between certain gauge theories and certain gravity backgrounds. I wasn't very specific in defining the gauge theory. So it's a 3 plus 1 dimensional gauge theory with maximal supersymmetry. Those who, if you want, I can explain it a little more. Or if you are satisfied with this,

we can continue with this. So it contains the Yang-Mills action we discussed, plus some other fields. And then we discussed this string theory in five dimensional anti-sitter space, that was the space I erased. That's the near horizon geometry of this tube

that we discussed. So of this kind of black brain. So if you go very close to this black hole horizon, that I described before, the geometry very close to it is the geometry of five dimensional anti-sitter. That's where this came from. And we had said that all excitations living very close to here would be described by the Yang-Mills

theory. I should mention that we discussed the boundary of anti-sitter space. So this boundary where the gravitational potential goes to infinity corresponds to moving towards the boundary,

corresponds to moving away from the near horizon region. And so if you are an object that has finite energy here in this near horizon region, and you are very deep inside, it will be very difficult for you to climb back out of this potential well. It's like you're living down here, and you don't have enough energy to climb out.

And so in a situation where this connects to flat space, this barrier is not infinite. But it's kind of saturated at some point, and then the space continues. And if you're living deep inside in the bottom, this barrier looks infinitely big, and it looks like you are in infinite space as opposed to the one which has the barrier. I have a question, please.

But you call it a black hole, but not a black hole in a term that is infinitely deep then, I guess. No, no, no, no. OK, yeah. So let me connect it to the ordinary notion of black holes so that it becomes a little clearer than what I'm talking about. So let's consider a Schwarzschild black hole.

So that's the black hole everyone likes or knows. Well, maybe. Not everyone. So it has this metric.

So this has some metric. So far away becomes the metric of flat space. This is the sphere, the radial direction. All of this far away becomes R3. And here, we have the redshift factor, which is what we were talking about, this factor that becomes 0 at the black hole horizon.

For a black hole, the only factor that becomes 0 is this one that becomes 0 very close to the black hole horizon, only one direction. Now, one could consider in four dimensions also charged black holes. And they have a metric which is very similar to this,

with the minor difference that, let's say, disappears squared. Well, there are black holes which are charged. And black holes which are charged have a metric which, roughly speaking, has the same form. Not exactly the same, but they have the same form in the sense that they have a single

0 at r equal to 0. So if we expand this metric around r equal to 0 for small delta around r equal to 0, so this is the geometry of the near horizon region. And by redefining delta, this looks essentially like flat space. So those are black holes, ordinary black holes.

Then you can consider charged black holes. And when you have charged black holes, you have the constraint that in order to have no singularities, the mass should be bigger or equal than the charge in some units. And when you take the extremal limit,

the geometry really develops a double 0 here so that the near horizon geometry has the form, let's say, delta squared d tau squared plus d delta squared over delta squared. Now, if we define, and then the sphere acquires a constant radius of d omega 2.

And if we define delta equal to 1 over c, this becomes minus d tau squared plus d c squared over c squared. The metric of a space, which we can call AdS2. So this is the two-dimensional analog of what we were doing before.

So the sense in which this black object is different is that it's extremal. It's close to extremal. It has this physically, what it would mean is that the Hawking temperature would be 0. So this guy has non-zero Hawking temperature, and this would have zero Hawking temperature. Now, another sense in which it is different

is that the black hole is localized in space. These objects are extended in space along the spatial directions, around, for example, three spatial dimensions. So in that sense, it's different. So it's a black brain rather than a black hole. But other than that, it's exactly the same as an ordinary black hole.

And in fact, this type of AdS spaces appear first in the context of these extremal black holes. They have, yeah. Any other question? No.

OK. OK, very good. Now, one more general thing I should say to connect to what I discussed in the very beginning is that we discussed these strings that

exist in large-n gauge theories, where the strings are just made out of sort of correlations between the colors of the quarks, or the colors of the gluons and anti-gluons. So we had the gluon and the anti-gluon,

and we had this chain. And there were these strings that were, in some sense, some kind of strings. And they exist in any theory, in any large-n theory, in the limit n goes to infinity for fixed lambda. So if lambda is fixed, lambda is g square n. So when we hold the coupling, then make n go to infinity

and make g very small so that lambda is fixed. In this regime, there will always be strings. And they will be weakly coupled. The string interaction constant will be a further 1 over n. And they are strings in the sense that the perturbation expands, so you

can expand their free energy or the correlation functions and so on. In terms of diagrams, you can run the plane and the doors and so on. And the idea is that those strings are the same as the fundamental strings of string theory, the same as essentially those 10-dimensional strings, but living on a curved background.

So the strings of QCD are not living in four dimensions, but they're living in five dimensions. That is one of, well, that's an important point here. They are living in higher dimensions in the same way that all the, in the same way that other excitations here live in higher dimensions.

They are made out of, so objects which are made out of, they are composite here. They are made out of many gluons and so on. And they really are living in this higher dimensional space.

Of course, the description in terms of the higher dimensional space is not super useful if the radius of curvature is comparable to the string scale, because we cannot solve it very easily. And it's more useful when the radius of curvature is large, which is the strongly coupled regime. So when you are at very strong coupling,

so g square n much bigger than 1. So then this IDCC. You can use gravity, forget about string theory, forget where it came from, and so on. So if you are only interested in Yang-Mills theory at very strong coupling, you can forget about string theory. You can just take this and do gravity calculations.

And never, ever think about string theory unless you run into some problems, and then you need to remember whether those string corrections are becoming important or not. So now I was planning to describe in a little more detail how to understand the mapping of states

that you have on this side to the states you have here on this side. I was planning to make a little more precise this notion that, for example, a graviton is a pair of gluons. So in what sense is a graviton a pair of gluons? So I was trying to make a little more precise this.

And we can make this more precise by explaining this in a slightly more abstract way, which is really the best way to understand it when you are dealing with, let's say, strongly coupled systems and so on.

And this is the following. So the first part of what I'm going to explain here is, I explained right now, is purely kinematics. So just how to describe the states. That's what I mean by kinematics. We need to find a convenient set of coordinates

and a convenient description for this so that we can think about it in the most clear way. And first, we need to think about the states of Young-Mills theory. So what's a convenient way of thinking about it?

So as I said, we create some excitation, and the excitation becomes very big. So it seems like we can't keep track very much of the excitations. They go to infinity. The energy spectrum would be continuous. So it's not, I mean, we could, in principle, describe it from that point of view, but it's a little less clear. When you have a system with massless degrees of freedom,

a useful thing to do sometimes is to put the theory on a finite volume. We put the system on a box, and then we get the discrete spectrum and so on. So in this case, the box is a 3-sphere. So we can consider the Young-Mills theory, or the theory on a 3-sphere, times R.

And in this case, we could consider that theory on this cylinder now. We could also view it as a cylinder. So this circle here is S3, and this line is just time. And now we are in finite volume.

The spectrum of the theory will be discrete. So for example, if you had a free massless field, and you put it on this S3, well, there will be a series of harmonics on the 3-sphere, and those will be a series of discrete energies, some energies. Now, this picture is related to the picture on the plane,

but it's useful to keep in mind this relationship, because it's useful to use it to label the states that propagate on this cylinder. So it will give us a way to construct states here,

in a way that is very independent of details. So the relationship is the following. So you can imagine that you have R4, Euclidean R4. So here, we could consider this to be Euclidean time or Lorentzian time. Let's just consider it to be Euclidean time.

And then in R4, we could view the evolution not as occurring in ordinary Euclidean time, which would run in this way, but in Euclidean time, which runs along the radial direction. And then essentially, we get the map to this picture,

because if we write the metric as dr squared plus r squared d omega 3, that's the metric of R4. And we divide everything by r squared. We divided everything by r squared, or we pull out the factor of r squared. So this metric inside here is the metric

of a 3-sphere times a line, which is log of r. That's tau here. Well, I didn't write what tau was. But this line would be log r. So these two metrics differ by an overall factor, scale factor. But if you have a theory that is scale-invariant, and in particular, also conformal invariant,

I guess I didn't explain what conformal invariance is. Do people need to explain it? No? They don't care. No? I hear more no's than yes. So in conformal theories, the trace of the stress tensor

is 0. So if we change the overall factor of the metric, if we change the metric by an overall factor, the theory doesn't change. So the observables and things we are computing don't change. And so anything we compute in R4 is the same as doing the same computation on S3 times R.

So in particular, there is a mapping between all possible operators we can have, we can insert here at the origin. So operators at the origin get mapped to states here.

So all different states that can run on S3, that you can have on S3. So the theorem S3 has a whole Hilbert space of states. For each state in the Hilbert space, there is an operator we can add here at the origin. Now, this looks like a lot of states and so on.

So let me just explain this in the context of a simple scalar field. Let's say you have a massless scalar field in R4. So phi. And you can insert. You can first don't do anything. Insert nothing here. So inserting nothing here corresponds to here having the vacuum of the theory propagating.

You can insert the operator phi of 0, phi at the origin. That would correspond to exciting the field here in the lowest Kaluza-Klein harmonic, a harmonic which is constant on the sphere. There is one little technical point I need to explain.

So when you have a massless scalar field here, here on the sphere, what you get is a scalar field that is just the usual Laplacian on the sphere plus one unit of mass. This one unit of mass comes from the conformal coupling to the curvature and so on. Let's say it's a technical thing, that in order for this to work, we need to add this mass.

There is a very good reason for adding it, but let me not explain it a lot. So this corresponds to taking this scalar field, considering it doesn't vary on the sphere. This has zero angular momentum. So it doesn't vary on the sphere. It's like a harmonic oscillator in the time direction. So that's a single harmonic oscillator mode.

You can consider the field. So that was this. So nothing corresponds to the vacuum state in the field theory. This corresponds to a simple, one of these harmonic oscillator modes. This corresponds to two harmonic oscillator modes,

phi squared of zero. Derivative mu of phi of zero, some other operator, corresponds to some other harmonic oscillator, mu, because now we are going to expand in spherical harmonics

on the S3. So for each spherical harmonic, we get a harmonic oscillator, and so on. And the energy of these various states, so this has some energy zero, let's say this has energy one, two, et cetera, corresponds to the scaling dimension of this operator,

how these operators behave under rescaling of coordinates. OK. So that's the story. Now what this enables us to do looks like somewhat abstract, this dictionary. This is something you can do in any field theory.

It has nothing to do with the gauge gravity duality. In any scaling variant theory, you can do this in the criticalizing model. You can do this in any theory that is scaling variant. This is a convenient way to think about this.

Now, if we are now in the context of the gauge gravity duality, then we can do one more column. Since this is something we can do in any field theory, in particular, we can do it in a theory that has a gravity dual.

So this is some cylinder, and it corresponds to the boundary of some way of writing AdS5, where it has a metric of the form minus, let's say,

cosh rho squared eta squared plus d rho squared plus cinch. OK. So here we are just simply writing the same metric that we wrote before, but in a different set of coordinates

where it covers different regions and so on. View it as a different way of thinking about AdS5 space here. And view this as the version of AdS5 space that describes best the theory on S3 times r.

So in this way of thinking about AdS space, the G00, the gravitational potential, rises very fast as you go to the boundary. So here I'm just drawing, now it's a solid cylinder. So the boundary is a hollow cylinder, only the surface of the cylinder. AdS space is the whole cylinder, the interior of the cylinder.

And the gravitational potential is a minimum at the origin, at rho equal to 0, and then it rises in all directions. So a particle that is here will oscillate back and forth. It will now go to infinity as it was going before. There is a relationship between the two AdS spaces that I can explain if you're curious, but for this point of view, view

this as two different ways of thinking about it. Here we see that the physics agrees with the physics we expect on a finite size box, on a sphere where the spectrum is discrete. So if you put in a particle, a massive particle in the bulk,

its minimum energy configuration is to put it here at the bottom. We can take the same particle and excite it and make it oscillate back and forth. And that would correspond to taking some excitation of this field theory and then oscillating this back and forth would correspond to having, so this

is the sphere on the boundary. And when this is, let's say, closer to this region of the sphere, the excitations are concentrated more on this region than more concentrated on this region and less concentrated in that region. And when it's oscillating in the other direction, it concentrates much more in this region than in this region.

And when it's sitting at the bottom, it's concentrated everywhere. I mean, it's uniformly distributed over the whole sphere. Good.

OK, and in this way, we can think about the various states. The various states are going to be created by operators. So if we have an operator such as the stress tensor, T mu nu, which is the trace of, let's say, F mu delta, F nu delta, et cetera,

this operator acting on the vacuum of the field theory will create a state. If we are very weak coupling, it will create just two gluons. It will create two gluons moving on the sphere. The two gluons will be moving on the sphere with minimal amount. If we set the stress tensor at the origin,

they will be moving with minimal amount of energy. On the sphere, it will create the minimal amount of energy in this state with spin 2, where the spins of the two gluons add up to spin 2. That we could have. That's what we have in the field theory. In the gravity theory, we just have a mode of the graviton, so a massless graviton, massless gravity

wave that is oscillating between these two gravitational potential wells, which carries the minimal amount of energy allowed by quantum mechanics. Naively, you might think that the massless particle can have zero energy, and that's the minimal amount of energy. But this is a particle that is forced to live in a space

with a gravitational potential well, so it has to have some non-zero momentum in the radial directions. So it has to have some non-zero energy. And its energy will be small and non-zero. And we can figure out exactly what its energy should be. And for that, we need a little more of kinematics.

So I mentioned that the energy here would be the time translation symmetry. So let me call this d tau squared. So we can measure energies with respect to this time on these pictures. That means we measure energies in units where the radius of

the sphere is 1. And on this side, that amounts essentially to measure energies in units of this time here.

And then if you have, for example, I need a couple of more formulas. So you find that the stress tensor here has some scaling dimensions. So stress energy tensor in three dimensions has units of energy divided by volume cubed.

Energy divided by energy density, energy divided by volume, or length cubed. So this is energy divided by length cubed. This is 1 over length to the fourth. So it has units of 1 over length to the fourth power. And something like that, there is something we call anomalous dimension, which dimension is how many units

of 1 over length it has. And for the stress tensor in four dimensions, it's 4. And what this means here is that just, and this follows simply from the fact that the stress energy tensor is a spin-to operator that is conserved, that obeys the conservation equation.

That, by some little mathematics that I don't have time to explain, implies its dimension should be 4. I mean, it's basically the same as this calculation we did here. So on this side, we have a graviton that will have a

very long wavelength. So the size of this potential well is of the order of the radius of curvature of AdS space. So the proper energy of this graviton will be of the order of 1 over R. And when we measure it with respect to this time, which differs by an amount of order R, we

find that the energy with respect to this time tau that we define here is exactly equal to 4. And to do this properly requires solving the wave equation for the graviton in the interior, which is essentially straightforward. Well, you have to deal with the indices. And it's a little messy, but it can be done.

But before you do it, you know what result you're supposed to get, because the result is given by the symmetries. Now, in the same way, you can consider other operators. And for each operator, you find the corresponding state.

Just as a simple explanation, let me say, imagine you have a very heavy particle of mass m, proper mass m here. Then the energy with respect to this time of that particle will be essentially m times some factor of R that comes from the redshift factor between proper time and this kind of time, let's say at rho equal to 0 where this

factor is unity. So we'll have R times m. So this will be the corresponding dimension of the operator in the boundary theory for a particle of mass m. So from here, you see the following.

So that if the masses are, if you have dimensions of order 1, like 4 or 6 or whatever, then it means that the mass will be going like some number divided by r. So this corresponds to particles whose wavelength is comparable to the radius of curvature.

I'm just trying to, the goal of what I'm trying to explain now is to explain the typical sizes or wavelengths of operators which have dimensions of order 1, like the stress tensor, which has dimension 4. And they are objects which are very extended.

They have a wave function which is like a massless particle in its lowest energy state. In a box of size r, it will have wavelengths which is 1 over r. That's what we're finding here, and it's all consistent with this. Different points of view. I'm just explaining the same thing from different points of view.

Is it clear? Now, if you had a particle whose mass is much higher than 1 over the radius of curvature, then the corresponding dimension would be very, very high.

So that's some kinematics. And so in particular, for example, let's discuss this

again as this question of the strain state. Imagine we want to trust gravity. So the strain is very small, and the masses of strains are masses which are 0 in 5 or 10 dimensions, and then there are the masses which are of order 1 over Ls. And they would correspond to dimensions or energies which

are of order r divided by Ls. So you see that to the extent where we can neglect the massive, so neglecting the massive strain modes will only be possible if the radius is much bigger than 1 in strain units. That was the condition for trusting gravity.

And that implies that the dimensions of those states should be much larger than 1. This is if gravity is good. Now, what types of states does gauge theory have?

So I'm trying to now explain the connection between the weakly coupled version of gauge theory and the strong

coupling version. Well, we discussed the graviton. Let me ignore the indices, OK? Just not to clutter too much. I'm going to ignore the indices a little bit. So this is like trace FF, OK?

That was the stress tensor. That had dimension 4 and spin 2. Now, there are also some other operators which spin 4, which roughly speaking could be viewed as f, derivative square f, OK?

They have a weak coupling. They have dimension 6, OK? This is when g squared is much less than 1. And we can ask, what do they do at strong coupling? And in a theory of gravity, there are no, you have

typically particles of spin 2, but no light particles of spin or higher spin. I mean, in fact, the interactions of particles of light particles of high spin are very, very strongly constrained, both, let's say, experimentally and theoretically.

So suddenly, if gravity is a good approximation, we don't expect any massless higher spin particles. If the dimension was exactly 6, its mass would be exactly 0 here in the right-hand side. And so if we want gravity to be a good approximation, one of the things that should happen is that the mass of

that particle has to be very large at strong coupling, OK? And you can compute it at one loop. So let's say at one loop, you get something of further g square, and so it goes in the right direction. So in other words, what you expect is the following for this state. So it's a function of lambda.

So we have the dimension. So let's say we have 6 here. And then it starts rising linearly from this correction. And at strong coupling, you expect it to be fairly high. Now, how high? Well, it would, the only, let's say, the first spin 4 states in string theory come from the massive string

states, which have mass of further 1 over the string length. And they have this dimension, r over Ls. And if you remember, but you don't have to remember, that went like some power. It's actually 1 fourth of lambda, right? So this is the behavior of a very large lambda that we expect, OK?

So this is what we should get from the string theories, weakly, from a string moving on a space whose radius of curvature is very large, we get this. From perturbative YAMLs, we get this type of dependence.

And you can ask, what do we get in between, OK? Now, one of the recent advances in the last few years was to be able to actually compute this whole curve exactly. So you can compute it as a solution

of an integral equation. You have to solve the equation numerically, but you can compute it. And it matches beautifully the weak coupling answer with the strong coupling answer, OK? So this is computed in the planar limit. So in this limit, where n goes to infinity with lambda

fixed, and you can actually do the calculation. So it uses the techniques of integrable models for those who know what that is. OK, so now let me explain one thing that is useful for understanding

when we think about these things. I tried to summarize the relationship between string, gravity, and the Lagrangian limit in a way that is as general as possible.

So in general, we have the first two points. First is that when the radius of AdS, the radius of curvature of these spaces where you have the gauge gravity duality, is much bigger than L-blank, then this

is if and only if n is much bigger than 1. Now, what I mean by radius of AdS here is any radius of curvature of the space that is dual to your field theory. And by n, I mean the number of degrees of freedom of your field theory, the n of the un matrix. So this is a general property.

And remember, a plank is what governs the gravitational interactions. And it's a n what governs this planar expansion and so on. So always, in order to have a gravity dual, a first necessary condition is to have n much bigger than 1, always.

So if someone comes and tells you I have a gravity dual of the Ising model, then you are going to be a little skeptical. Because the gravity dual of the Ising model will be a very strongly quantum theory of gravity. Because what governs the classical approximation to the theory of gravity is how different Ls is from L-plank.

This is what governs the extent to which gravitational waves are weakly coupled. And second, then we have the radius of AdS much bigger

than Ls. And in general, when the string length is defined at all, this will always be much bigger than L-plank. This will always be whenever there is some kind of string. And then this implies that lambda

is much bigger than 1. So whenever the radius of curvature is bigger than the string size, which is also necessary in order to have a gravity approximation, you need that the coupling should be very strong. Now you might have strong coupling and still not

necessarily have a space that is weakly coupled everywhere. You do have it in n equal to 4, but in general not. But this is definitely a necessary condition. So any theory that has a gravity dual will necessarily be strongly coupled. And the reason it has to be strongly coupled

is because operators like this that have higher spin have to gain large energies, have to have large dimensions. And the only way they can have large energies is that the interactions are strong. I mean, the weakly coupled picture of this thing is just two gluons moving, sloshing back and forth on the sphere. So the two gluons are moving independently.

And as you increase the spin, it means that two gluons are moving completely independently of each other. And as you crank up the coupling, there is a stronger interaction, a stronger color interaction between them that pushes their energy to higher value.

So the mental picture you should have is that theorists that have a gravity dual are strongly interacting theorists where many of the states in the spectrum have been pushed to very high energies. And you're left with a small subset of the possible excitations of the theory.

The smaller subset will always include the graviton because the stress energy tensor has protected dimension. And maybe some other ones that, due to the symmetries of the theory, will also have. OK, other questions about this?

OK, maybe I'll repeat this once more because I think it's important. So theorists in this regime with large n gauge theories

will always have a string dual. The string dual might not be easy to analyze. Only some special circumstances are met, like high coupling. The string dual will be approximated by gravity. Now QCD is a theory, even large n QCD, so QCD with a large number of colors,

would be a theory where lambda is probably going to be a further 1. Well, lambda runs with the energy. But lambda never gets, lambda is either very small or probably a further 1. So the approximation of gravity might or might not work because we're in the regime where lambda is a further 1.

You just do the calculation, and you see whether it does work or it doesn't work. You don't have an a priori reason for believing that it would work. And it works well for some cases and doesn't work for some other things. So there are some things like the bulk viscosity

that gravity gives you a beautiful value for it in agreement with experiment. And there are some other things like the spectrum of mesons where you really need the string, as I mentioned in the very beginning, where gravity doesn't do a good job.

Now I'd like to, in what remains, show you how to do some other calculations. So far I've discussed general properties and so on. I'll try now to fill in some what they're usually

called entries in the dictionary. So how, given a field theory problem, how you translate to a gravity problem. I'm going to discuss a few examples just so that you get an idea. You get the idea. Well first, I've discussed it very quickly.

Maybe I'll just repeat it in words. I discussed how to calculate correlation functions of a stress tensor. It was done by sending in gravity perturbations from the boundary and letting them interact in the bulk. If you want me to discuss that in more detail, I can do that.

Not a lot of enthusiasm for that. Well, we could engage theories. One sometimes is interested in calculating expectation values of Wilson loops. So these are some operators which are defined as the path integral of the gauge

field over a circle. Physically, what they are is just expectation values of these things. What they are is essentially the phase factor that is accumulated by a heavy probe quark that moves along the contour C. So in the presence of the gauge

field, this quark factor due to the interaction with the gauge field will get an extra phase factor of this form, which will be influenced by all the gluons that are running around and so on. If you wish, it contains all the effects

of the radiation of this quark and the interaction of the radiation of the quark with the Yang-Mills fields, with the color electric field. In QED, this is trivial. So you have just simply the radiation you can take into account by calculating, essentially, this diagram, the exchange of one photon

and exponentiating it. And that just gives you the whole interaction of the trajectory with the radiation. But in a theory with interactions, it's more complicated because you have all these kinds of diagrams. So the radiation itself interacts with itself and creates some other radiation and so on.

So this is related in the gravity or strength dual to taking this contour now defined on the boundary. So this is a contour in R4. So this is R4, some contour. And then you take a surface, a two-dimensional surface sigma

2, that ends on this contour at the boundary of AdS, of the space. And you find the minimal area surface that ends on this contour. And the expectation value of this Wilson loop

to leading order is given by e to the minus tension of the string times the area of the surface. And this has a behavior which, generically, goes like square root of lambda. That is the behavior of a string tension and strong coupling times the area in units

where the radius of AdS is equal to 1. So this is a purely geometric quantity that depends only on the geometry of the contour and nothing else. And this is the whole dependence of the coupling and strong coupling. Of course, there will be further corrections here of order 1 plus terms of order 1 over square root of lambda.

This is the behavior for lambda much bigger than 1. And the behavior for lambda much less than 1 is the same as the QED result. You just take this leading interaction, and that would be when you neglect the effects of radiation on itself. OK. Now, one thing you can calculate, one related calculation,

is the calculation of the quark and the antiquark potential where you take the quark line and the antiquark line as two points separated by some distance l on the boundary. Then in the bulk, it corresponds to a surface that goes between these two positions.

And the potential is computed as the energy of this string or the actual length of the string, but with the AdS metric, measured with the AdS metric, not measured with the flat space metric. And if we measure the distance between these two with the flat space metric, we would get a potential which would be linear in l.

But if we measure it with the AdS metric, we get actually a potential which goes like 1 over l. So the reason it goes like 1 over l is because as you separate them more, they go deeper and deeper into the interior where the work factor is smaller and smaller. And actually, you get the opposite dependence

than the one you naively would have expected. And the dependence on lambda for strong coupling is square root of lambda as it was here. And for weak coupling, it's lambda over l. So that's the same as the QED result multiplied by a factor of n because there are n possible quarks, n possible color charges for the quark and the antiquark.

Yeah, maybe I should elaborate a little more on this picture. So maybe if you have a quark. So we discussed that in QCD, for a quark and antiquark,

we get this color electric flux that does this with constant energy. And then this would lead to a potential which is the tension of the QCD string times l, if this tube starts having constant energy density.

In a scaling variant theory, the picture looks more like the one you have in ordinary electrodynamics of the flux lines spreading everywhere. And when you start separating these charges,

the part of the energy that depends on the separation decreases like 1 over l, always, of course, with a minus sign. Now, the difference between these two, in the gauge gravity duality, will be related to the geometry of the space that you consider. So in the case that the theory is scaling variant

and you have AdS space, you'll have this behavior always. And the theory will not be confining. But you can have this gauge gravity duality also for situations where the boundary theory is not scaling variant. So scaling variance was useful for explaining the simplest case. But in cases where you don't have scaling variance,

no scale invariance, then the behavior of the work factor, recall we had this redshift factor that was very large near the boundary, pushing things toward the interior. But in confining theories, what this will do

is it will rise again, or the space will terminate. And what's important is that there will be a minimum value for this work factor. It's similar to what was happening with the work factor, or the, this factor is sometimes called work factor, sometimes called redshift factor, sometimes called gravitational potential.

They are all closely related things. For the purposes of this talk, they are all the same. If you want, you can ask me the difference. So you, so any object will be one to sit at a particular value of c. For the case of a particle, what this means is the following. So recall that the distance from here is the size.

What this is saying is that in a confining theory like this, if you have an object, it will minimize its energy by acquiring a specific size, this particular size. It doesn't want to be smaller, doesn't want to be bigger. That's similar to what happens to the proton in QCD,

where it has a given size, the preferred size, given by essentially the dynamics of QCD, just the confining dynamics of QCD. I mean, the mass of the proton doesn't come from the mass of the quarks. It comes from the confining interactions of the strong interactions of QCD.

And the same happens for the string. Imagine you have this picture, and now we do the same thing, but for a string. So there is an extra direction here. And again, so if we have a string that

is stretching along that spatial direction, the whole string will want to slide down up until it reaches this minimum size here. And if it's reaching some points on the boundary, well, OK, it will reach the points on the boundary here. But in the middle, it will want to settle down to this minimum size. So that minimum size, or that position

in the radial direction, corresponds to this equilibrium size of the string. And at long distances, it will give you a confining potential, similar to the QCD string. And finally, I would like to end up by discussing some black hole thermodynamics issues.

The first point I'd like to make is this great gravity duality does not

describe only empty AdS space. It describes this AdS space with anything it can have inside. So it only describes spaces that are asymptotically far away. They look like empty AdS space, far away where the gravitational potential rises. But in the middle, it can be anything. It can have gravitons.

It can have black holes. It can have whatever you want. In particular, it can have black holes. And an object that is interesting to consider is to have a black brain, the non-extremal analog of the extremal black brain we discussed before. And I think I probably will not write down the metric.

So it's some metric where we have the boundary. So now, if we draw again the gravitational potential, g0,0. So it rises near the boundary. And then at some point, at some particular point,

it goes actually to 0. And in terms of proper distance, it goes to 0 linearly. So it has a finite Hawking temperature. So pictorially, we can say that this, we have the boundary. And then we have the black hole horizon at some distance,

c, let's say, c horizon, c horizon from the boundary. And the new features compared to the previous one is first that we have a temperature. So the previous space had 0 temperature. This has a finite temperature. And it also has a non-trivial, first of all,

this configuration will not be boosting variant. If we make a boost, there's a preferred rest frame, which is the rest frame in which this object is at rest. And this in the bulk, corresponds to the Yamman's theory at finite temperature.

So you have the Yamman's theory where the gluons are moving around, jiggling, like if we wish a plasma or a fluid. They are strongly interacting. So calling it a plasma is a bit of a misnomer. But there are strongly interacting gluons at finite temperature. And they are supposedly the same as this black hole

at finite temperature. And the entropy of this plasma is supposed to be equal to the entropy of Bekenstein and Hawking of black holes, which is the area of the horizon divided by g Newton.

This is the gravitational entropy of black holes. It's just purely geometric quantity that depends only on the area of the black hole. Well, do I need to explain where this comes from? Should I explain it? Yes.

Well, should I explain why black holes have a temperature also? OK, that would take more than five minutes that I have. Well, let me just mention one thing. So black holes have a temperature which is related to essentially the slope of this

or essentially the gravitational force you have here at the horizon. If you write down the first law of thermodynamics, dE or dM equal to d free energy equal to TdS,

so the amount of change in the total energy or free energy, the amount of total change in the internal energy is equal to this change in the entropy. You know what the mass of the black hole is as a function of the temperature, and then you can calculate the entropy,

and that gives you this formula. It gives you a very simple formula. So the formula for the temperature is a little more complicated, but the formula for the entropy ends up being nicer and more geometric. I haven't explained to you where the temperature comes from. Well, do you guys want me to explain the temperature of black holes?

Some yes, some no's. Let me finish this story, and then at the end, you can ask me. So we have this entropy which is purely geometrical and classical. It comes from just the classical action.

And this is supposed to be the same as the entropy of this gas of gluons that if you were at weak coupling, you would calculate as, let's say, t cubed, and that's just dimension analysis. And then a factor of n squared, which measures the number of species you have, because you have n. Each gluon can have two indices that run from 1 to n,

so you have n squared gluons. So you have this plus some simple factors of pi and so on. OK, so when you compute the area, you find that the area is some geometric quantity, and ends up being, again, t cubed, and divided by this g newton.

And the newton constant, the coupling constant of the string or gravity theory, as I mentioned, went like 1 over n, or the coupling goes like 1 over n, and the newton constant, the square of the coupling, goes like n squared. So this is, again, also goes like n squared. And the two have a very similar form, but they have a different coefficient.

So if this is at weak coupling, so this is lambda much less than 1, it has some coefficient. And at strong coupling, it has a different coefficient. So if we were to draw, for example, beta times the free energy as a function of lambda,

then at weak coupling, it has some value, let's say 1. And at strong coupling, it has some value of 3 quarters. And what's really ever computed are these aspects of the curve. So the value of strong coupling plus the first correction, the value of weak coupling plus the first correction. And in between, it's somehow supposed

to go between one and the other. Now, so if you're interested in calculating something at strong coupling in this theory, you can calculate it using this gravity description. So if you believe in the relationship, you just calculate it using the gravity description.

If you want to check the description, then you have to do a hard calculation and calculate this whole curve. That's not known with present methods how to do. But if you believe in the relationship, then you just calculate it using gravity. And you can calculate a bunch of things related to thermal of this finite temperature

density of quarks and gluons. For example, what I've discussed so far is the equilibrium property. So we have the black hole completely straight and translation invariant. But you could make a small perturbation

on this black hole horizon. Put a little wiggle. This little wiggle would represent a wiggle of density. So here, the fluid is a little more dense than at some other position. And this wiggle will expand and dissipate. That's what you expect from quarks. This is the black hole picture. And of course, the corresponding picture

at finite temperature is the same. So you have the Yang-Mills theory with a bunch of stuff which has some density. This is the profile of the density. It's constant, but you can add a little wiggle. And if you add a little wiggle, this wiggle will dissipate and will eventually go back to equilibrium.

And it will have a non-zero. So this wiggle will not oscillate, as it would do if there was no dissipation. Because there is dissipation, this oscillation is dumped. And there is characteristic damping time. And here on this side, what happens is you have this wave near the horizon of the black hole. And the wave falls into the black hole.

And so after a while, it has fallen into the black hole. And the system goes back to equilibrium and to a uniform density. So falling into the black hole is the same as equilibration in the boundary theory. And this equilibration time for this long wavelength citations and so on is dominated by the bulk viscosity,

by the sheer viscosity. And the sheer viscosity can be translated here into the absorption rate, essentially, of these waves near the horizon. And you get an actually very simple answer for this absorption rate, which is the absorption rate is

essentially proportional to the area of the black hole. And the entropy is also proportional to the area. So if you compute the difference with the ratio between the area and the viscosity

and the entropy density, you find that this 1 over 4 pi, which is just simply a number. And this is a very low number, which represents the fact that we

have very strong interaction. So the viscosity in a fluid is very small when the interactions are very large. OK, so you need to, if interactions are very small, you can always transfer a lot of momentum from some region of the fluid that is moving at one velocity to a region that is very, very far away moving at a very different velocity. And this is a very efficient mechanism for transferring

momentum across a velocity gradient and will give you a very large viscosity. So very dilute gas will have a very large viscosity. But a very strong interacting fluid, as strong interacting as to give you a gravity dual, has this very low value

of the viscosity. It's interesting it is non-zero. And it's also interesting that when people do experiments at RIC and so on, they see also a very low value of the viscosity within this value to within an order of magnitude, let's say. Of this value. While any approach that starts from the weak coupling side

will tend to give you larger values. Because you start at the large value at weak coupling. OK, so maybe I'll end over time.

So I'll probably end here and open for questions or comments. Thank you. So the easiest way to understand the temperature of

the black hole is Euclidean space. So I'll explain it from that point of view.

Well, maybe I'll also explain it from a Lorentzian point of view. Yeah, maybe I'll explain first from a Lorentzian point of view. Qualitative way from the Lorentzian point of view. And then so the temperature of the black hole is related to

a simpler physical phenomenon, which is the fact that if you are an accelerating observer, something called the Unruh effect. But anyway, the name is not important. So the point is that if you are an accelerating observer in flat space, you will see a temperature.

So imagine you're in flat space vacuum. You're in a rocket. And you turn on your rocket. And you start accelerating. You put a thermometer through the window. You will see some temperature. And this temperature will be proportional to the acceleration. And it's actually equal to acceleration in units where c

and h-bar and so on and k and all the units we normally set to one are set to one. So it's a very tiny temperature in ordinary units. Just to give you a more physical idea is if you translate temperature to wavelength, to one over

wavelength, and again acceleration also to one over wavelength, then the temperature and the wavelength are the same. Well, that's the same as saying that they are the same order. But for ordinary accelerations, these wavelengths are very, very large because you have the factor of c, which makes

them very, very large. But anyway, so the idea is that if you have an observer who accelerates at any constant rate, this observer will see a temperature. Now, how can you at least more or less qualitatively

understand why you would see a temperature? I won't explain. So I'll explain it from the point of view of an observer who is in Minkowski space. So you're observing Minkowski space. And you have this guy here following this accelerated trajectory. And he has a thermometer. And his thermometer has, let's say, two levels.

Well, the thermometer has a system with many energy levels. Well, let's say many energy levels, or you could have two levels anyway. So many energy levels. And so he has this physical system that is moving across with him. And it's moving along an accelerated trajectory.

And if you have some system that couples to the fields outside and is moving along an accelerating trajectory, you expect it to emit radiation, to emit the branched-solarity radiation. So if you have a charged particle that is moving, you expect it to emit radiation.

You have a couple of positive and negatively charged particles that are moving. You also expect them to emit some radiation. They will emit a bit less. But they still emit some radiation. And that is roughly the mechanism for the mission of this radiation, is that you have some system here with

several levels whose energies can change as you emit radiation. And so due to the acceleration, you'll emit Branched-Solarity radiation. That same Branched-Strand radiation is seen by the person who is here as exciting or de-exciting. They are the energy levels of this system.

And it's actually seen as a temperature. This requires some calculation that I'm not explaining. Perhaps the simplest way to explain it is to, again, resort to the Euclidean story. So here, in order to make this accelerating trajectory more

manifest, it's convenient to choose the coordinates of flat space in this way. OK, this is just flat space written in unfamiliar coordinates where trajectories of constant rho are

disaccelerated trajectories with various constant accelerations. And again, if you fix tau and move to different bodies of rho, we select the trajectory. So here, an observer who is accelerating uniformly has a

world line which is rho equal to rho 0. So it's fixed at some particular value of rho. This is just some choice of coordinates. These are called rendered coordinates. You have never seen them. Well, they are related to, if you write t plus x equal to rho e to the plus minus tau. So this is the explicit relation to the ordinary t and

x coordinates. Anyway, so that's this. And now, if, so this is a time translation invariant situation. So you can go to Euclidean time, and this becomes d rho squared plus rho squared d tau. Sorry, I put the minus sign in the wrong place.

So this is the time direction. So now we go to Euclidean time. We have d tau Euclidean. And in general, this space would be conical depending on the period of Euclidean time. So we can, and we will remove the conical singularity

if tau Euclidean becomes identified with tau Euclidean plus 2 pi. And in this case, we remove this conical singularity. And then we get flat space. So there, we just get ordinary flat Euclidean space. Now, this is the correct thing to do if we are in

Euclidean space. Because recall that if we were to analytically continue, forget about this coordinate. Just go back to t and x coordinates, and we continue to Euclidean time. Then we just get the ordinary Euclidean space. And the ordinary Euclidean space, the angle is identified with period 2 pi.

But this ordinary identification of the angle from the point of view of these coordinates looks like a finite temperature. Because in Euclidean time, you have a Euclidean time with a finite period. So whatever you calculate here, like a correlation function of fields at various times and positions, will give

values which will be consistent with the Euclidean time periodicities. And such that when you translate them with Lorentzian time, you'll get finite temperature results. So if in Euclidean time you had a circle, when you translate back to Lorentzian time, you'll get finite

temperature behavior. I'm sort of explaining this perhaps more direct ways, but that require the knowledge of you should know that if, well, this fact that if Euclidean time is periodic, then the Lorentzian time behavior would be finite temperature.

Yeah? In which the AdS if t-correspondence is valid? Yeah. You didn't put any conditions on the t-arm meals specifically? Well, I expressed them in terms of lambda, which is g square n. Yes.

Yes, yes, yes, yes, yes. The reason I explicitly expressed lambda because that's the combination that sets the ratio between the string tension and the radius of AdS. And that's a very physically important ratio. The g square yam meals is the 10-dimensional string

coupling. So in this 10-dimensional space, remember we had AdS5? I never talked about the sphere, but we really have the sphere. And in this 10-dimensional space, there is a 10-dimensional string coupling. And that's g yam meals.

By the way, I also never talked about the theta angle. So the yam meals theories have a theta angle, which is a second parameter. That's the second massless field that we have in this, the expectation value for a second massless field that we have in 10 dimensions. The viscosity was very low because it's very efficient

momentum transfer from the coupled system. No, no, not very efficient. The opposite of efficient. Oh yeah, so there's a small resistance, right? Yeah, yeah, yeah. I mean, what you want to have low viscosity is that two layers are moving very fast with low transfers momentum that is flowing from across a velocity gradient, right?

You have a fluid that is flowing fast here. And as you fall in this direction, it's flowing less and less fast. You want to have low momentum transfer from here to there, right? If you have weak coupling, this particle, let's say particle has a tiny momentum in this direction, well, will travel for a huge time up to here, and then it

will interact. And the difference in momentum between here and here is huge because of the difference in velocity. And you will transfer a lot of momentum in that direction. At first sight, you might. Yeah, you could also say that due to the strong coupling, they all accelerate at the same velocity, and then the other way around.

Yeah, but what happens at strong coupling is that, so this gets bumped to something that it hits this guy and so on very close to here, where the velocity is also very similar. So it's not so efficient at transferring the momentum because the velocities were too equal.

I mean, this ratio here is equal to 1 over essentially the mean free path in units of the density. And questions or comments?

Check, yeah.