 Please find your place. Before the lecture by today, we have an announcement, and for that I would like to have the slide. As you may know, precisely today is Salon's birthday. The family of Salam, several years ago, in 2014, they decided to give an award, which is called the Spirit of Salam Award. It's a very nice idea, because it recognizes what the Spirit of Salam is. People who have been dedicated, dedicating their part of their life and their careers to support scientists in difficult conditions. I see the example of that, but we have people from different institutions, scientists and non-scientists who have been dedicating part of their life and their careers to support scientists in difficult conditions, particularly from developing countries. It's a tradition now for us to announce this prize, precisely on Salam's birthday. Today is Salam's birthday. This year, I would like to announce the recipients of the Spirit of Salam Award for 2019. Here we are. You may recognize some faces. The first one is Professor Jakob Pales, and I will read. Stanis Jakob is a good friend of ICDP. You may see his photographs over there. He is the third one from the back. He was the chairman of our scientific council. He was the president of Tuas, the Academy of Sciences from developing countries. He has been a key person creating the institute of mathematics in Rio de Janeiro, Brazil. The statement that the Salam family wrote to give him the award is to Jakob Pales for his extraordinary contribution to the cause of science worldwide. As a renowned mathematician, a mentor or young researchers, a leader in key international organizations and in the fatigable promoter of scientific advancement, especially in the developing world. We understand that Jakob may be connected following this presentation online, so we would like to congratulate Jakob for this award, and please join me to give a round of applause for him. Congratulations, Jakob. Thank you for looking at us. The second awardee, as you can say, is the Marikuri Library. This is an original case because the first time is given to part of the institution rather than a person. I think it's very good because the library is a key part of ICDP, and we all remember when we came here the first time that I was going to libraries like OACS of Peace, and looking for wisdom and papers and so on. I think it's a very good way to recognize the work of the library. All the librarians have been very much supporting all the scientists that come to ICDP for many years. I'm trying to find the statement. The statement of the prize for the ICDP Marikuri Library is for its important role in supporting ICDP's mission of building sustainable science by providing the services and space conducive to the acquisition of knowledge. Please join me to congratulate the people from the library. I hope that someone from the library is looking at us, and following us, so were Nicoleta and her team, for we were too shy to come. But anyway, so we congratulate them. And the third awardee is here. So, Sandro Rochela. Sandro is a good friend of ICDP for many years. We have been working here for so many years. Salama asked him to come and join ICDP more than 25 years ago, 30 years. And he has been a key person to develop our activities in applied science. Now it is ICDP for Development Group, the General School of Ironomy Unit. And he has been not only creating this group and building it up, but also participating in many, many, many activities to support scientists and science in developing countries at a very practical level. I have been with Sandro several times in many activities. In particular, I remember going to Nigeria together. We had a meeting with the Parliament of Nigeria and at the end of the meeting, we came out with the vice president of the Parliament. If I remember, I came out with the vice president. There were a lot of photographers and nobody looked at me. Then Sandro came out and everybody was taking photographs with Sandro. And I know that in many places they call him Papa Africa, which I think is very well deserved. And he has been very much supportive of science from developing countries and it's a pleasure that he has been recognized. Let me try to find the official statement from Salam's families. Somehow I managed to mix up all my papers here. So you give me a second. Okay, Sandro Odechella, for his dedication in fostering research in ionospheric physics, radio science and its application in digital communications throughout the developing country. So please join me to congratulate Sandro. Maybe you don't mind standing up and we can just, everybody can look at you and see. And maybe the cameras can also say thank you. And his tradition, we have this ceremony. Oh, here is a person from the library. So he would make a magic trick for us. Very good. So at least one representative from the library, very good. So you don't have to say any words, you have to just make a magic trick for us. Okay, don't worry. Very good. So join me to congratulate the whole library. At least to say hello to everybody. In fact, I was not aware. And so as I was saying, the tradition we have is that we have the award ceremony precisely on the day that the students from the diploma graduate. And this year is the 29th of August. So please reserve that day. We have the ceremony for the speed of the salam award. And then we have the graduation ceremony for the students. So let's congratulate all the winners and well, thank you for all your support for developing countries. Okay, so this is the announcement. So now we'll let Juan to continue to his talk. We keep track of the time. It's quarter to five. So again, as I told you, this is the second salam lecture of 2019. And it's kindly supported by generous funding from the CAFAS Foundation. And the title of Juan's second talk is, I think it's toy models for black holes. And he will give the first part today and tomorrow give the second part a bit more technical. So let's welcome Juan again for the second. Thank you. Can you hear me? Okay. Today we'll be talking about ADS-2, SYK, and wormholes. So I'll explain what this means if you haven't heard it before. So, we'll be discussing some quantum mechanical systems with a finite but very large number of degrees of freedom. When I say degrees of freedom, that will be qubits. So it will be systems with a finite dimensional Hilbert space. And these systems will have a 1 over n expansion. So we can compute things to leading order in n, sub-leading order in n, and so on. And there will also be strongly coupled. So these degrees of freedom will be strongly coupled, those spins. And they will develop an approximate scaling variant behavior at relatively low energies. Now, the systems cannot be exactly scaling variant because exact scaling variants will imply a continuum of energy states. But being a finite dimensional system, there will be a discrete set of energies. But it will be approximately continuous. And that approximately continuous distribution will be almost scaling variant. And we'll focus on some universal features. And this system will have a universal feature that is found actually in two systems. One will be this SYK model that is this system, a simple quantum mechanical system with a finite number of degrees of freedom. And the other system is a near-extremal black hole, which we also think it should be described by a quantum system with a finite number of degrees of freedom because the black hole entropy is finite. And this will become a relatively simple gravitational system. And we'll see that the gravity part of the system will be common, will be something that is common to both systems. So that's the first part of the talk will be to explain... Well, to introduce these two systems and to explain that common universal behavior. So let us first introduce the first system. The first system will be the quantum mechanical system. It's called the Sachs-Devier-Gittier model. It was first introduced by Sachs-Devier to study superconductivity. Then it was further analyzed by Gitaev, who noticed some of the similarities to black holes. And also these two authors analyze various properties of the model. So it's a system where we have N-Mairana fermions. So the Hilbert space is generated... The set of all operators in Hilbert space are the set of all operators you can build from these Mairana fermions. The Mairana fermion is roughly speaking the square root of a qubit, the square root of a quantum spin. So if you take two Mairana fermions, they generate a two-dimensional Hilbert space, the same as the Hilbert space of a qubit. And the three spin operators are... Yeah, so they're related to this fermion. So Psi1, Psi2, Psi1 times Psi2. So that's... Okay, so... Now, in this Hilbert space, we define a Hamiltonian, which is a sum of many terms. Each term contains a product of four of these Mairana fermions. And we have couplings, which are random. So we take these couplings to be random. So this Hamiltonian is, in some sense, generic. It's not just a single Hamilton. It's a family of Hamiltonians that we could have. And these random couplings are taken to have a Gaussian distribution. So each of the coupling itself is taken from a Gaussian distribution with a variance which is given by some energy scale squared. So this Hamiltonian has dimensions of energy. These variables are, well, dimensionless variables. So we have an overall energy scale that is set by these couplings, which we call J. And then we put in here some factor of N that is chosen so that the model has a nice large end limit when N gets large. So to lead in order, we can treat these couplings to be essentially like an additional field, but an additional field which is independent of time. So we'll take the computations and we'll average over all these couplings in a time-dependent fashion. Now, there are very similar models which have no random couplings, which have, where the Hamiltonian contains no random terms. It's just a simple Hamiltonian. They are called tensor models and they were introduced by Grau, who noticed some of the interesting properties of this model, and then we didn't realize that they have the same physics as the SYK model. And there are many other people like Klevon and collaborators who studied further those models. So this order part here is not essential. So if you are confused by this, ignore your confusions, set them aside. If you don't like this order, models without this order have the same properties. So we will be interested in the strong coupling region. So there is an energy scale, J, and so if you look at the model at very high energies, then it's a very simple model. It's topological essentially. There's no dynamics. And then at low energies, this interaction becomes important. This is a relevant perturbation from this trivial UV fixed point. So then you go to energies such that this interaction is large, but not too large, not too low energies. The energies will not be so low that we distinguish the individual energy eigenstates. So low energies in this context will mean inverse temperatures or times such that compared to J, they are much bigger than one, so we're in the strongly coupled region, but much less than N, so that N is really the largest parameter. Now, more precisely, what that means is the following. So imagine we take some realization of this model with some random couplings. We can take a particular N, let's say 32, and then we have a finite-dimensional Hilbert space, the dimension 2 to the 16, and you find all the energy eigenstates. You put this on a computer, you diagonalize the Hamiltonian, and you find that there is a bunch of energy eigenstates. It's statistically symmetric because there is as possible to have positive J as negative J, but this is the distribution. So there is one state here at the very bottom that is the lowest energy state, but when we say that we are in this regime of low energies that we're interested in the talk today, we don't go to energies that are so low that we pick the lowest one. We focus on a region here near the bottom of the spectrum, where there are still lots of states. There are still an exponential amount of states, an exponential in N, amount of states. So this is some distribution which has a factor for the e to the N in it. So we have even for 32, this is very large, even though N is 32, which doesn't look like a very big number, the number of actual states is 2 to the 16, which is a pretty big number. And so we're interested in this bottom of the distribution. And so for the purposes of this talk, the distribution of states here can be approximated as continuous. So that's the regime that we are interested in. Now, it turns out that the whole physics in this regime can be understood in terms of a certain effective action that describes the large N limit. Actually this effective action is valid throughout the spectrum in the large N limit, so in the limit that N is the largest number. Now, I will explain what this action is, but for now I want to focus only on one aspect of this action, which is the fact that N appears as a parameter here multiplying the whole action. So it's an action where we traded the N fields or the N operators that we had by a single function of two variables of two times. I'll explain how that arises later, but the important point is that it's a single function and N appears as basically 1 over h bar of this theory. That's the quantization parameter of this theory. And in the large N limit this action becomes classical. So this is somewhat analogous to how we expect things to work in let's say gauge gravity duality as we saw last time that in the large N limits and in this limit of large numbers of colors and so on, we expect to have a classical gravity theory. So this is the analog of the classical gravity theory for this case is of course not a gravity, well at least not obviously a gravity theory. And I'll explain how it arises in the following. So before explaining how it arises, let me say that this variable G of T1 and T2 of two times is roughly like a fermion bilinear. So morally this action is roughly similar to the action you have let's say in a superconductor where you start with a bunch of fermions and then you get an effective action in terms of a bosonic variable which is related to Cooper pairs, to pairs of fermions. And then you describe the physics in terms of that action. So if you know your superconductivity theory then that's an analogy. If you don't know it, don't worry about it. We'll see what this means. So here I outlined the derivation of this action. So how you go from the original variables to this action. And so you start with this action. So this is the Hamiltonian that we had and this is the Lagrangian expression for fermions that would quantize the fermions to give the commutation relations that we discussed before. And then we'll average over the couplings with the Gaussian distribution. So this is the Gaussian for the couplings and we'll integrate over the couplings. So this is a path integral over time. So the functions psi are functions, grassman functions of time. J is just an ordinary integral. They are not functions of time. So then we have the integral for the J is Gaussian. So there is this manifestly Gaussian term and there is the linear term. So we can complete the square. And after we complete the square and we integrate it over J we're left with basically the square of this term. But this term had an integral over time. So we get the integral over time all squared and we end up with two integrals over time. So one for each of the terms that we completed the square for. And inside that will have a contraction between fermions. So when we integrate over couplings we get the delta function of the first index, second index, etc. And that will give us a bilinear fermion, bilinear to the fourth power because there are four indices. So now we have to do this integral over fermions and we do this integral by a trick that is common in ON-like theory. So this is a theory that has now at this level it has an ON symmetry where we can rotate this N fermions into each other. And in such theories it's convenient to introduce a variable which roughly speaking is this. So you say that G is the same as, so this G is the same as the G that appeared in the previous transparency is the same as the fermion bilinear. And so we insert one which is the integral dG of delta function of G minus the fermion bilinear. So this just would give one. And we can write the delta function as the integral over sigma so this is just the Fourier transform of this delta function and it's sigma times G minus the size. So we insert this one in this partition function we multiply here by one and then once we have this delta function we can replace this term by this G to the fourth term so that this term becomes this term here. And then we have this term here, this term over here. And then we have now the action for the fermions so this whole thing was designed so that the action for the fermions is quadratic so we had this quadratic term and this term is also quadratic and then we integrate out the fermions and we get the determinant and the determinant is over the operator d time plus sigma. So that's sigma and G where things that we introduced that's our auxiliary fields here but after we integrate out the fermions we get the mass dynamical fields so that's the derivation of this action. Maybe that was too quick for if you've never seen this type of derivations before we will not need the derivation in what follows. What is important is that we have some action that you can explicitly derive so you could go through this derivation in more detail if you want but you just derive it from the original action. It's somewhat similar to the steps you take to derive the action for superconductivity from the fermi c and the small interactions there's also a quadratic term it's very similar to this story in fact this story was motivated by superconductivity by trying to understand high DC superconductivity but here we don't have spatial dimensions everything is happening in one time and each fermion is interacting with all the other fermions so it's simpler, it's a simpler model. So we got these terms now the action is non-local in time so the reason we got two times was because when we integrated over couplings the couplings were independent of time and so similar actions were obtained for ordinary ON models and because n appears as an overall constant here for large n we can approximate the path integral over sigma and g by simply the classical solution of this action so you can calculate the classical equations of motion the Euler Lagrange equations of motion for this Lagrangian by taking the functional derivatives of this Lagrangian with respect to g and sigma and you get some integral equations and they are relatively simple to write down I won't write them down but what is notable is that we can just solve them numerically we can just take the time direction and discretize it and say I don't know we have a thousand steps for time and then we then g becomes if you wish a matrix of thousand by thousand so the two time arguments you put them in these equations you're just solving the computer you get some function that's the solution that gives you the large n behavior of this model you can calculate from these the free energy you can calculate various things now so far we haven't taken the Lagrangian limit but when you take the Lagrangian limit then these solutions become simple they take a scaling form of the form tau minus tau prime to delta and this is some ansatz which you can also if you make these ansatz you can check in the equations of motion that they solve the equations at lower notches so you can check that analytically but well even if you couldn't check it analytically you could numerically check that this is the right behavior at lower notches so there's some simplicity at lower notches and some kind of scaling symmetry and you find that the exponent here is one quarter and it's relatively simple to compute this one quarter so normally when we have a quantum systems or statistical systems that develop scaling variance it's typically pretty hard to compute the scaling of exponents like for example for the icing model that describes liquid vapor phase transition people work very hard to determine this exponents here in this simple modeling one line you determine this one quarter I'm not showing you the calculation but you can believe me that it is so we found the scaling variance now usually systems that have scaling variance they usually also have conformal invariance so conformal invariance are more general transformations of coordinates which so scaling variance you're simply rescaling the coordinates but conformal invariance are more complicated angle preserving changes of coordinates and in one dimensions conformal invariance would be invariance under arbitrary reparameterizations so we can ask whether that holds for this system whether it develops a conformal invariance in this kind now it turns out that indeed this type of conformal invariance of full reparameterization invariance turns out to be a symmetry of this action if we neglect this term in the action so that this was the original kinetic term for the fermions it turns out that when you go to low energies this term is sub-leading I'm not explaining exactly why but you can believe me that there is some... we're taking a lower energy limit of some action then you look at the action and you decide which terms scale which powers and it turns out that the important part to know here is that this action has a reparameterization symmetry which you can explicitly check so you can go to the action and say well I drop this term that is not important in low energies you explicitly check that it has this symmetry now how does this symmetry act so if G let's say was a solution of this action and we are given an arbitrary function of time then the idea is that we can generate another solution which is taking the first solution that we had G of t and t prime and instead of t we put the function of t and the function of t prime and we also need to rescale it a little bit by this factor that has to do with the infrared factor of the normal dimension so if you are used to conformal field theories this is how normally primary fields correlation functions behave but if you're not it's just some rule and so if you can check that if the questions of motion are obeyed for the original one the questions of motion are also obeyed for this one so this symmetry implies that we have actually a whole family an infinite family of solutions of the equations at low energies so we can call this an immersion reparameterization symmetry that the model has at low energies and so just as an example of what this might be useful for we can take this scaling variant solution that would be like a very low temperature solution for the model and we can apply a particular a particular transformation which is just the tangent of time with some parameter beta and then we get a solution which is a reasonable solution for finite temperature so this tangent maps a finite period of time into an infinite period or infinite Euclidean time so it goes from finite temperature to zero temperature and applied to this function it gives us this one and so we get the finite temperature solution from the zero temperature solution that's one example of what this is useful for now so we had this if we start with this one we have all these other solutions and all these solutions have the same action in the infrared limit because it's a symmetry of the action but recall that we were doing a functional integral over g so we were doing a functional integral over functions of g and this is some sub-familious functions that is parameterized by f so this is not as general f is a function of one variable so it's a little less general than complete general functions of two variables but we get a big family all with the same action so if we did the path integral then we would get an infinity and this would be in contradiction with the finiteness of the Hilbert space so originally we found that the Hilbert space was finite so whatever we do we should get the finite answer so you can view this as some kind of ghost on modes and we'd give a diversions but the solution is to remember that the symmetry is only present in the very extreme infrared limit but let's say we're not in the extreme infrared limit then this symmetry will be broken slightly and then we'll have some small action for these variables this is somewhat similar to having a particle in a box in a very big box then as really if you're far from the walls of the box you can think that you have a translation symmetry but at some point the translation symmetry is broken and you have a discrete set of states right, the finite dimensional Hilbert space now then you can say well the symmetry is broken but the symmetry is broken when we go to high energies and high energies here means that when you look at the functions of two times you look at two times that are very close to each other and so all the effects of breaking the symmetry will come when the two times are close to each other and from those terms in the action and those terms in the action will be an integral over a single time so we expect that the leading term that breaks the symmetry should be a term that is an integral over a single time so some action, some Lagrangian which depends on this function that is a single integral over time it should have the n because the original action had an overall power of n and then this term should have a certain symmetry certain set of symmetries that come from the fact that there are certain functions that do not change so when we apply this if the function is a global conformal transformation then it does not change this g so it does not generate an inequivalent function so that means that the original action could not possibly depend on such on functions that differ from each other by such as two transformations and that determines that the simplest Lagrangian that you could have is this so-called Schwarzian Lagrangian so in a few words the idea is that from the symmetries of the problem the scaling symmetries of the problem you can conclude that the leading term that violates this reparameterization symmetry should be this Schwarzian action and this action governs so that's what the action should be the actual numerical coefficient requires more detailed calculation which if you solve the equation numerically before you can figure out what it was or and so on now I explained the emergence of this action in a very somehow abstract way perhaps would sound a little abstract but just to emphasize how it is derived from the symmetries of the model but you can actually directly expand the action around this infrared solution and derive the expansion of this action and match them and so on so you can do this explicitly we did this explicitly in a paper with Douglas Stanford now at these low energies there are also other fluctuations of the function G which are not of this form and those fluctuations are not suppressed that the action is not suppressed and so they have a well-defined action and they behave like let's say conformal fields in this quantum mechanics theory so in this low energy limit we have two degrees of freedom one is this function F with this action and the other is a whole set of degrees of freedom which are organized in conformal representations so that's one system so then we'll have a second system and that second system consists of near-extremal black holes so at this point this is not directly connected to the previous one we'll see how they're connected before we see how they will be connected a bit later so for now we consider an near-extremal black hole now what is a near-extremal black hole so that's a charged black hole so you have a black hole with electric charge then you find that as the mass gets lower and lower there is a minimum mass for which the hooking temperature of the black hole goes to zero and the geometry develops a scaling region near the horizon the geometry near the horizon becomes very long develops a long neck which has a geometry of two-dimensional anti-acetyl space times a two-sphere so the two-sphere around the horizon acquires a more or less constant size and then you get a deep throat that in the exactly zero-temperature limit becomes infinite and has roughly this geometry this is the geometry of two-dimensional anti-acetyl space two-dimensional space with negative curvature so it's the analog, let's say, of a sphere of hyperbolic space but with a time direction and here there is a scale symmetry so a scale symmetry corresponds to rescaling time and also rescaling this variable C this variable C is roughly the radial position along this throat so if we sit at some position on this throat we can rescale to a deeper position inside the throat and then all the processes in the two positions are related simply by rescaling of time so they will happen slower if you are in the bottom of the throat now this is a picture of Euclidean ADS-2 or otherwise known as hyperbolic space and we say that these near-extreme black holes the Euclidean version of these black holes, for example will have a piece of the geometry which will look like hyperbolic space now we won't get all of hyperbolic space because all of hyperbolic space is infinite when you go to the boundary but we will get the portion which is the portion inside this red line so the horizon would be the center and the Euclidean space would be the center and then this red line is where this two-dimensional throat joins with the four-dimensional space so that's this region here in the geometry so that's roughly what we get so we don't get the outside here we only get the piece that is inside the region inside the red line now this problem actually has there are many other configurations that also match to this four-dimensional space which are configurations where we draw the red line we have we draw other red lines with some fluctuations and locally, very near very near the line the geometry looks the same as for the perfect circle and if we are very near-extremality these fluctuations might look large in this diagram but locally the line will be very very straight and will be very very similar to the circle so we have all these configurations and this is analogous to the infinite family of configurations that we had in the SYK model so that here we see the first analogy and we can parameterize these perturbations by function that gives us the relationship between the time coordinate so there is a time coordinate here Euclidean time coordinate which is just a circle coordinate here which is anti-sitter space and then we have some other time coordinate which is the proper time along this curve and we can parameterize these curves by giving the relationship between these two times so when we are at the proper time along the curve where we are in the angle here in hyperbolic space a proper time or proper distance on this curve here everything is Euclidean so we keep and so we have this infinite family and we would therefore if we did the path integral or the Euclidean gravity path integral we would get again some infinity but we get the finite answer because we have to remember that this throat is not infinitely deep it's just some depth that is finite we are not exactly at extremality we are near extremality and we have to keep the leading effects that perturbed away from ADS2 and it turns out that that amounts to keeping so we have an effective two-dimensional action which looks like the Einstein term in two-dimensions and it's multiplied by some function which has to do with the radius of the two-dimensional sphere if we approximate the radius to be constant then this function here would be constant and we would get something which is simply the two-dimensional gravity action which is in two-dimensions this is topological so it doesn't depend on the geometric shape at all and so the only thing that this gives us is the near-extremal entropy but in order to have some non-trivial dynamics we actually need to keep the leading deviation of the S2 from the extremal value so these two can have some small fluctuations around the extremal value and so this phi is just the deviation from the extremal value and then when you do the functional integral with this action then integrating over phi sets the metric to be exactly AdS2 so the metric is completely rigid and it's AdS2 and then this term in the action just drops out because this sets r to be exactly minus 2 and so when you put that back into the action you get nothing and then there is a boundary term that you always get when you do these things which is the extrinsic curvature and so the important thing is that this extrinsic curvature is just an integral over time so it's a one-dimensional we started with a two-dimensional problem but now it got reduced to this one-dimensional problem which is the extrinsic curvature of this curve in AdS2 and it turns out that this extrinsic curvature is essentially well it's basically the same as this Schwarzian action so if you parametrize the curve in terms of the relationship between the proper time and the boundary time you find that the extrinsic curvature reduces to this action another way of saying this is that this Schwarzian action is the same as what you get by taking let's say this circular trajectory and then making small deviations away from the circle the variations in the extrinsic curvature so we introduced two systems and in both systems we could derive the existence of one mode that in this case this mode is related to the non-trivial gravitational dynamics of the system this mode will well we derived it from a gravity action from the Einstein action and it captures all the gravitational back reaction in the system in addition we could have of course particles that propagate in this AdS2 space and so on some matter fields etc what this mode describes is the gravitational aspects is what remains of gravity if you wish we have gravity in two dimensions and you know that there are no S wave gravitons there are no gravitons in two dimensions so all that remains is this boundary mode if you wish a quantum mechanical degree of freedom that lives on the boundary of space and one of the things that this degree of freedom does is it keeps track of how much energy you throw into the system so if you throw energy into the system the mass of the system changes turns out the mass is also proportional to the Schwarzian action so the mass changes and so on and it keeps track of that but it also keeps track of other things that will be important and we will show them in a second now what is the relation between these two systems so the relationship is that they are somehow in the same general class of systems so an analogy would be like saying that we talk about the 3D ising model and the second order super fluid critical point there are two different conformal field theories they are both conformal they both have a stress tensor so they have something that keeps track of energies but other operators are different so they have differences between them so it's a little maybe busy graph so we have two systems one is the SYK model and the other are near extreme black holes described by Einstein gravity they are not the same this in the low energy limit is described by nearly ADS2 gravity and low energies so and fields that propagate so this nearly ADS2 gravity consists of quantum field theory on a fixed ADS2 space and then some dynamics of this boundary of where you join ADS to the four-dimensional flat space that's where the gravitational degree of freedom lives we also had some conformal invariant part that had to do with the fluctuations of the function g in the directions orthogonal to f of the reparameterization and also we had the reparameterization mode now the idea is that these two degrees of freedom are the same and they have the same action they are both governed by this Gaussian action and the form of the action and so on comes from the symmetries of the problem comes from the fact that the problems have reparameterization symmetry that is both spontaneously and explicitly broken so it's characterized by some of the pattern of symmetry breaking of both theories and this action governs a bunch of physical properties so for example it gives us the form of the low temperature entropy it turns out that the low temperature entropy of these systems has a constant part plus a part that is linear in the temperature the fact that you have something linear in the temperature comes from this action also if you calculate if you calculate the chaos exponent so you calculate the commutator we were talking about yesterday so yesterday we talked about the square commutator that we could compute in a general gravity theory or in a general quantum mechanical model so we can calculate that commutator for late times and the late time computation comes from also from this action and then we'll discuss some other things that have to do with wormholes which also come from this action and all those things will be common to both theories in particular yesterday we discussed that the chaos exponent is maximal for gravity it's also maximal for this simple quantum mechanical model so this is an example of a quantum mechanical model that has the maximal chaos exponent okay so I discussed the two models and some of the relationship between the two models now I will discuss some aspects of the physics that are governed by this single model and I'll use in the pictures the ADS2 language to describe this physics but everything I will say has a corresponding statement in SYK, the SYK model so so the two are essentially the same so we could consider Euclidean black hole or the Lorentzian black hole and the idea is that this Lorentzian black hole again has two sides the same as what we talked about yesterday and that in the quantum mechanical theory corresponds to the thermo field double or this particular entangled state that we also talked about yesterday okay good now let's talk a bit about the dynamics so in the dynamics of this gravity theory we have some bulk particles that propagate on a rigid ADS2 space so it looks at first sight that in this type of gravity theory there is no bulk reaction of the metric that does not curve the spacetime so the curvature of spacetime is fixed it's just this ADS2 the matter just propagates there without producing extra curvature in the geometry but what it does is this matter is the trajectory of the boundary so this boundary should be viewed as a dynamical particle whose dynamics is fixed by that Schwarzian action we discussed and the presence of matter can change the dynamics of the boundary so how does it change the dynamics well it changes in the following way so imagine that the boundary we emit some particle or we had a particle that bounces from the boundary then it gives a little kick to the boundary trajectory and so if the boundary trajectory was going to go along the dotted line then instead it will go along this violet line so that's all it does that's all the matter does it only when it hits the boundary it kicks it a bit it kicks it outwards so now let's first focus on the cases that we had no matter so the red line would follow along this dotted line and it would hit the it would reach this point in the global ADS2-Penros diagram so this whole strip here is representing ADS2 space and ADS2 space has a Penros diagram that corresponds to that of a strip with an infinite here time direction I'm not sure if I wrote the metric anywhere I think I didn't maybe I'll write it here you can think of this as a strip where sigma goes from 0 to pi and then you have the square of a sign of sigma so tau is a time direction that goes from plus to minus infinity sigma goes from 0 to pi and this is some function that rescales the whole metric so the trajectory of light rays will be the same as the trajectory of light rays in the original strip metric and this factor here is multiplying let's say the time direction so it implies that when you get to the boundary of the strips there is a big gravitational potential so if we drew a relativistic version of the gravitational potential it will be a big well and it will have a sine squared so any particle inside here by the force of gravity will be constrained to be here near the center okay so that's the metric of ADS2 and we anyway so then we have this boundary trajectory it hits the ADS2 boundary here so that means that if you are an observer that lives on this boundary or if you are a four dimensional observer beyond this boundary so here you can have a four dimensional space that connects to this two dimensional space times NS2 then you will not be able to receive any signal that comes from this region so if you have some particle that sends you a signal from this region you will not be able to reach you here that you are living on this boundary or you are closely connected to this boundary if you are to the right here of the diagram this orange line would be the horizon of that black hole okay it divides the region that can send signals to the region the region that can send signals which is this one from the other region imagine you throw an extra particle to the black hole okay so we expect that if you throw an extra particle to the black hole the black hole would grow right here the growth of the black hole is related to this kick that we talked about before so now this boundary particle will follow the blue line and the the violet line and will hit this boundary of ADS earlier and so the horizon will have moved to the position now of this other of this other dotted line okay and so there is the region between the two dotted lines before we send the particle could send the signal but after we send the particle cannot send the signal so this region now ended up in the horizon of the final black hole inside the horizon of the final black hole so that's the growth that's the growth of the horizon the growth of the black hole now an important property of this the difference between the two trajectories is that these are two trajectories in ADS2 space which more or less like hyperbolic space and we know that in hyperbolic space two trajectories when you produce a small perturbation the two trajectories will the distance between the two trajectories will grow exponentially in time and the same will happen here even though in this diagram they seem to be very close to each other because there is this very large factor this factor 1 over sin2 is becoming very large the actual distance between the two trajectories becomes very big and it grows exponentially with the boundary time and the growth of this of this distance is just given by the geometry of ADS2 and that fixes what the exponent is here and the exponent is just simply related to the temperature of the black hole and that's why essentially you get the maximal exponent you get the maximal exponent because this whole calculation is determined by the symmetries of this two-dimensional space and the symmetries determine both the temperature of the black hole as well as the the as well as the shift in the these trajectories now the fact that the trajectory moved is not very easy to determine if you do local measurements because if you are on this trajectory or you are on the other trajectory you end up on one of the two trajectories but you cannot see where the trajectory would have been if you had followed the other trajectory if you had not thrown in the matter for example so in order to see the motion between these trajectories you need to do some more sophisticated experiments like for example these commentators that we discussed before which roughly amount to saying that you do some perturbation and then you ask what's the value of some other observable in the state where you did the perturbation versus the state where you didn't do the perturbation and that way you can determine that but an important point is that the chaos the chaos that was quantum chaos in the SYK model so we had the anti-commutator let's say of two fermion operators that is growing so yesterday we talked a lot about quantum chaos so we can do the quantum chaos computation in the SYK model by taking two operators let's say two different fermions and one at one time and the other one at a different time and then these commentators would grow in time with this exponent but in the gravity picture this is just a simple geometric motion of a particle in ADS-2 and even in the SYK calculation the growth of chaos has to do with the trajectory the behavior of solutions of the so-called interaction that we discussed so this particular collective mode of the problem so what was quantum chaos becomes just classical chaos of essentially a particle in hyperbolic space a particle in ADS-2 becomes the simplest type of chaos that you could have so this is the same thing that I said before and this motion in hyperbolic came from the structure of quantum to group and I should mention that there are both proposals for realizing SYK model in the lab here I just quoted one paper there are other proposals and there are also proposals for measuring the chaos exponent to measure the chaos exponent is a little tricky because you need to compute this commentator and if you expand this commentator this commentator has various terms that looks like you do you calculate an operator time 0 then an operator time t then one at time 0 and one at time t and in order to do this you need to evolve the system backwards and forwards in time and so in order to compute this quantity even if you want to measure it experimentally you need to evolve the system back in time now this looks impossible that we cannot evolve a system back in time however if you realize that if you have a really simple system like the Hamiltonian and the signs of the terms in the Hamiltonian are under your control you could imagine flipping the signs for some time and so you change all the signs of all terms in the Hamiltonian and then that's the same as evolving backwards in time so the proposals for measuring the chaos exponent involve doing such things and should mention that some of the people in these papers are experimentalists who actually do experiments so this is not very theoretical people okay so now I'll how much 10 minutes okay so now I'll discuss something similar to what I discussed yesterday of teleportation but an entanglement but I'll discuss it in this context of ADS2 so anything that you do with one boundary so if you send particles to one boundary or they bounce from the boundary and so on they always move the particle, the boundary outwards and when they move the boundary outwards you can see less of the interior than you could see before then you could see with the empty black hole so anything that you do to the black hole will make the black hole grow you can never somehow make it shrink and see more of the black hole except if you do things that involve both boundaries so if you put for example some interaction between the two boundaries and that would be some interaction between the two copies of the quantum mechanical system so we had this picture corresponds to an entangled state in two separate quantum mechanical systems and you can put an interaction that couples the two quantum mechanical systems and gives let's say an attractive force between the two boundaries then you can make the boundaries bend inwards so you give a kick to the boundary inwards and then it would allow you to see more of the black hole, in particular it would allow you to see part of what you would what before was the black hole interior so the black hole interior was the region inside this orange lines this region here but after you do this perturbation then you can see what was happening inside here so if you had a particle here you could not see it but now you could see it so this involves coupling the two systems and by coupling the two systems you can indeed, if you send the signal here, the signal can go in and appear on the other side and this is closely connected to teleportation so this coupling does not need to be a quantum mechanical coupling, it can even be a measurement so you can do a measurement of the Hawking radiation like a fire, this is the hotness of Hawking radiation so you take a picture send it to the person here and the person here can do some transformation that again gives a kick inwards to this trajectory and will allow you to extract the signal you send here the signal here is a little cut but anyway I discussed this already yesterday and I just explained how it works here and I would like now to explain how you can use these ideas to analyze some black hole paradox some paradoxes that arise when you think about black holes and I'll discuss only one of the one paradox which is called the Hayden Prescott Paradox and this paradox is roughly the following so this picture is from their paper so they said let's imagine we have an old black hole and that you've been observing this black hole for a long time so then they said that if you have this black hole perfectly entangled with your quantum computer and then some general quantum information theory ideas tell you that if you send some information into the black hole let's say Alice sends her diary then by collecting a relatively small amount of Hawking radiation then Bob who has his quantum computer a very powerful quantum computer can decode the contents of the diary and this they derive by assuming maximal entanglement between the black hole and the quantum computer and some general properties of quantum information theory and assuming that the black hole dynamics is chaotic it's maximally chaotic let's say then relatively quickly you can decode that but relatively quickly I mean you don't have to wait until the whole black hole evaporates you have to wait only a few so-called scrambling times which is a time which is logarithm so it's the time the light takes to cross the black hole times the logarithm of the black hole entropy since it's just a logarithm it's not much much bigger the black hole could be big but the logarithm could be not so big and now imagine that Alice sent the diary and arranged for a machine to send the message the message that originally the diary contained send the message here to the right side so then Bob after decoding the that information he could jump into the black hole and then he would see two copies of the information it would be the message that Alice sent and the information that he decoded then this would be a violation of the no-cloning quantum if this information was quantum information that would be a violation of the no-cloning theorem so that's the paradox now there are different ways out of the paradox you could say oh well maybe Bob dies and okay fine he can't tell anyone that he saw the violation you could say maybe this message had to be sent with ultra-transplankian energy and then okay maybe it's outside the regime where we can trust things there are ways out that people had proposed but actually there is an interesting way out when we think about this problem in the following way so the setup of the problem says that you have this old black hole that is maximally entangled with Bob's computer now let's say we take this Bob's computers and Bob has a very powerful computer so he can produce a second black hole that is maximally entangled with the first one and it's in this thermofill double state that we discussed so this is a very hard thing to do but you can do it and then now Bob in his quantum computer has a copy of the black hole maximally entangled with the first in the thermofill double state and let's just further assume that there were 80s two black holes then this would be the picture that we would have for those black holes we would have the original black hole is on this side then the black hole that is in Bob's computer is on this side so this is the exterior region of Bob's black hole this is the exterior region of Alice's black hole now Alice sends her message this is the message that Alice sends and now Bob will decode the message so first how is it possible that Bob can decode the message using a relatively simple method what can happen is that Bob can collect a little bit of Alice's Hawking radiation of the Alice's black hole and then he can feed it into his black hole and produce this attractive force that we were talking about before and then now the trajectory of the boundary here of his black hole will be such that it moves a little bit inwards and can now see the message coming from Alice before that was not possible the message comes in and it's behind the horizon but it's just behind the horizon so a little tiny kick can let him see the message and now Bob actually sees the message so this is how he can decode the message so that process which Hayden and Presky said were from some abstract quantum information theory ideas we know that we can do it so there is an explicit way to do it if we have this particular entangled state so Bob got the message but now the message is only on one side so if Alice were to if we were to evolve the black hole here backwards after having extracted this and without putting it back then the evolution would be this dotted line and it would be behind the past horizon for the black hole so the message has really switched sides the message switched sides from Alice's black hole to Bob's black hole and it's only on one of the two sides it was never in this diagram a duplication of the message ok now since I kind of ran out of time maybe I won't discuss this more complicated setup so in summary the process of extracting the message puts it out of reach from Alice and the message is not duplicated in the bulk and we don't have to invoke any transplankian issue we don't have to involve Bob's death or anything so we have to use our standard from standard loose of gravity in the wormhole geometry it is assuming this connection between the Einstein-Rosenbridge and entangled states so we are assuming that and something we have not given is what is the geometric picture of the process that that leads to the formation of that geometry when you entangle the two systems in this particular way so if you have some generic entanglement the particular one of the thermo-fill double what is happening really in the geometry and that we don't know but once you have that geometry then there is a simple picture I think a general lesson about this is that when we do the complicated quantum operation that extracts a message out of the black hole when we think about the black hole information problem and we are thinking about doing a very complicated quantum information calculation on a quantum computer we cannot ignore the space-time that is generated by the quantum computer so one of the lessons of gauge gravity duality is that strong interacting systems with complex interactions can generate a space-time so if you are thinking about the black hole information problem and you are doing some quantum computation to try to check whether information is preserved or not then you will be doing a quantum operation a very complicated computer and that computer itself will generate a space-time and that space-time can be connected to your black hole of understudy through the interior and so the things that you do on your quantum computer can send messages behind the original black hole horizon in some sense it's a bit like the Maxwell demon so if you are thinking about thermodynamics paradoxes for example the Maxwell demon is a paradox because it suggests that you could violate the second law of thermodynamics the Maxwell demon is this demon that looks at molecules in a gas and lets the cold molecules go in one half of the room and the hot molecules in the other half but once you include the fact that the demon itself is subject to the loss of thermodynamics and in doing this operation you need to do some computation and the computation takes some memory and you generate some junk and so on then you will preserve the loss of thermodynamics right so here is in thinking about this Hayden-Preskel problem we need to remember that the quantum computer generates its own space-time and that could be connected non-trivially to the interior of the space-time that we were studying and that's important for thinking about the paradoxes I'm not saying that this solves all the paradoxes it just solves this particular paradox or this particular form of the paradox and we still have yet to understand how all the paradoxes are resolved and there are various ideas for how to do that yeah so this is more or less what I said so far okay so now I think I've run out of time and I'll just finish so the conclusion is that the SYK model is a nice solvable model it has many features in common with near-extreme black holes and that's why it's interesting for us working in quantum gravity the SYK model is also interesting for people studying superconductivity and so on because it's a model with strong interactions it's interesting for people who are interested in quantum chaos for example because it's a model that is maximally chaotic so it's interesting to various people but for us it's interesting due to this connection to near-extreme black holes in both cases we have low energy action low energy dynamics which is almost conformal invariant it's maximally chaotic and the chaos is described by simple classical variable sometimes called the scramblon but it's this motion that is determined by the SL2 symmetries which is a model in ADS2 or hyperbolic space we saw how we can apply this to wormholes and entangled states and so on and it's related to traversability of these wormholes and teleportation once we put some interactions between the wormholes so in particular you could make an SYK model in the lab and then send signals from one model to the other and the signals would travel in the way we discussed here I discussed everything here with a picture describing how that happens I think I have a formula some formulas you don't need to just believe that there are some formulas and with these formulas you can tell an experimental is what he should see if he or she simulates this SL2 model builds a system that is described by this model in the lab ok thank you one more questions here yes you said quantum computer generates space time now what kind of quantum computer is it turning machine or something else or beyond turning machine because as a computer scientist they will not understand what exactly it means to generate space time by a machine which is a chain of states so the idea is that a quantum computer will have some memory the memory will consist of a bunch of qubits of many qubits n qubits it could be a thuring tape or a quantum thuring tape these qubits will be entangled with each other in some entanglement pattern the idea is that the space time is connected to the entanglement pattern of these qubits and that's how the quantum computer well we don't understand how it generates it in general so the computer is some system that evolves with some Hamiltonian but here the important part is that it generates a highly entangled state as a process of the computation and that highly entangled state is the thing that could generate in certain cases a space time so in general we don't know what the space time is so we only know it for very special entangled states so if you had a quantum computer that on its memory it builds the quantum wave function for example the model of of harmonic oscillators well unharmonic oscillators as I described yesterday then it would generate this 10 dimensional space time so the state of those qubits could be in principle computed by that space time geometry I think I was still curious it generates something but how do we know it generates space time well you only know it in some cases so the statement for a computer scientist it's probably not interesting if you're interested in your computation you don't care about any of this but if you're interested in space time then you care because then if you are starting to think about black hole paradoxes and things like this then you need to care about this connection start sending signals to the interior of black holes by doing operations on your quantum computer and that's that's why we care you only care if you care about space time if you don't care about space time then don't worry about this just care about your quantum bits hopefully maybe at some point in the future the fact that it generates a space time might be useful for a quantum computer scientist but for the moment it's not but for instance if people think that space time is an emergent concept can this be a way to say well this is something that is beyond the plan scale that will be a fundamental duty from which space time will be an emergent thing can you think in this way yeah that's right if we found the quantum system that describes our universe and if we could describe our universe with the quantum system then you could in principle simulate it in a quantum computer of course it wouldn't be a quantum computer that is built in our universe because it has to be but it's something important for understanding the nature of what goes on in our universe and of course you would be interested in if you're interested in space time you would be interested in understanding the particular quantum systems that generate a big space big space time governed by Einstein's equations in general when I when I made the statement that a complicated quantum state will have some geometry in principle it could be true but it could be a very quantum geometry something we don't understand and for which we don't have an alternative description we only have a connection between states and space time only for very special states very special states of very special systems I described so today I described a system that doesn't look too esoteric I mean you could build it in a quantum computer if you have a general purpose quantum computer you could build the SYK model actually some of the people in this in the experimental proposal have a quantum computer in their labs so like more and looking so if you could build an SYK model then you could discuss all these processes that have an alternative quantum computer exactly hardware you are talking about I mean it has a few qubits I have seen the one that Moore has and that they have a few qubits and they can do some computations with it but if you could you not only have to have the quantum computer you have to input this particular Hamiltonian and then do some computations but that's an example where we could understand more or less what aspects of gravity you are reproducing so it's not too far I mean it's similar to the gravity of near-extremal black holes you don't reproduce all aspects of gravity and all aspects of space-time you just reproduce only this particular modes governed by this observable it's an example you mentioned two experiments you mentioned the second one just to make sure the the Lyapunov exponent second is a proposal for measuring Lyapunov exponents that doesn't have to do with gravity or anything it's just in general but what kind of system will that be well again some systems of spins and so on that you control enough to be able to evolve the system backwards in time so it's crucial to have a system that you can change the sign of the Hamiltonian so that you evolve it forwards in time and then backwards in time the questions here I have three questions please one is that do I understand correctly that your resolution to a hard and press scale paradox is that Bob actually measures only one message rather than two yes the second question is kind of related it's related to the yesterday's Romeo or Juliet tragedy so imagine that imagine that Romeo is not doesn't have a very good education in quantum gravity so he's not sure that if they collapse their cubits into black holes there will be a geometry a wormhole connecting them so he wants to test it so he drops in a couple of rabbits and applies the same transportation protocol and then asks Alice if she recovers some baby bunnies on the other side if I understand correctly your claim is that the answer is actually yes so these rabbits are not frozen into some quantum circuit they actually have materialized through a space and is that correct and this seems to be some falsifiable statement yes yes yes I mean Romeo could jump himself and see that he has he doesn't feel anything bad yeah now one tricky thing is that if you are someone who lives on the boundary you could always say okay maybe Romeo died and then he was resurrected and with no memory of having died that also agrees if you only ask the question at the very end it also agrees with the idea that there are let's say firewalls but these firewalls are things that Romeo falling in would not feel so I feel that so that the whole picture that you get using this is consistent with the idea that when Romeo falls in or the rabbits fall in they don't feel anything special and moreover at least the result that I see at the end at the other side the result is the same at the other side and there is also one little thing which is that the Romeo will have age a little bit by falling in because it takes a little time to go through the horizon and which you can calculate and so you have this little aging going from one side to the other this feature is also present in the SWK model there is a little bit of extra evolution the third question is that can you elaborate a bit more yesterday at the end you said something about understanding the nature of singularity like the black hole singularity or the big bang singularity so in usually or sometimes in physical systems when we get singularities like in hydrodynamics we know that it's a breakdown of our description so time continues to run it's just our description that is not valid and we are looking for alternative description that allows us to go beyond while for instance in the case of the Schwarzschild singularity the time ends there too so is there do we have a paradox a more concrete I think it's a it's a problem for which I don't have an answer I don't know anyone who has an answer or an answer that is generally believed by people that makes sense I mean it's an open problem I mentioned it because it's an interesting problem and it's connected to the big bang singularity which is also what is the most interesting problem in quantum gravity but there is no more mathematical paradox that is connected to that that you have in mind well I mean it's something that we don't understand it's a process physical it's a process we wish that we don't understand and we would like to understand it we would like to be able to have a similar to what you said in hydrodynamics some description that doesn't break down right we think that the full theory should not break down should give us some description the question is how to get this other questions something when you were talking about the manipulations you were doing for the SPI K model it looks similar to the normalization somehow it's like a duality from fermions to bosons or so but can you see a strong to weak coupling relationship can you see the correlation is the same as that you do for O n models that you introduce some kind of Lagrange multiplier and then you integrate out the n fields and then you get the simple theory for the Lagrange multiplier exactly so that's interesting this is very similar it's almost identical essentially like in bosonization you have the you go from fermions to bosons and then you have a strong coupling for one and weak for the other ones or can you say something like that well here the thing that controls that gives you a weak coupling in the end is n so I don't know how to think about it in bosonization language okay n is the coupling of the bosonic variable the bosonic variables okay I just say also that the reparameterization invariance was spontaneously broken or explicitly broken both, yeah in which the fact that we get the ADS2 geometry breaks the symmetry spontaneously that's why we had these zero modes and the fact that we had this explicit Lagrange that had the Schwarzian breaks it explicitly so it's a combination of both things it's like when you have let's say you put a heavy particle in a big box so the translation symmetry is broken spontaneously where you put the particle you put it here and so on but it's also explicitly let's say we could put it on a very shallow potential then it's also broken explicitly by the potential and this very shallow potential is analogous to this Schwarzian action I don't know if I was waiting for your question Hi I work, I do my research in string theory and often I face this question that we are working on this for long time and there is still no experimental evidence what's the point of doing research in this field and I don't know how to respond to the people who doesn't work on this field so if I ask you that question how would you respond right well we know the theories of physics that we know are very highly constrained so quantum field theory the theories of interacting particles is the formalism of quantum field theory is very constrained by special relativity and quantum mechanics and that lead led to very simple very general predictions like the existence of antiparticles and various aspects of quantum field theory now the idea is to do the same but with general relativity put together general relativity and quantum mechanics and the main goal of this is to understand the big bang which is also something we don't understand and the idea is to develop a theory that can put these two things together have some theory that works even if it's a toy model from which we can then extract general lessons so the minimal thing you could say about string theory is that it's a toy model that the usual string theory rules give us a toy model where quantum mechanics and gravity can be put together and from where we can learn more general lessons maybe it's the right theory but at least we might get to the right theories studying it now the idea so that's what this particular thing called string theories but I think of string theory more like an acronym rather than a particular theory so where we say strings I would like to say that this solid theoretical research into natural geometric structures so let me try to explain so solid means that we try to look at formulas that make sense the formulas might not agree with some experiment but at least they should be self consistent so it should be self consistent gravity and quantum mechanics the formula should make sense so that's a requirement that you apply as a philosophy as a community we generally apply this requirement it's theoretical so far hopefully in the future it will be experimental we will be able to test some of these ideas and then there is this idea of natural geometric structure so of course general relativity is a theory of geometry famously and geometry plays an important role in string theory there is some string theory generalizes geometry in some way that we don't fully understand we've seen geometry and entanglement are connected in a way we cannot fully derive and cannot fully understand entanglement is also some natural if you wish geometric structure in Hilbert space or feature of Hilbert space that is natural it's natural in two senses it's natural in the theory and it's present in nature so these are things that are present in nature and we are trying to study them better study much of the activity in the field involves studying better the things we understand quantum field theory that we know experimented very far quantum field theory gravity and so on and by putting them together people have developed interesting connections to other areas of physics that you wouldn't have expected like for example quantum chaos that we discussed and so on so the fact that we try to do things that have solid formulas I think it's what enables us to perhaps make contact with other areas of physics and also learn from other areas of physics and apply them to the problem of quantum gravity what we are doing is somehow trying to realize this dream of Einstein of putting together all the interactions of nature together and have the unified theory by unified theory it's a theory that unifies quantum mechanics and gravity it's a multi-year process we don't know how long it will take it might take it took 50 years so far it might take many more years but you should remember that for example understanding the Schwarzschild solution took 50 years just understanding the classical aspects of the Schwarzschild solution took 50 years just the theory and then it took even more time to find black holes in nature that we are really convinced of the cause so I think it's an exploration that is going on and it will hopefully will continue for many years and we are moving so the formulas we are studying are not the same formulas we were studying many years ago we will learn some things and we will learn the problems are maybe perhaps more complicated than we expected before but maybe we'll hopefully see some connection I think the connection to experiment will be very important and it will lead us to understand things better can I add something so there are other theories like loop quantum gravity and other theories which also tries to address this question then I mean both of these things are theoretical and none of them as far as I understand has any experimental evidence then the question will be why string theory why not any other theory well string theory is the more developed of the approaches and it leads to some theory that really does have gravity it predicts the correct value for the black hole entropy for example and so it has past certain non-trivial consistency checks that other approaches have not now in the past there were some alternative approaches to gravity where it seemed to be incompatible with string theory like let's say 11 dimensional supergravity or tensor model and then it was found that they are connected in some way and it might be that any other consistent approach will be eventually connected I think the important thing, the philosophy is not that well okay my theory is string theory is correct we are not working we are not wedded to the particular theory we are wedded to doing this philosophy that I discussed here of finding correct formulas and that makes sense and then exploring them in detail even if they don't necessarily agree with nature as we understand it now but at least they are closely connected in the sense that they are describing a quantum gravity system perhaps not the one we have in our world so what will be your criticism in terms of string theory what is it that it is less developed than what will be the open questions for them that are not open for strings well I mentioned one of them the value of the black hole entropy so there was a calculation of black hole entropy in loop quantum gravity even for the leading order term there was some discussion you had to introduce an extra parameter to get it to agree but even then the calculation was computed for the sub-leading term the term that goes like logarithm of the area and that also disagrees with the calculation you can do in effective field theory that is ordinary gravity now maybe there were some errors and the error will be found and it is a smaller set of people working on it and perhaps then they will find some agreement and when you do the calculations in string theory you do find agreement I mean you get the sub-leading term they agree in a somewhat non-trivial way and you well at least double curve here has done some calculations of all the sub-leading terms of black hole entropy and you get even the number number of states and integer and so on so that's all of how you could make it work but he has done this calculation at least for example he has done this calculation for supersymmetric black holes in supergravity theory so you could say well that's not described as standard model okay that's true but at least within that theory it gives the right answer so that's the idea of doing at least some correct calculations in perhaps a restricted models that at least you know that they are correct in that case and try then to extract lessons for the more complicated real world Any other approaches like discrete gravity or council theory? Well yeah so the discrete gravity is an interesting question so so for example we know that if you take planar diagrams in gauge theories so those planar diagrams can be viewed as discretizing a two-dimensional surface and they are building a two-dimensional random geometry and so you can view them as describing two-dimensional gravity in some way and this is some connection which has been studied very clearly and cleanly and it works and so on and it's two-dimensional gravity theory is the theory that describes the string wall shape and that's related to what we discussed yesterday of the connection between large engaged theories and spring theory but then people said well why don't we consider diagrams, sort of similar diagrams but not with objects that have two indices but let's say objects that have three indices right so objects that have three indices could be viewed as a little tetrahedron and then we put the tetrahedron together so the Feynman diagrams of that theory, the leading order in n diagrams could build perhaps a three-dimensional space time and then people explored this idea and then they found that indeed those systems might have some critical points where there are lots and lots of diagrams that contribute and so on and that Gourav did some of this work and these are called tensor models and it really doesn't work to build a three-dimensional geometry at least that version does not work but it is something that got connected to gravity in some other way, I mean those models were actually the same physics as SYK and they were describing some kind of theory of gravity but in two dimensions so in ADS2 type gravity and they were useful for that and they recently useful results that they obtained studying these models now maybe someone in the future might find a way to get three-dimensional spaces out of this dynamic and out of this triangulation out of these models I'm not saying it's not useful but I'm just saying that it hasn't happened what has happened that is interesting is connected to gravity well it's connected to gravity in some other ways So you will say that any theory that is solid in this regard it will be a string theory given your definition Yeah, so I think what Grau was doing is a string theory It's there somehow But this is the definition It's acronym I'm happy to consider it More questions? Yes When we start calculation we always have this variable space and time Right? So when you come to this conclusion in your theory that you have got emergent property as space time So what is the connection between those variables which we started with and the one we have emergent? What I mean is if you start with a background So what you would like is to be able to start with the original variables and derive space time and do some transformation on these original variables maybe integrate them out or do some average variation of variables and derive the space time variable We don't know how to do that in general So even in the cases that I discussed yesterday with some connections to 10 dimensions we know that two are related but we don't know the explicit transformation The closest we are to an explicit transformation is what I showed you today for the SYK model So for the SYK model we start from the fermions and from the fermions we derive this action for the bilinears which morally is like gravity that out of it you can derive this Schwarzian action and this Schwarzian action we also saw it in gravity So that's an example where you can get closest to derivation of something that looks a bit like gravity Ideally we would like to do much better, we would like to be able to take a very strongly coupled N-equal to four supersymmetric QCD and derive the 10 dimensional space time that's supposed to be dual to it as we are asking general questions what is your general opinion about the landscape and the swampland Well I think it's a very important question whether quantum gravity is consistent with the question is whether whether quantum gravity implies any constraint on low energy physics Right, so are the parameters of the standard model or the parameters of low energy physics we can measure in any way constrained by quantum gravity or is it that there are so many solutions of quantum gravity that anything that you could imagine at low energy that is consistent could also be described by quantum gravity and I think it's an interesting question and there are some cases where there are some evidence that some things are not allowed but I think what happens is that the closest you get to something that is experimentally there are some things we know theoretically pretty solidly and there are some things that are very relevant experimentally and we know less solidly theoretically so it's hard to make a concrete statement for me the statement that in this general area sounds most interesting is the question of whether the inflaton can have large field range so this is a question so naively from this criteria of connecting to quantum gravity it looks like it would be difficult to have an inflaton that has a very large field range very large field range means practically that whether we will see whether we will serve or not the gravity waves produced by inflation but for this you would need to really put a constraint, a numerical constraint on the field range from theory if you could say from theory that the field range cannot be bigger than M-plank for example then we would be able to tell people who are doing C and B that you will not be able to see gravity waves so it would be a falsifiable prediction because if they actually see them unfortunately we don't understand the theory well enough we don't understand whether we can put this with the right constraints I mean there are pretty solid arguments that the field range cannot be very very large I don't know, 10 to the 5 M-plank but there is no very concrete statement that it cannot be just one M-plank or just precise number that you can so yeah I think it's an interesting way to try to connect ideas of quantum gravity to phenomenology what about the existence of the sitter or not the sitter well this is an example where the theoretical evidence actually contradicts the claim so there was a claim that there are no solutions of classical metastable solutions of the sitter space on the other hand there were some old scenarios for how to construct solutions in the context of string theory for the sitter space so I believe the old constructions and actually there are new the new thing is just a conjecture that is contradicted by some pretty reasonable constructions so I just simply think it's wrong just think it's wrong very good, ok so I'm pleased with your answer so so I think the best way to finish so let's thank Juan again for the question I would like to ask some people who were supposed to join us for a photograph to come down now while the students get ready for the only the people who were asked to come so and the students were still gathering here just for and everybody are welcome to join us for the drinks