 Thank you very much for the invitation for this conference to celebrate this bold work. Igbo has worked on many aspects of gravity, which has heard of his impact on measurements, or from his work, works ranges from very concrete, astrophysical observations to more fundamental aspects of gravity, such as the behavior near singularities or the interrelation between gravity and string theory. So my talk today will actually be related to a paper that Igbo wrote on self-gravitating strings and black holes and the connection between strings and black holes. And it's based on a paper we recently wrote with Yimin Chen and Ed Whitten exploring further that thing. So we will consider here black holes in weakly coupled string theory. And in that theory, when the black hole size is close to the string scale, they are supposed to turn into highly excited strings. And we'll discuss some aspects of this transition. So that's what the talk will be about. And surprisingly, we'll find that the situation is different depending on the detailed type of string theory we are dealing with. So throughout the talk, we'll be considering what's called the weakly coupled string theory, so where the interaction between strings are relatively weak. So in those theories have a scale which is the set by the tension of the string. We can choose units over that is one. We'll choose it to be one in some formulas. And in the Newton constant, on the other hand, is set by the string interaction scale, the string coupling, and will be very small. So the plank length will be much smaller than the string length. And the string length is the length at which the Einstein description of the system says it's to be valid. So for distances bigger than the string length, we can apply ordinary gravity. And for distances smaller than that, gravity is modified by string theory, but it's still weakly coupled. And we can consider two types of string theory. It's called type two string theory or the heterotic super string. The two means that there are two supersymmetries in 10 dimensions. And the heterotic there is one supersymmetric in 10 dimensions. And we will mostly be discussing problems in four dimensions where we have four compact dimensions and six compact dimensions. Now let's introduce the various actors in this story. So the first is a highly excited string. So this is simply a string that is oscillating, an oscillating string with a lot of energy, so a large amount of energy compared to its tension. And such strings have some entropy which comes from all the possible ways in which it can be oscillating. And they have the peculiar feature that the entropy is linear in the mass. So it's proportional to the mass with the proportionality constant, which is so further again, the string scale. And we can think of this proportionality constant as a certain inverse temperature. It's sometimes called the Hagridon inverse temperature. And this is an interesting temperature because if you have the thermodynamics of these strings, this thermodynamics is well-defined only for temperatures less than this particular temperature or inverse temperatures, of course, beta bigger than beta H. Because in the partition function, we have a factor of minus beta M from the energy. And then from the entropy, we get the factor of beta HM. So only for beta bigger than beta H, this will be convergent as we sum over the very massive string states. So there is this maximal temperature after which that we have in a string there. Then we have also black holes. And of course, you're very familiar with black holes. I don't need to tell you what they are. But in this context of string theory, these black hole solutions are well-defined as long as the Schwarzschild radius is very large compared to the string scale. And you can compute the first, the corrections relative to that large limit. So when there is an expansion in terms of the string length over the Schwarzschild radius and the first terms in that expansion were computed. So the corrections due to string theory. And however, we can wonder what happens when they approach the size of the string there. We cannot really compute it per terratil. We have to do something better. Now, another indication of what might happen is based on looking at the entropies as a function of the mass for both the black hole and the string. We just saw that the free string has an entropy which is linear in the mass. And I put it here in blue. On the other hand, for a four-dimensional black hole, the entropy goes like the area and that's quadratic in the size and the mass goes like the size. And therefore the entropy is quadratic in the mass. So we have a curve like this. And so the curve really, each of the curves can only be trusted. For example, the black hole curve can be trusted for very large masses. But if we naively extrapolate the curve, we see that it crosses the extrapolation of the free string curve. Also the free string curve cannot be trusted for very large masses. It's valid for masses where you can neglect the self-gravity of the string. But if you naively extrapolate that, you see that these two curves cross. So this has led to the idea that perhaps what happens is that we just simply need to pick up the, for a given mass, we need to pick up the configuration that has the largest entropy. And that when the black hole is small, it will turn somehow into this highly oscillating string. And this point where the extrapolated curves meet was sometimes called the correspondence point. So where the two give the same mass and the same entropy. So the question we'll try to address is whether there is a smooth transition between the black holes and these highly exciting strings. And one motivation for revisiting this question is that in the string picture, the microstates are explicit. So we can understand that the entropy comes from the various states of this oscillating string. On the other hand, in the black hole, what the microstates are is not completely obvious. I know, but on the other hand, the black hole has some interior, but in the string picture, there's nothing analogous to that. So if one understood this transition better, maybe one can understand how these microstates are represented on the black hole side. So this is just some long-term motivation. We are not going to answer this question, but this is some motivation in the back of our minds. So we've seen this curve. This is the curve we said before. We said that there are some modifications here in the black hole due to string corrections. And I'm now going to discuss some modifications from the string side. First, we need to discuss a little bit some comments about the string theory at finite temperature. So if you are at finite temperature, then a natural way to think about that is to compactify the Euclidean time direction. So you go to Euclidean time and then you consider Euclidean time to be compact with some length. And the length of the Euclidean time circle is the inverse temperature. Now in a situation like that in string theory, there is a new feature that appears, which is the fact that a string can wind around this Euclidean time circle. And so there is that particular string, sometimes called the winding mode. And the mass square of this string well has a piece that is proportional to the square of the radius. This would be the naive expression for the mass just the mass is just the tension times the length. But there is a correction due to the quantum fluctuations that is negative. So some kind of Casimir energy on the surface of the string. And that leads to this formula where whatever that energy is that depends on the number of ways in which the string can oscillate is some constant. And that constant determines this inverse higher than temperature or maximal temperature. So when this mode becomes massless that's when we are reaching this special temperature. If beta is bigger than that special value than beta H then this mode is massive and the whole partition function is well defined. If their hand beta is smaller then this mode will be tachyonic and the partition function, the fluctuations around this so then the solution becomes unstable. So thermally reasonable solutions are situations where beta is bigger than beta H. So we're always going to be considering that situation. And if we think of dimensionally reducing the higher dimensional theory on this circle then in the extended dimensions we can write down an action for this winding mode. We can think of this winding mode as a field that has some action with a particular mass given by this. Now in addition, we can have some gravitational interactions and so this mode can have some self interactions. You could also have a kite to the fourth term and so on but it turns out that the most important interaction is the one mediated by massless fields and that is actually gravity will be the most important one and it is attractive and it gives rise if we integrate the gravity out it gives rise to a term that's basically a negative term in the sense that well that's just a familiar effect that the gravitational potential is negative. And when beta is very close to beta H this mode is light and we can approximate this whole discussion in string theory by a simple discussion, much simpler discussion that involves gravity in one less dimension. So gravity plus this extra field in one less dimension. And this approximation leads to an interesting solution. So that is a solution of this action and that's a solution first found by Horowitz and Polchinsky and it was the subject also of this paper that I mentioned of the Moran Veneziano who analyzed the sort of the same configuration from the point of view of what from more Hamiltonian point of view and giving a real time representation of the solution. So this is a solution that is localized in let's say three spatial dimensions. So we have the Euclidean time circle that's relatively small but then we have the three spatial dimensions and we get a localized profile for the winding mode where it's spherically symmetric and this is roughly a profile of the winding mode and it describes a self-gravitating string in thermodynamic equilibrium. So it's a kind of string star. So it's a finite temperature and self-gravitating. Now you can solve that equation and find how the size behaves. And it turns out that as the mass increases the size of the object decreases and well then the, and actually the mass is proportional to the deviation from this higher than temperature. So this size has to be larger than one for us to be trust the gravity approximation. So this description is only true somewhat close to the higher than temperature and for situations where G squared times the mass is relatively small so that this size is large. So and so this implies that this description is valid below that correspondence point we were talking about before for masses below that the mass corresponding to that correspondence point. Now this is a classical solution and something interesting about this classical solution is that it leads to a non-zero entropy. So normally classical solutions don't have an entropy an exception to that is black hole, for example that has a classical solution with an entropy and that's due to the fact that the circle the Euclidean time circle shrinks at the horizon. But in this case there is also classical entropy that comes from a different mechanism. It comes from the fact that the mass square this binding mode depends explicitly on the temperature. And so if you take the usual derivative of the free energy or the logarithm of the free, yeah. The logarithm of the partition function or the classical action that is given by the classical action then we find that because the fields are obeying the equations of motion the only dependence on the temperature comes from the explicit dependence of the temperature of this mass term and that leads to a relatively simple expression for the entropy which is just given by integrating this winding mode profile. This winding mode profile square roughly gives us the amount of strings we have in each positioning space. And when we do this then we get that to lead in order it gives an entropy which is beta times M so proportional to the mass and with some subliminal correction. So this is similar to, and beta is close to beta H so this is similar to the entropy we get from the free string point of view. So this is consistent with the picture that this solution is describing an oscillating string after taking into account the gravitational self interactions. So briefly mentioned a couple of things about the geometry of the Euclidean black hole. So we have the Euclidean time circle that shrinks to zero at the horizon. Now when we have a black hole we can also have a condensate of this winding mode. So if you try to calculate what this profile of this winding mode is at some distance from the black hole it can be calculated in terms of a string ball shape that wraps the horizon. And this calculation is going to give you something of this form that is exponentially small especially if the temperature is very low as it needs to be in order for us to trust the black hole. So the winding mode is present and this implies that in particular that the following thing. So if you look at the circle far away there is a symmetry and the translations on the circle but in string theory this is also a symmetry associated to winding the string around the circle but that symmetry is broken by the fact that the circle shrinks and you can unwind that mode. So and also it's represented by the fact that we have a non-zero winding condensate. And we can think of this as a thermal atmosphere of strength. So this is the Euclidean picture but in Lorentzian signature we should think of the black hole as containing a thermal atmosphere of strength. So this is really the Hawking effect but for strength that are bound near the horizon. And this effect gives a classical contribution to the entropy which formally is a further one over G string squared but it's not really calculable at least to my knowledge because it's concentrating near the horizon where we cannot do this calculation honestly. So notice that the winding symmetry is spontaneously broken both on the black hole phase and also in the highly excited string phase. So in both cases there is some kind of winding condensate and this is consistent with the idea that they might be continuously connected. So the idea is that if we have this horizontal axis represents the inverse temperature beta of the system for large beta we have the black hole description and then for beta which is very close to this lowest possible value of beta we have the Horowitz-Polchinski description and the question is whether there is some interpolating solution between these two. Now both configurations have a non-zero classical entropy and so we can ask about this interpolating solution. So how we would find interpolated solution? This is a classical solution in string theory and it's described by two-dimensional conformal field theory. So two-dimensional conformal field theory describes the string wall sheet moving in this space time. Here we're in four dimensions so we are thinking that we are always talking about the conformal field theory which describes those four dimensions and there's a separate one that describes the motion of the string in the six dimensions and this extra factor we always keep it fixed the six extra dimensions could be torus or simple circles so or it could be something more complicated but we're only focusing throughout the rest of the discussion on this four-dimensional part. And so we expect that there should be one only one parameter beta that we vary and we go between these two. So that would be the naive expectation and that's who would like to understand. And the answer for the type two string theory is that such interpolation as a classical string solution cannot happen and the argument is the following. So consider and we are going to consider certain the main problem is that we will not be able to construct this solution explicitly. So we're going to simply argue that there cannot be continuously connected by finding some invariant that we can compute something that we expect that should remain invariant as we change beta. So there are quantities in these two-dimensional field theories that are invariant under the formation. And one such quantity, yeah, I should mention that these theories, these Walshet theories are supersymmetric. So because we are talking about the super string and a candidate invariant is the so-called with an index of this supersymmetric Walshet CFT. So that is defined to be the partition function of the Walshet theory on a cylinder inserting minus one to the F where F counts the fermion number. So this is a quantity that is where bosons and fermions cancel when they have non-zero energy but when you consider states with zero energy you can have a non-zero value for this. So it's roughly counting the number of bosonic minus fermionic ground states of the system. The string language, they are also counting the number of Ramon-Ramon ground states. So on the Horowitz-Polchinski side we find that this index is zero because we can consider a flat target space. It can be deformed to a flat target space. And on the Horowitz, on the black hole side we find that the index is actually two. Now this index is for the situation that the field theory is described by a so-called sigma mole that is a map from the two-dimensional Walshet to some target space. Then the index is equal to the Euler characteristic of the target space. And the Euler characteristic of the black holes to the political invariant is actually two. And since they are different there cannot be a continuous connection. So the simplest possibility might be that at some point in the middle there are some breaching of the geometries becoming strongly coupled so that the string coupling is becoming large in some breaching of the geometry. That might be what happens but maybe there is a first order transition. We don't really know exactly what would happen in this case. Now in this heterotic string theory the story is slightly different. So now we have a Walshet theory that has only a right-moving supersymmetry but no left-moving supersymmetry. In this case the index can be computed and it's actually equal to zero on both sides. So the index again is computing the number of so-called Ramon ground states of a system or really the difference between the Ramon ground states with positive and negative parity and because both configurations are parity invariant we get that it is zero. Another known invariance are also the same and in fact there is a so-called linear sigma model analysis which suggests that they are continuously connected so I will explain what this is. So in situations where we cannot explicitly construct the conformal field theory we can view the conformal field theory as the infrared limit of a simpler theory. So we start with a theory involving free fields and which is not conformed as the theory of free fields and then we add a potential. So this theory with a potential is not conformal invariant but the idea is that the infrared it will become conformal invariant. I'll try to explain that a little in just a second. So in this particular case there is some supersymmetric action involving super fields that we can write down. I will not discuss it in detail but it involves a certain function of some super fields and by the time you go to the component action you end up with a potential which is the square of that function that appears in the Lagrangian. So in other words supersymmetry implies that the potential should be the square of some function. Now for our problem what we want to do is we are in four dimensions we are going to introduce five fields so we have which are in two groups so we have a group of three and a group of two and we're going to pick up the function that has this particular form and this is chosen for the following reason because now when we set W equal to zero then for example we can solve for X squared and we can think of so X so this formula is saying that for each value of Y we have some value for X squared so we can think of this as a circle so Y were X were two dimensions and when we set the square to be something we have a circle and this you can think of as the point in R three for example. So depending on the sign of some combination of these coefficients so for example if B minus C over A is bigger than zero then we can set Y equal to zero here that's the lowest value that the radius can have and that value will be positive so in this case the radius of the circle never shrinks in this three-dimensional space and we have a solution which has the topology of the forward spot changing type. On the other hand if the sign is opposite then there is a minimum value of Y where at which the circle shrinks to zero size and the solution has the space where W is equal to zero has the black hole topology. So this is just one equation over five variables so the space of solutions of W equal to zero is a four-dimensional space with these two topologies depending on the parameters that we choose. So the idea is that as we vary the parameters the classical vacuum manifold so that what we get by setting W equal to zero is similar either to the Horowitz Pochinsky or to the black hole topology. So the idea is that it's the following so we start with a theory with three fields plus a potential and then we look at the classical vacuum manifold given by W equal to zero and this is some curved space and that will not by itself define a conformal field theory so in fact there will be a flow to a conformal field theory that will adjust the metric on this space to be rich and flat for example or to solve the appropriate equations. We did not analyze this step in detail but after that step we're supposed to flow to a true conformal field theory in the infrared and that's the solution we are looking for. So this is just a mechanism for thinking about the solution and for studying properties of a solution which we cannot explicitly find. Now in the UV theory we had three parameters this A, B and C in the exact infrared theory in this conformal field theory we expected just one marginal parameter which is beta but we also expect one relevant deformation of this CFT because that in principle throughout this process should be fine tuned and the reason is that both the black hole and the HP solutions have one negative mode when we think about off-shell deformations. And so we expect that the third parameter should be irrelevant so no matter what its value is we should flow to the CFT so that we have to adjust only one parameter. Now the interesting aspect here is that nothing special happens when this is equal to zero in the original here in the original UV theory. So there is no new branch happening there so we expect that the flow to the infrared theory does not need to show any surprise. So that is at this intermediate stage the two theories are continuously connected. Now when they flow all the way to the infrared it might be the case that they're continuously connected but what we are presenting is not a proof it's just an indication. It could happen that after flowing for a long time that it become again disconnected. Now this business of the linear sigma model can also be applied for the type two theory. And well, in view of time maybe I'll skip this discussion and only mention that this fact that the two solutions are different in the type two theory is also related to the fact that d-brains are different on the Horowitz-Polchinski and this black hole side. So on this HP side we can have a d-zero brain so a particle, d-particle but on the black hole side the same charge is carried by some electric field in the black hole and there is no explicit particle. Okay, maybe I'll, well, I'll just briefly mention that everything that we said was for neutral black holes but there's a special kind of charged black holes that for which this discussion also rises, also. So the conclusions are that we discussed the possible connection between the black hole and the self-gravitating string of Horowitz-Polchinski that was also further studied by Damor and Veneziano. And for the type two case we showed that the two could not be continuously connected as classical solutions but for the heterotic case that could be connected. And there are many questions for the future whether we can say a little more about the CFD at intermediate values of beta. Notice that this description is just really the description of the analog of the Schwarzschild solution in string theory. So the Schwarzschild solution was found one year later after GR was invented, was discovered but in string theory, well, the rules of string theory were already set up 50 years ago but we don't have the analog of the simple spherically symmetric non-trivial solution and that's what we've been talking about today. And we can track the picture of the microstates through this intermediate region. So this should have been a question. So the question is whether we can do that. And I think we haven't explained why from a more conceptual way why the heterotic and type two pictures are different. So this is still a puzzle. But well, I'd like to finish wishing happy birthday to T-Ball again. And the questions from the audience. Short questions. Or is there from the net question? No question. Maybe T-Ball has a question. Just a short comment. All the solutions you are talking about here and the transition is a Euclidean solution, not- Yes, yes. Yeah, your time. Yes, yes, yes. We consider the Euclidean case because it's simpler and in particularly in four dimensions the situation is particularly simple. But yeah, so of course understanding the problem in Lorentzian signature is also... It's not just a V-quotation. You want us to, if you add this solution explicitly. Yeah, so in principle, so that Horowitz-Walschinki solution we discussed has this winding mode and the meaning of the winding mode wave function is not clear in Lorentzian signature. And however, if you calculate the stress tensor of this solution, you can continue that stress tensor. And as you showed in your paper with Veneziano, you can interpret that as a gas of strains. So you could directly think about the solution in Lorentzian signature, but you wouldn't have this winding mode profile. And I think it's an interesting question. I probably should have put it in the questions of exactly what the interpretation of this winding mode profile is in Lorentzian signature. Thank you. Sure. Okay, I'll ask one question more. It's of the self-gravitating strings you wrote and there's this, you use some kind of homogeneous operator, beta, del del beta. So it's kind of automatic or it's kind of, or you put it by hand. The action in the self-gravitating strings. Yeah, so there was an action that involved the winding mode. So that action is derived with the assumption that this winding mode has low energy in the three-dimensional theory. So we started with a four-dimensional theory with Euclidean circle. When the circle has a special value close to this algorithm inverse temperature, then there is this winding mode, which is very light. And so then it's included as an extra light field. And so all we wrote was just the quadratic action with the mass term. Is that the question you were asking? Yeah, it's so good this time. Okay, good. Thank you. It's fine, it's fine, it's fine. It's agreed. Thank you. Thanks again. Thank you very much and happy writing.