaneously. Bjorkian I think did it first. Many other things, many other people tried. But all this looks to me extremely contrived and I don't think it ever went very far. It has been revived over the years, including more recently by Horava. But I don't think this manages to go all the way. And of course, the most plausible thing that people have been saying is that, well, if you are going to have the graviton emerge as a bound state, then probably spacetime should emerge too. Well, these are words. They sound like reasonable words. But the amazing thing in some ways, and that's the only point where I will put the word amazing or miraculous, is that somehow perturbative strength theory has evaded these first stumbling blocks in a much more conventional way. You know, it has circumvented these original no-go statements in a very conventional way, actually. And one may say that Dirac is really the father of this theory, both because he wrote this marvelous little paper where he tried to compute the mass of the muon as the first excited state of a membrane, and also, of course, because he never believed in a renormalization. You know, he thought that there should be some well-defined theory in the UV and one should never take infinities and limits, you know. Okay, now, so again, three words about strength theory. I'm not going to explain, of course, all the mathematical details, but essentially one way to state it is the following. There is a deformation of geometry, of classical geometry, that one may call the alpha prime deformation or the sigma model deformation. So the starting point is a two-dimensional field theory. This is the quantum mechanics for a simple string that propagates in some background metric and extra fields. This is called the dilaton. And now, for consistency, one has to actually assume scaling variance of this two-dimensional sigma model. Basically, the two-dimensional auxiliary metric has to completely get out of the picture because it's not part of the degrees of freedom of the propagating string. Now, one of the most beautiful discoveries, I think starting with Friedan's thesis, is the fact that indeed the beta function equations for this two-dimensional sigma model give gradient flow derived from some action or C function, and I have written down the first few terms of this function, which turns out to play the role of the classical deformation of the Einstein-Hilbert theory. So it has the Einstein term, and then you see it has a series of higher curvature corrections with very well-defined coefficients. People compute them. And this is simply the beta function or sigma model deformation, which in math language, if you want, is a rich flow but corrected by the appropriate higher order terms. Anyway, the upshot of all this is that geometry or classical Einstein theory is in a sense replaced by conformal invariant sigma models. And I think this is an example of a deformation. You know, when you come from some other directions, one tries to deform consistently Einstein's theory. That's a deformation. It does all sorts of nice things. It has no ghosts, and this can be shown. It always has a spin-2-massless state in the spectrum of this string. So in a sense it is a bound state, but not of a finite number of quantum fields, but of some string. And by the way, the fact that this should be massless uses very crucially quantum mechanics, namely the fact that when you compute the mass operator for the string, it starts out as minus one. That's due to quantum fluctuations of the string. You know, there is some kind of Kazimierz mass, and if it weren't for this minus one, you would never get a massless spin-2 particle out of a fluctuating string. Of course, when you said no ghosts, in which sense? Because when you have alpha prime, I mean you sigma, etc., I mean you sigma. Up to this stage there is ghost, but if you resum the infinite series, and of course you are not doing it like this, you basically go directly to the spectrum of the string, and then you can check that there is no ghost. Now these are many segments, there are some kind of collective degrees of freedom. I mean, each are coming from these other sides. Yes, absolutely. They are a collective degree of freedom, and if I solve an equation just after alpha prime, I am getting the term section which became my ghost. Absolutely. So what's the source of this? Well, you have to resum. The ghost up to this order would appear at the scale alpha prime, the string scale. Very good. If you resum the infinite series, it goes away. You don't see it directly, but could you, because nobody resums the series which you would resum? Yes, because that's not how you do it, but you can do it differently. You just quantize the string, calculate the S-matrix and so on, and then you see that the S-matrix has no ghost. The S-matrix has no ghost. The S-matrix has no ghost. Yes. It's okay, but it's okay. But so it means that in this kind of extra mode, you should not trust the GPS from the system. You shouldn't trust this truncated action up to R squared if you are going to talk about things at the scale of the string. So anyway, this is clearly a fascinating deformation of classical geometry. It has given rise to all sorts of beautiful mathematics. You know, mirror symmetry came out of this. No geometric backgrounds, things that have absolutely no geometric counterpart. Also sometimes called asymmetric corpifolds have come out of this. And it's clearly a very rich mathematical subject, and it's also something that allows you then to go on and compute the things that a naive quantum field theorist wanted to compute, things like one or two loops scattering of gravitons that give you finite answers with no ambiguities. And again, this has been developed by many people, in particular Pierre van Hove and Michael Green here. There are beautiful mathematics in calculating all this. And that's essentially where I will stop about general string theory. I just want to make one more remark that it is a theory where no external sources are allowed because only on-cell quantities are non-divergent or finite. As soon as you try to go off-cell or put some external source, you don't know anymore how to calculate all these finite things. So in a sense, the no-mass of quantum mechanics that Thibault mentioned the other day is also a no-mass of string theory. It's claimed that finally victim proved that it completely will provide a finite. Is this believable? Well, the claim is that if you find any divergence, it can always be converted to an infrared divergence. Infrared divergences we are used to, we are not worried by them. They simply tell us we don't understand something about the ground state and we don't understand many things about the ground state. Now, I don't think there is really more... I mean, this is something you can prove by your hands. You deform Riemann surfaces and you show that the dangers come from boundaries of modular space of the Riemann surface and those can always be interpreted as infrared problems. And, you know, Wheaton has tried to indeed write down very carefully the Docher and Funk first, the measure of the supermodular integrations. I don't believe there is a proof. No, if you ask me there is no, even for me there is no proof. But it sounds like a very reasonable statement. But isn't it also that ambiguity is in defining this measure that sort of starts growing after five loops? Yes. And he claims that... But the ambiguities are due to gauge fixings? Yeah, okay. I mean, let's not get... You know, in a sense, if you had said to someone 40 years ago you computed two-loop gravitons scattering amplitudes and you will get a finite answer, a physicist would be perfectly happy with these, might even tell you we never see anything beyond two loops anyway. But it's true that that's something that has to be proven and it would be a great achievement if there is finally a proof. But, you know, for someone, again, more heuristic and I've reminded, like me, you may say, okay, all this is marvelous, why aren't we done? What's the story? We have a nice deformation of geometry. It seems to do all sorts of other things. Well, there is a pragmatic and the foundational answer to this. The pragmatic one, I think, is clear. The connection to the observed low energy world is incomplete. And this raises very hard problems. Super symmetry is essential. When you break it, the divergences can creep up again. The sitter's space doesn't arise naturally, so our accelerating universe doesn't arise naturally. And there seem to be indeed problems of how to select a vacuum that is stable and resembles our low energy world. So in a sense this, you know, for a physicist, clearly the big problems, if one wants to one day make contact with the real world, are essentially there. But from a more foundational process. What is a vacuum in what you stated here? Well, yeah, okay, so you know, this is a theory in ten dimensions. Our world looks four dimensional. You have to therefore find some compact manifold. My question is, how is defined a vacuum state in this theory? Yeah, so a poor man's way would be something that looks like Minkowski, empty space, or the sitter with an extremely large radius of curvature. Again, I'm not going to talk about this because there's a more foundational problem, which is time. And in the sense, in the rest of the talk, I think I will mostly talk about this, but I will go to you what I mean really. But you know, since Slava mentions all these great Russian physicists, and I'm Greek, I thought I should find some Greek here, and in a sense the first guys that try to do a phenomenology of time were the Prisocratics. And there are these marvelous sayings of Heraclitus, who basically notice two phenomenological things about time that are absolutely true. There are some flows, and there are no closed time like geodesics, and this is the Greek translation for the phrase that you never get into the same river twice. Yeah, right. It's not obvious. Yes. Putamisitisin, aftisin and venusin, etera ke etera idataipiri, which means sort of, you think you get into the same river twice, but the water and you have changed. Yeah, right. So that sounds trivial now. It's not Erasmian. Yeah, I'm sorry. Anyway, so now time. So just to summarize, you know, string theory, I think avoid some of the first traps on the road to theory of quantum gravity, but as I will now try to argue, issues with time are crucial and not at all understood. So let me start with singularities, actually. So string theory has been indeed very successful in resolving singularities. I will give you a few examples of those. These are places where the geometry, the geometric Einstein description breaks down, and in GR one would have to supplement the quantum field theory that lives on it. One would have to supplement the theory by ad hoc boundary conditions. For instance, here is an orbifold. It is simply Rn modded out by some discrete group of the rotation group. It has a conical tip singularity, and if you try to do general relativity and QFT on it, you need to give some ad hoc prescription for what happens here. There's only time-light singularities. These are exactly, yes. Now, string theory resolves this, and the way it does is that it gives you a very precise set of so-called twisted states, basically strings that go around the singularity. You can quantize them. They live near the singularity because it takes a lot of energy or mass to get away. So there are these localized degrees of freedom. They can be massless, and so the singularities that you think you get are completely resolved because there are these new massless states that you didn't see in general relativity, and if you do any scattering or anything else, you may excite them, and again remember the motto is that if you see singularities, they are infrared, it's because you forgot some massless infrared degrees of freedom, and that's exactly what happens here. Of course that's okay. In this sense it resolves. In a sense the string goes and then doesn't fill this singularity, or in a sense that beyond this singularity you can unambiguously solve equations for the background. No, no, in the sense that in this space I can calculate for you unambiguously, say as matrix elements, even if the incoming particles seem to propagate from the cone. No, but on the side of the background okay, with this singularity you can say also that the equations which determine your background for them cache problem is completely well defined. Yes, you see, of course I could do the following. How you would explain your space? There's nothing beyond, there's none, that's what space has. So space is with singularity, just for this matter it's problem, you see if I were doing quantum field theory I would have to round off this thing and then define my story but there would be a total ambiguity on how do you round it off and you get different answers for different smoothings and so on. So string theory gives you a prescription, that's what one means resolved, basically it tells you what to do at this point. Now there are various other examples, maybe I won't describe them all, one of the most famous ones are so-called deep brains and the deep brains are again singularities in space-time but where instead of these localized twisted modes you get open strings, that's the thing that sticks to this space-time defect, you know it's something singular in space-time it is characterized by the fact that an open string sticks to it, the open strings also can be massless and again that's how you resolve the deep-brain singularities in a very precise way in particular again you can do scattering and you get exact answers if you allow for the possibility that open strings can also get excited in the scattering then there is something called the conifold maybe I will skip this the novelty of the conifold that's simply some singular calabria manifold the novelty is that the massless degrees of freedom are not strings as in the previous two examples but wrapped membranes or the two brains and various things have been checked also here but here is an example that probably anyone that has worked on geometry knows it is the so-called taubnat metric so this is one of the very nice metrics in four dimensions with the topology of R3 cross a circle and this metric you see or maybe you don't see and here is the explicit metric and if you think of it by dimensionally reducing to three dimensions so you forget this compact circle actually the manifold looks like a cigar as one says that's how it looks where x equals to zero is the tip of the cigar there is no singularity there it's simply locally R4 but if you try to dimensionally reduce this circle what you will find exactly is essentially the Dirac monopole so some singular solution of three dimensional euclidean geometry but of course in one more dimension there is no singularity at all this is a perfectly smooth manifold and what are the massless modes localized modes the singularity here they are simply Kalucha Klein modes so modes that course sorry singularity singularities which are resolved singularities in which person may be interested it's not singularities like in black hole it's not singularities like in union but I'm coming to this now you are running, yeah these singularities are also singularities from the point of view of curvature invariance like scalar curvature is infinite well yeah that's a good point so of course in this example the answer is no but in the deep brain example the answer is yes you can't get curvature diverge if you just take naive gravity or super gravity solution this singularity because you know I don't trust singularity itself because when curvature becomes of the plane curvature you know that these solutions are not trustable because all terms for instance your alpha prime expansion becomes of the same order yes so you should not do an expansion that's what I'm trying to say you should directly that's what strength theory does you don't do an expansion but hold your question maybe we can discuss it because the key thing I want to say is what Tibor already said all these singularities are time-like actually whenever we resolve a singularity in strength theory time is an idle spectator so these things are things that you know stay there for all time and with space-like or light-like singularities strength theory has had remarkably less access actually I might even say no success and it's not because people haven't tried you know people have tried all sorts of things for instance orbifolds but where you just mod out by a Lorentz transformation say indefinite metric spaces or simply weak rotating say tau-nat solutions or things I will mention them in a moment called S-brains or the so-called the Sitter CFT correspondence nothing seems to work at least in any reasonable manner so clearly there is a problem here is actually an amusing analogy it will show to you a little bit what I mean we wrote this little totally unnoticed article with Marc Enno on Claudio Buster just think about two plus one dimensional electrodynamics and define a source to the Bianchi identity you know this is df equals to 0 usually but let's put a source that plays the role of the Dirac monopole in three plus one dimensions so if I was in three plus one and time was a spectator here this would be a monopole but I am in two plus one that's crucial and now you may say what is this that's something singular for a Dirac monopole we know or resolve the singularity by going for instance to some non-Abelian theory writing down a Thouft monopole but how about this simple theory well first of all if you think about what this source does it does the following thing we called it an event it simply deposes g-units of magnetic flux at some spacetime point and then this of course follows the three equations and simply expands in a light cone and there is the anti-event where you basically focus a beam of light in these two plus one dimensions to a spacetime point and there it disappears so now you may say well can't string theory resolve this kind of thing the answer is it does but it does in a very bizarre one first of all you can you can check the little paper that's the essential no trivial statement there that there is some analog of Dirac's quantization condition what you can do is try to calculate in this simple model what happens to a charged particle in the presence of the event and there is a topological answer it doesn't depend on the details of the event here only on the long range fields and the topological answer is that the particle acquires angular momentum which is eG over 2π and if this is quantized the coefficient G has to also be quantized but you see it comes there is no Dirac's string here it comes from a very different argument anyway so string theory resolves this singularity as follows and you will see the difference it resolves it with the help of an extra dimension so that's exactly what happens there is this object I call the d2 brain and the equations on the d2 brain are precisely the equations of 2 plus 1 dimensional electrodynamics it turns out so believe me for this but now this d2 brain lives in some higher dimensional space and actually magnetic flux is confined outside it but can be carried by a particle that's called the d particle and therefore the process I just show to you corresponds to some d particle coming in from outside the flux is confined it cannot spread it hits the surface and then it spreads by the way you can actually try to do this in an experiment in a Josephson junction the words are exactly the same in the Josephson junction we are not very good at convincing experiment maybe someone should try maybe to convince them to do this experiment I can tell you more about this the point I want to stress here is that in this example singular events are generic a d particle can hit while anti-events require absolute fine tuning of initial data one can easily generalize this to higher dimensions but the main point is that the resolution unlike that of time-like singularities is very different you see a time-like singularity here would be resolved by embedding in an Orabilian theory which does the resolution but this doesn't help here you can check the resolution depends on initial conditions in an extra dimension so how to transplant this to a gravity theory and does it make you any wiser I'm not sure but this is the kind of problem that people that try to talk about S-brains and the pyrosis and all sorts of things are facing the resolution seems to depend on initial conditions in a non-trivial manner so to summarize this string theory is all sorts of time-like singularities but space-like singularities pose a different challenge so before I go on I'm happy to stop and yes I just want to make a comment that somehow the distinction between time-like and space-like is not so drastic if you have a Schwarzschild black hole the mass of the galaxy and you put on electrons the singularity becomes space-like becomes time-like I'm going to come to this so maybe we should discuss but mathematically it looks very different but physically it doesn't matter I don't agree of course but let's discuss it I'm going to comment on this now precisely I want to come now to entropian horizons so the singularities I talked about are generally namely in a sense there are not many quantum states there and one way to say it is that all these localized modes I talked about are generically in their ground state or if not they just escape to infinity and you are back into a non-degenerate singularity so what happens to degenerate singularities now the simplest example are the so-called extremals to charge black holes so I'm going to come to where your sieve is pointing here is an example so suppose you take a string now it happens to be an atherotic string here it winds around some compact circle a number of times call it w but it also has some momentum in this same direction which is quantized in units of the inverse radius because the wave function has to be single valued so the momentum is n over r and n and w are the two charges I referred to here so this is you know some kind of string it has these two charges that are obviously conserved you cannot undo the winding you cannot destroy momentum it's a conserved quantity and one can check that the mass of the string is bigger or equal than some lowest possible value it's called the extremal value which is n over r plus 2 pi r the winding number times the tension of the string and now one can also easily check that this has there are a number of degenerate quantum states of this system and the entropy in the extremal case is exactly given in the large nw limit by 4 pi square root nw so that's simply you know quantizing a free string would give you this answer so the number of states grows as the charges become larger and larger now when we say well here is a particle that has entropy and two charges let's think about the corresponding gravity solution in this case I only had a compact circle so I mean nine non-compact dimensions super is there because it's always there for convenience in string theory otherwise things get ugly but I will get to that too and you write down the spaghetti solutions and they happen to be singular actually and the reason why they are singular is that it's not only metric unfortunately here or fortunately there are many other fields including scalar fields like the radius of compactification you know this cannot be of course a constant it's a field in gravity it can change from one point to the other and the dilaton is another one and it turns out that with two charges you can never equilibrate the scalars to finite values just to understand why you know think of this system if there was only winding it would tend to shrink to zero if there was only momentum it would tend to blow it up radiation pressure but both of them managed to stabilize it to some optimal value so the radius is stabilized but the dilaton is not by these two charge solutions and actually it's really to the force that several people including atives have the Volcker have succeeded in doing precisely that's one case where they managed to sum up these alpha prime corrections in a precise way and to show that even though it's a singular solution of Einstein theory it is a non-singular solution of this infinitely corrected theory by this higher order terms and not only this but by using waltz generalization of the entropy formula you can match exactly this 4 pi square root nw by just calculating the horizon area of these black holes so these are sometimes called small black holes they are of course very interesting in a sense but I don't want to to continue with them if you sum up your correction is there explicit formula for the sum in R yes there is not for all quantities but for the entropy of an extremal black hole so you cannot write equations but how you know the singularities are removed because for singularities you have to have equations which you are solving yeah yeah no you are you can see that the square correction suffices to smooth the solution in the sense of creating a horizon you know if you put the first two terms but then you know the whole point is can you exactly calculate the area or actually the equivalent of the area in this system and that's what these people have done and if you do it you find but the ground without singularity I mean corrected the ground which satisfies all these corrected equations which take into account of alpha prime corrections without singularities by what is replaced the ground if you would have it so you know you go near the singularity it's the usual solution of Einstein's theory and then very close to the singularity it changes and it creates a horizon that's what I'm telling you you can go I don't want to go behind I'm not sure because you will see that going behind is an issue but what I'm trying to say is that here is a singularity that's the summary is a singularity that has degeneracy it has entropy it is apparently being resolved by string theory but it's all happening again at the string or the plank scale at very very short distances so it's you know a better version or a more rich version of the resolutions we have already seen but nothing drastic yet well it's something you know that I'm not sure actually maybe you see if knows this time like I think it's time like because it's extreme but what about large black holes and now that's where things start becoming more interesting because if we now repeat this thing but to a system where the horizon grows to very large size then why on earth would Einstein theory break down or if it does what happens here you see it breaks down because curvatures are large but in a very large black hole the curvatures are almost zero at the horizon so that's where things start becoming more interesting now the famous three charge example is the strong javafa black hole doesn't matter you know how you make it up here is in a sense a cartoon version of the icing model of this black hole it has the five brains the one brains and kalutsa client modes that move up and down and the whole thing is wrapping a five torus four torus times a circle and the three the number of five brains the number of strings and the number of kalutsa client modes or the momentum are the three charges and again believe you solve the supergravity equations there is now a big macroscopic horizon for large charges and there is now a real Beckenstein Hawking formula area over four gh bar which is given by these charges and this is an example where you can count states of a black hole ok so it tells you that at least the microscopic picture cannot be totally crazy now this kind of black hole so now I'm going to get close to what you see if is saying this is really basically a generalization of the charged black hole in four dimensional Einstein plus Maxwell theory so here is the metric for a charged black hole it looks like the Schwarzschild metric the scale factor F or the redshift factor F has a somewhat different form you see it's quadratic in one of the there are two places where it vanishes they are called outer and inner horizons they are given by the mass and the charge in terms of this formulae and again there is an entropy the area of the horizon of a G and the temperature which is the difference of these two radii so T equals to zero is the extremal case where the mass is equal to the charge and the square root vanishes and that's in a sense the ground state of the system ok now here is indeed a very interesting proposal and you see if is one of the people who have worked a lot on this which has occupied people for almost ten years now and one proposal which goes by the name of Fasbol is to say the following so you know I showed you some deep brains and we could do some counting that measures exactly the number of microstates of the system but could these microstates be simply encoded in the geometry ok now this goes against folklore theorems called no hair theorems which are basically valid in four dimensions and which tell you that outside the horizon of a black hole once it has settled down to the ground state there is not much you only see the electromagnetic fields that carry the charge or couple to the charge but nothing else now this folklore theorem clearly fails actually in higher dimensions so the folklore says that there are no gravitational solitons other than black holes there are many many invasion windows and pitfalls with a very illustrious prehistory apparently this is the only wrong paper of pauli I am told pauli wrote a paper with Einstein on the nonexistence of regular stationary solutions of relativistic field equations in 43 and they had in mind the kalucha Klein theory in five dimensions and I just told you right before that there are monopole solutions that are obviously smooth and that are not black holes they are just regular solutions solitons of kalucha Klein theory now the argument the reason why the argument was wrong is that they used some wrong scaling argument you can look at the paper which is always amazing and reconforting to know that even these people can make a mistake but the theorem of lichnell which was which is also wrong absolutely now the key to evasion is basically non-trivial topology and there is a very nice recent paper of Gibbons and Warner that puts the dots on the eyes in the particular case of five dimensional supergravity with two supersymmetries I am not sure how I am doing with time who is counting time then I want to maybe skip this proof but essentially the way what they show in a very general setting is that if you consider a five dimensional gravity theory well supergravity it has some extra stuff and time a stationary or static solution this means that you assume a time like killing vector kappa then the existence of non-trivial second homology so non-trivial two cycles on the spatial sections have to give you stable solitons and they exhibit a large number of them and basically you can avoid the simple no-go theorem because of some Chen-Simons term you can look at the proof if you want afterwards so one can avoid this and a lot of hard work has gone into trying to generate enough of these solutions to account for the entropy of the three charge black hole now at the end of my talk maybe your sif will want to add things or make a comment let me just make the following statement you know remember these would be now solutions that look perfectly like black holes but when you get close to the horizon space will end somehow in some smooth manner so in a sense there is neither horizon nor interior of the black hole in the usual sense now of course it would be a great mathematical achievement if the entropy of these things could be accounted for by smooth 11-dimensional geometries supergravity geometries I'm personally skeptical and indeed because I'm not sure I agree with the earlier comment of your sif in a sense a baby version of this are multi-center Taubnath geometries in five-dimension these are perfectly smooth solutions of five-dimensional Kalucha Klein theory the mimican extremal four-dimensional charge black hole from far away there is no doubt about this but does this really imply a breakdown of effective field theory for some in-falling observer that should normally see a very flat almost empty horizon as general relativity usually tells us my take is that in most of these general relativity theorems or statements one always makes an assumption of genericity as for thermodynamics and things that happen here look extremely non-generic another way to say it is precisely if you draw the Penrose diagram of a charged Reissner-Nostrom black hole let me again not explain it but to go to the last part but the singularities of this geometry are indeed time-like they are hiding behind the second horizon so this is the outer horizon this is the second horizon this is the region in between and the thermos and others have shown that a priori this Cauchy horizon as it is called is completely unstable to a small perturbation and should collapse to space-like singularities so I think but I cannot prove it that we are still in a situation where we don't have space-like singularities and one would have to really exhibit cases with space-like singularities to make the first ball case more convincing now again I'm holding off your shifts probably a rebutal to this the last ten minutes go to the last question which I think poses a very sharp problem that many people have been pondering the last two or three years which is the question of emergent geometry ok now all these issues come into very sharp focus in the context of the so-called ADSFT so I will tell you exactly what I mean now even if you are not a strength theorist you must have surely heard that there is this conjecture which says that on-shell quantum gravity with asymptotic anti-desitter boundary conditions at spatial infinity is equivalent to an ordinary relativistic conformal field theory or quantum field theory in one less dimension or on the boundary of this space the most famous and studied example is quantum gravity in ADS5 xS5 and an equal four-dimensional Young Mills theory now Young Mills theory is unitary everyone would agree even though again maybe proving it won't be always necessarily easy so if the correspondence is right one should have states that resemble black holes that's I think obvious so here we have a very well posed problem how does the geometry of this anti-desitter space emerge from a regular conformal field theory in particular of course one should evade Wienberg's theorem but this we know can be evaded in a strength theory now to start and to see that this is a hard question clearly this is a very hard question but here is what you can do consider first empty anti-desitter this is empty anti-desitter you see it's flat spacetime and one radial dimension so that things get blue shifted or red shifted as you move uniformly as you move in the radial dimension z is one of the coordinates of the gravity theory but not of the conformal field theory so the conformal field theory indeed only has operators that depend on x okay let me take one of these operators it's a conformal theory so it has some scaling dimension delta and now there is a dictionary that I won't describe but which allows you to formally write down something like a free gravitational field in the bulk so you take this operator you Fourier transform it then you sum over these are solutions of the free wave equation in anti-desitter space with appropriate mass related to the scaling dimension you see I write this as phi CFT because this is entirely defined in a quantum field theory there is no gravity however it's easy to show that's very simple by just looking at the two-point function of this operator in CFT that this has exactly the properties you expect of a free quantum field in anti-desitter but actually in the limit of the number of colors to infinity this has only two-point functions that's part of this large n factorization so it looks really like a free field but now what does it mean to find locality in the bulk geometry of this theory well we know that what this means is that if say you do a four-point scattering of these fields you should find an answer that falls off very rapidly unless the wave packets that you formed come into the same point in spacetime of five dimensions right so I mean these are free fields now I'm going to make them interact to leading order in one over n ok so just regular perturbation theory and I want to understand why this interaction would look local not in the x-muse only but also in z you know z was a made up coordinate it didn't appear a priori in the conformal field theory now that's a very nontrivial question actually and it hasn't been answered really you know there is really marvelous mathematical progress to calculating exactly the four-point functions of super young mills for all values of the couplings this would answer this question if we had the answer but all this technology is completely on-shell in the conformal field theory so you can only take the external operators to have light like momentum and clearly here we need off-shell quantities in the conformal field theory to calculate on-shell scattering in the gravity theory so even this simple fact empty at the desitter has not yet been fully tested but there are very indirect but very convincing arguments that if the conformal theories of a special kind namely it has a gap in the scaling dimensions for operators of spin bigger than two and while just looking at the structure of conformal blocks there is convincing evidence that this theory gives you local physics in five-dimensional anti-desitter even though it's a four-dimensional CFT and of course these statements have been checked, can be checked in any called four super young mills so these are not assumptions once you assume them once you prove them and from the structure of conformal blocks you can essentially convince yourself that locality is there in the vacuum now I used here Poincare coordinates to go to global coordinates where anti-desitter is a box it confines things in the middle because there is an infinite so the boundary is not anymore a plane but a sphere times time and if you try to go to the boundary there is infinite blue shift so you come back in and now let's think now about this problem and I'm done in a few minutes so we still have now super young mills defined on S3 times R the gravitational description tells us that radiation is reflected at the boundary of the sitter and if you work it out there are two kinds of black holes in this system there are so called small black holes that are like the swatches of black holes they come in and evaporate but there is another kind of black hole if you put enough energy into this you drive the temperature very high since the radiation cannot escape that's a trap you will reach thermal equilibrium and for a high enough temperature this thermal equilibrium geometric state is a so called large radius black hole with a big horizon it's not simply a free gas of gluons I'm sorry of gravitons in anti-desitter it's called the Hawking-Pales transition so at a sufficiently high energy you may then say that the typical state of the quantum field theory made out of four different square operators that's the correct energy should resemble a large radius black hole and now we have a problem can we reconstruct this geometry from normal quantum field theory and this is now a geometry that has a horizon and the space like singularity here is the geometry there is no way out here there is a horizon and you just crunch if you just solve the Einstein equations that's what you find that's the Penrose diagram actually this is an eternal black hole this is the exterior this is the black hole interior so if you create the black hole an infalling shell you don't get this full diagram this full diagram is so in a sense the crucial maximal extension of the black hole and it has a white hole in the mirror but the main point I want to make is that there is a horizon and the space like singularity so what will conformal field theory do about that now in a sense we are really stuck with the problem so since the black hole does not evaporate gravity is unavoidable but if this is strength theory or super young mills one must be it must be possible to understand why and how the effective field theory breaks down it better break down because there cannot be such a singularity in super young mills notice that the Hartle-Hottin assumption or things like that will not help here because the singularity is a crunch I mean you are just ending there it's not some initial condition so what can you do again people have tried all sorts of exotic and that's a fair game when you are faced with such a question why not try anything here is a real exotic for instance it's called post-selection or a final state boundary condition or the singularity here is a formula that Jörg loves very much this is basically the probability of successive measurements in time a1a2 up to an giving specific values a1a2an and this proposal says well we have to stick in also some kind of density matrix for the final state but that's got you know even I cannot swallow this that's catastrophic because you violate the one rule you want to keep namely that the result of an experiment of n-1 measurements doesn't depend on the nth measurement you may want to make later in time I mean if this was true then all notion of causality would go so for instance this is an exotic proposal that I find very hard to accept well let's take a closer look actually long after the black hole has formed near the horizon the geometry as usual looks like empty localiminkovsky space and in terms of the coordinates of an infoline observer these are the Rindler coordinates the vacuum somewhere here looks like a thermal vacuum you can you know make this very precise so there are the modes that are defined say at some given time t there are the right and the left modes the right modes are the only ones that can escape out to infinity so they are the only ones that propagate to the boundary where the conformal field theory is defined so if you are going to make an effort to reconstruct the geometry beyond the horizon you have to say something about the other half of the modes the left modes are these made of operators in the same Hilbert space or not here are three answers ok no that's in a sense leading you to the unavoidable conclusion called the firewall why well you can see it in the figure if these are totally independent modes who tells me that this state has been prepared so carefully that all sorts of quanta from this side into the black hole interior and then when I am falling in from this side I will be hit by all sorts of radiation so to avoid the firewall if the L modes were completely independent acting in a different Hilbert space than the R modes would require a very big fine tuning of the state of the system and we know nothing about it so the other answer is yes but then you see if these L modes were acting on exactly the same Hilbert space as the R modes you would get into another problem then the infalling observer can act on the same Hilbert space states as the guy outside and then why can't he send the signal to him and if he has access to exactly the same Hilbert space three minutes and then and he can modify things then he can send signals but remember this horizon a guy in here can never forever communicate anymore with the outside so you know both run into trouble and now here is the third and I think probably most reasonable proposal proposed by these two young guys Papadodimas and Raju in a series of papers and the answer is yes in a restricted sense and unfortunately the construction is state dependent I'll tell you what I mean and then I'm basically done let me simply say that the idea looks promising but it still must overcome I think many many hurdles state dependence causality and still it won't really tell us what happens at the final singularity but the idea of these people is to essentially mimic in an approximate way the Tomita Takesaki construction of observables so they say the following let's consider a typical highly excited state psi of the conformal field theory that's a typical state that has enough energy to be a black hole now the algebra of observables and this is something you would absolutely agree is really a very small algebra you don't want to use any possible operator because in particular there would be operators that destroy the black hole but the only experiments we do are basically perturbing a little the state so let me think of an algebra that does not alter drastically the state in particular the operators never annihilate the state of the black hole this is the state dependence of the operators we are going to look at now this cannot really be an algebra either and that's where you know more mathematically stronger people in my opinion can come in and try to settle some of the points in particular we want to avoid n square fold products of these operators because these are precisely what would annihilate the original state so it's an algebra in the sense that you can add things but you cannot arbitrarily multiply things now if you assume all this then by mimicking essentially this construction you can show indeed that one may construct an isomorphic algebra a tilde which commutes with three and such that the o tilde and o operators are exactly entangled thermally in the particular state psi ok it may sound total logical but it tells you what does it mean to have entangled operators oh it means that if you expand out the state in eigen states of this operator you will find exactly what you expect from a thermally entangled state so remember you will basically reconstruct a particular state that I flashed by you maybe somewhere here here since I see David getting up let me just finish and then we can discuss it more but in a sense what this tells you is that you can try to construct the modes behind the horizon which are thermally entangled with the modes outside but it will only be an effective description and more importantly it will depend on psi so every time you change psi it will be a different set of operators now you may say that's it no point in considering it but in a sense maybe that's what we need to describe in a background independent way the geometry of an infalling observer so I'm done I think you know the merit of this is that it helps to sharpen a little the puzzles in some very definite setting put them in a context where in principle with unlimited technical prowess they could be answered but actual strength theory calculations have been very hard to use in this debate and some people have still while have been brave enough to try them it is a well known fact that high energy scattering of strings is non-local in the sense that strings stretch out to arbitrary transverse size and many people feel that they should still play a role in the resolution and it doesn't at all in this Papadodima's Resolution so I'm not sure what the final story is but I think it's a well defined problem for once and something should come out thank you very much we have done a brief question I want to make a remark because Costa has discussed about this geometry so the point is for extremely supersymmetric black holes at this point you have constructed enough states but they believe to be enough states to account for the entropy so there I think it's established that you have this singularity which is time like for the free charge black hole but the reason is very simple singularity is a string theory it contains low mass degrees of freedom and because it's a low mass degrees of freedom it affects the physics of a large scale so when you approach a singularity you can say I have a singularity the curvature blows up I can stay 5 km away and I can do physics this is wrong when you go near a singularity if the curvature is large here you can have a low mass degrees of freedom which can affect you but you never have a singularity the only if I may say a comment I mean of course that would be marvelous if it's true but then one would have to understand why in this room effective field theory in approximately Minkowski's space time is valid if there is a singularity in Andromeda or in the other side of the universe and that's the real issue right I mean if one gets some understanding of this then it would be marvelous