AdS/CFT Correspondence, Part 1 - Juan Maldacena

thank you for the kind introduction unfortunately what he said is wrong I'm not the best one to explain it so again you're just wrong yeah I'm sorry so yeah the I think Pedro will be the best one to explain it anyway so we'll start today with the basics of 80s 50 so today there will be no supersymmetry no integrability so very simple as try to explain the basic idea behind the idea safety and then probably for the next two lectures if well there are many if you have many more questions about this I can try to answer them in the next lecture so otherwise I'll go into describing model how to use integrability to analyze some problems in a DSST and there we will definitely use supersymmetry in the theme of this code okay so 80's 50's it's really I think a bad name so the the subject should be called the quantum field theory / quantum gravity duality but sometimes it's also called the gauge gravity duality for reasons that will be clear in a second and the basic idea is an equality between when you take a space-time and you consider quantum gravity in this space-time in interior of the space-time so this whole interior and then this should be equal to a certain quantum field theory that is defined on the boundary of that space-time ok so that's it is this quantum field theory / quantum gravity duality in just this statement the statement that these two things are equal now we have various examples where we'll give a specific space time here and specific gravity theory and a specific quantum field here on the boundary and there are various examples of this so many of the examples involve string theories so that string theories is a way of quantizing gravity and when we have strings so if we have strings here in interior in particularly in particular if we have weekly couple strings which is one regime of quantum gravity then typically the gauge theory that this dual twist will be UN gauge Theory you engage theory usually in the large n limit so and you need to be in the large n limit because here strings have a coupling and the coupling between between the strings turns out to be proportional to 1 over N squared now the reason this is the coupling comes well it due to an old argument of toast that if you have a UN gauge Theory you can divide the diagrams the Fineman diagrams of that gauge theory into planar diagrams and non-tenure diagrams and non planar diagrams are suppressed by parts of 1 over N squared and this amounts to and turns out to be exactly the same as the division we have here in string theory between string worksheets which have the topology of the sphere or string which have more complicated to polishes which represent string interactions ok that was a very quick review of the soft argument but basically since since that argument it was expected that large n gauge theories should be related to string theories of some kind and people thought well maybe those are string theories that somehow live in two in four dimensions number of for example he studied with gage steering for dimensions people expected those strings to live in four dimensions but actually those strings should be thought of as living in some higher dimensional space in fact they live in this the interior of this this other space-time now in gauge theories we usually have a gauge coupling and so there's the yang-mills coupling and not to be confused with the string coupling and the effective coupling in a large n gauge theory is Chi square n and the reason for that is that is that we can have n colored objects and gluons propagate in between that that can transmit the force and so interactions are n times bigger than you would otherwise expect so so in other words if you have the propagation for example of a gluon and you have this type of diagram there is a color line here which can have n different colors and for that reason this diagram gets enhanced by a factor of n so really the effective coupling of these theories is Chi square N and so if you have a large n gauge Theory this this will be the coupling so whenever lambda and this is sometimes called the toast coupling lambda and when lambda is small the gauge theory will be weakly coupled and it turns out that in this relationship the radius of the space here so some measure of the typical curvature radius will be more concrete in a second the radius of curvature of the space that is somehow the size of space in string units is typically a power of Chi square n to some positive number here now the fact the string theory can be well approximated by gravity whenever this number is very big so whenever the size of the space is we're on the string scale so in other words the picture is something like this so we have some very big space and the graviton is a little string whose size is a further the length of the string and in order for for you to be able to approximate the string state by a point-like particle such as the graviton of einstein's theory you need that the size of your space is much bigger than the string scale okay so gravity will be a good approximation whenever this is much bigger than 1 and I said before that perturbative gauge here is a weakly coupled gauge theories delusion intuition of gluons moving almost really interacting now and then is good when this is very small okay so now the regime where these two theories this two approximation theory is perfectly incompatible and that's a good news because otherwise we would have ruled it out immediately because the theory here doesn't look anything like perturbative gauge Theory and that's also why it's called the duality so that's where the word duality comes from is the fact that the parameter space is divided into two regions and in one region of parameter space one description is simple in the other region of parameter space the other description is simple okay so here one one regime vertebrate ebh Theory simple when lambda small when lambda is large the string theory description is correct okay so that's the basic statement and will give more and more will specify it more and more so that's somehow the basic story now in order to understand this a little better one needs to understand better what kinds of spaces we're going to be talking about here and the spaces are spaces which have something called the word factor [Music] warp factor is just simply a gravitational potential so there are spaces where for example if we start with a theory which is defining our four so we have the time this is our 3 comma 1 and they have an extra dimension that's AC squared and then we have here a gravitational potential which depends on the coordinates see it's just some a role factor of the metric when a factor multiplies the metric in this Lorentz invariant form is sometimes called quark factor you might be also used to read ship factors which are factors such as multiply the time direction and so on now what happens when you have a situation like this well just let me so the types of metrics will consider this factor will be such that the divergence force equal to 0 so some region where it blows up and then it in some examples the simplest examples like the ones that have most symmetry they just decrease continuously but for the purposes of illustration let's imagine a situation where this four factor just goes up again okay I imagine you have a gravity in a situation like this we have a warp factor that has a minimum at some location okay this is just simply gravity this is a gravitational potential well so if you have if you put in a particle somewhere its energy let's say you put it addressed at some position C its energy which has simply be the rest mass of the particle times the local values before were factored WFC right and the particle will want to go to the region where the word factor is minimized so the particle will want to sit here at the bottom okay that's a gravitational potential one now imagine we have a situation like this and we concentrate on the lowest mode so we concentrate on the modes where the particle is stuck here and moving purely in the four dimensional directions so then clearly the physics of this particle I mean this particle will behave as a four dimensional particle right now if we do anta mechanics of course this particle will have some wave function and it will have a wave function localized near the minimum of the potential and again from the five dimensions here we go to a four dimensional theory now this is nothing surprising you of course know that in many situations to think on this matter for example you have electrons moving the chain a two dimensional confined to a two dimensional layer by a potential in the theta C direction and they behave as two-dimensional particles so that's that's fine nothing surprising here now imagine we start including the higher modes so there is this wave function and then there are some other wave functions for it oscillate right and so on right so these are all the higher modes representing the radial oscillations of this particle right this is more like a particle in a harmonic oscillator will have all these various modes so then you would say well you have a certain four-dimensional thin theory with an infinite number of particles you would say because you have an infinite number of particles because you have all these various colors like line modes right okay now this is what you would naively say naively you have an infinite number of four-dimensional particles and now this fact that you have an infinite number is the reason why if you have a quantum field theory let's say on a box and you want to include and you want to treat that the fields here in one less dimensions well it's useful as long as you consider actually scales less than the box when you consider initially scales higher than the size of the box it's convenient to go to the higher dimensional field theory and it's not convenient to think of it as a field theory which includes all this infinite number of particles okay now what happens in a theory of gravity is that there is a kind of truncation and I will a kind of truncation that allows you to keep put essentially a finite number of fields so you are able to describe this whole configuration this whole set of and so on everything you need in terms of a finite number of fields and well there are various qualitative reasons why this happens one is the so-called holographic principle that does you can describe the degrees of freedom in some region of space by a number which goes like the area another more practical is that if you try to excite here very high closer climb modes you might end up forming a black hole and that truncates them so and in fact in this duality to the disability we are talking about this UN gauge Theory has large by finite n so it's a finite number of degrees of freedom which is describing the five dimensional space-time okay and this finite number of fields this number of fields is turns out to be proportional I will explain later wide it is proportional to this am now just giving results it's proportional to the radius of curvature of this space well it's basically is proportional to the inverse of the Newton constant the five dimensional Newton constant in units of the radius of curvature of the space so that's the number of finite number of fields you need to include now a very special case of this relationship now you shouldn't be confused by thinking perhaps my explanation was a little misleading in that you shouldn't think of these fields that will describe the field theory has been directly this Colusa climb mods will see that this fellow circuit mods we are talking about here are really bound states of the fundamental fields all that I wanted to illustrate here is just this truncation in the number of fields and why it's important to have gravity actually in one of the exercises I propose that you demonstrate that if you have you have a here an ordinary mountain field theory then it cannot be dual to a quantum field here in the boundary okay so it's crucial to grab it in the bulk now now let's consider a special case and a special case is a situation where this war factor W is equal to 1 over C okay so in this situation we have a gradation of potential of this kind and geometric in this situation pass an extra symmetry so let me so we have all the four dimensional four dimensional directions plus the extra direction divided by C squared okay so that's metric and this metric has a rescaling symmetry so if X goes to lambda X and C both lambda C this metric remains I mean the metric remains of the same form so that's an isometry of the metric and this will translate into a symmetry of the boundary theory so in this case the boundary corresponds to these directions the four-dimensional directions so the boundaries whatever seats at C equal to 0 so in general the boundary occurs whenever the work factor diverges and when it diverges in this particular way the theory is conformal in the traveler so if it behaves like this in near the boundary you'll have a field theory which is scale invariant because this bulk symmetry so the fact that we have the symmetry in the bag Theory translates into a symmetry of the field theory of the field theory and that symmetry shows symmetry under under rescaling ok now let let me continue talking about the physics of this word factor I mean the most important thing to understand in the gauge gravity duality or one very important thing to understand is the significance of this scale factor what it does to the physics and what what happens under this very big gravitational potential that we have so let's try to understand what happens to a massive particle so we discussed before the fact that if we put a massive particle here at some point in this potential will have it will have an energy measure with respect to that time so here so there's the time T and we have time translation symmetry and their and their translations of this time T and if we measure energies conjugate to this time translation so that we have the operator that implements those translations in the Hilbert space blah blah blah if we measure energy conjugate to this we will will find that the energy of a particle at position C is just given by well the same formula it drove written before M R times divided by C zero okay so if this is a particle sitting at some position C zero that will be the energy at that position of course if we put a particle at this position C zero it has a gravitational force and will start moving in right so a particle put that this position will start moving to the interior of ideas and if a particle has this energy and it comes from the interior will reach to this point and then bounce back okay so a massive particle will never be able to get to C equal to zero okay so we'll always bounce back before it gets to this boundary so we said that sequel to zero was a boundary notice that this is a place where you can't quite reach with massive particles you try to send a particle carry on higher energies you get closer and closer to C equal to zero but you bounce back before you get to the three equal to zero with massless particles is slightly different so massless particle in principle can a massless geodesic can get to sleep at the zero and this is the way to see that is to look again at this at this metric and notice that massless particle doesn't care that there is this factor 1 over c-squared and then you find that you have really half space I mean you have in the C direction you library can come and then go back out okay but really I mean they can do it but at the cost I mean when they get here they are really really very redshifted and by putting a simple boundary condition here and will will expesive I later the boundary conditions also massless particles go back in and internal will have to supply some extra boundary condition here at the boundary of ideas to tell us how this massless particles are reflected okay now notice that as we as this particle has as we move this particle to smaller and smaller values of of off C node it will have higher and higher energies and this is the first manifestation of something which is with the see later in more detail which is the fact that approaching the boundary of the of ideas a corresponds to going to very high energies in the boundary especially very high energies of shell energies in the boundary theory now this can also be seen by noticing that if you take the scaling symmetry and you perform performed rescaling that movie two smaller values of C that is you choose here lambdas which are smaller than one then you're also going to access that are smaller than one so on the boundary here you are shrinking their distance you're examining the UV behavior of the theory now in fact when it turns out that if you have a particle at some position C 0 so massive particle at some positions is C 0 it corresponds in the boundary theory to some object some objects and bound states some states in the field theory which has some characteristic size which is also proportional to C 0 and again this is this is also due to this just simply application of the symmetry so if c 0 corresponds to some object of some size then as we scale it will be stayed together with C 0 so size is here on the boundary on the boundary Theory correspond to 2 this distance from the boundary in the bulk so a point-like particle in the bulk corresponds to an object that has some nonzero size in the boundary and as we approach as I approach the boundary this object rings and become smaller okay now this is also another one way to think about the motions of the extra dimension the fact that in X conforme scale invariant theory in order to specify the state of some object you need to give not only the center of mass positions but also its size and so you need one more coordinate that you would naively have expected and this one more coordinate is this extra position in the Z direction okay now let's talk a little more about this space it's called this place it's called on ties either specifying some I think I already said it is the simplest space with constant negative curvature and because it has more symmetries that than that then they are apparent in this in this way of writing the metric so I've written the metric in such a way that you see the Poincare symmetries plus the rescaling that I've just but it has a few other symmetries which are called special conformal transformations and so on so by writing the metric unfortunately there is no coordinate system that makes all the symmetries manifest so you have different coordinate systems make different symmetries manifest okay so but there is another set of coordinates that are particularly useful for thinking about anti-de sitter space and these are coordinates where the metric takes the form D tau square cosine hyperbolic of Rho plus 0 squared plus sine hyperbolic of Rho squared V Omega 3 squared we are talking about the five dimensional place either space now there is some overall factor which is telling us the radius of curvature of the space we have the metric of the sphere this would be the radius of the sphere this just setting us to set in the scale of the space and so in this coordinates the space can be actually in fact the Penrose diagram of the space can be represented as a cylinder with one time dimension and then this circle here is really the s3 and then we also have the radial direction that starts from the origin of the cylinder okay imagine we consider a Euclidean field theory for the time being so Euclidean fil theory will be related to Euclidean ad space and Euclidean idea space also it's known as hyperbolic space I will it's the same you can represent in the same way but with a plus sign here yes a hyperbolic cosine I'm sorry I learned with Russian books which right right this yeah sorry sorry about that okay so the so if you have the Euclidean fin theory then you come up the Euclidean field theory to a cylinder by considering something called radial quantization so you start with some origin and you think of time not as slicing the theory in this way but slicing the theory a different radii okay now here the size of the sphere is changing but we have of course the conformal symmetry that allows us to change the role scale factor in the metric so this is now the metric of the four dimensional filter in Euclidian space we can multiply this matrix divide this metric by R squared and so we get here log of R squared now we have a cylinder Euclidean cylinder okay and then the theory on this Euclidean cylinder is dual to the theory and the Euclidean version of this space which we could also now we could analytically continue the time coordinate on the cylinder to Laurentian signature and we would have this Lorentzian signature space now the reason I mentioned this this relationship in detail between the plane and the cylinder is the fact that if you start with the quantum field theory on the plane and do this mapping here you get the theorem the cylinder on its ground state so you get the ground state of the theorem the cylinder the state with lowest possible energy but by inserting local operators here of to insert some operators at the origin and you do the same mapping then here you get some excited States on of the theory on the sphere so you have this this mapping between operators in our four and states of the theory on well on s three times time okay so this is something that is true in any conformal field theory in four dimensions but it's actually true in any dimensions you have to only change the dimensions here the cylinder and it follows from the simple argument about the value scaling symmetry of the theory there is scaling in a conformal field theory you can change the metric by any overall factor and the physics will be the same okay up to something called the conformal anomaly which is not important for what I'm saying here okay so now we have this mapping between States operators and States of course here we had this idea space now we can view the field serious living on the boundary so this is just a boundary cylinder this is a cylinder with the interior and this is the cylinder without the interior that's where the filter lives and different states here correspond to different states here in in the bulk so in particular imagine you have particle here sitting at the center at the center of ideas so you have some particle and we can compute the energy of this particle with respect to that time the time coordinate that appears in that metric and so the energy of that particle will be M times the radius of ideas if it's sitting at the origin now and that will be the energy of the corresponding state here on on the boundary theory and such energies are related to the behavior of the state and their translations in this direction which is the same as three scalings of this coordinate R and so that eigen value is the same the conformal dimension of these operators so those synergies are the same as the conformal dimensioning usually denoted by Delta is the dimension of the operator well it's how the operator scales and their risk aliens of the coordinates so we have this relationship between the operator dimensions and the energies of particles now of course when we did this if we did this for a massive particle and I've ignored quantum effects so I ignored the fact that this particle can move and have some wave function so in other words notice here that this gravitational potential again is like some harmonic oscillator potential but it's not really harmonic and harmonic listen but it's some potential and the two quantum effects there will be some quantum contribution to the energy of the particle and you can do that when two mechanics and it's just simply corresponds to solving the wave equation for a massive field on this curved space and you can calculate that energy precisely and you you'll get the precise expression which for scalar field has the form 2 plus square root of mr squared plus 4 so I'm not telling you how so this is the whole exact answer so it's exactly in the regime where you can calculate well it's not quite exact I mean it's only exact in the approximation where you can see it you take into account the quantum mechanical effects of the particles but you do not take into account the back reactions of particles on the geometry ok um good now this formula shows a couple of features that so first is consider a massless particle so particle M equal to zero that would correspond to an operator of dimension Delta equal to four so first thing to notice is that a particle with zero mass does not lead to a state with zero energy because of the ground in that case the only contribution is just the ground state contribution from the harmonic oscillator potential so there is that contribution to the energy and that contributes to a nonzero energy an energy equal to four in this units not notice by the way that the spread of the wave function here is if this has some curvature scalar further one the spread of the wave function will be again further the radius of the radius of curvature right just even in the original formula that we use some energy which goes like the gradient terms you need for curving so changing the wave function at scale so further the radius of curvature that contribute with a mass or inertia for the one over R that give a contribution to the inertia further one these are the Corrections we have to take into account when we do the quantum mechanical calculation anyway so mass equal to zero will still have nonzero energy Delta equal to four now Delta equal to four in a fourth dimension since here is a special dimension because this is a dimension of so-called marginal operator so we can add this operator to the theory at least to lead in order the theory will remain conformal so this is a perturbation which at least at first order in the strength of perturbation it does not change conformal symmetry now there are many situations where we have scalar fields with a completely flat potential in the bulk the delay zone for example in some of the examples in string theory it will be an example of that and in such situations where we have no potential for the dial tone and the load not have any value so the scalar field can have any value so the theory will be conformal for any value of the de Letran and che in the van the values relation would correspond to changing the constant value of that scalar field everywhere in space okay that that's what that corresponds to yep yeah that requires more calculation that's why I said if you have a potential which is exactly flat then it will be exactly marginal yeah exactly that would require more calculation there are many situations where you have Delta code for states but not they are not all marginal not not all exactly marginal another interesting case is the case of the graviton so the graviton again is a mass particle which in the bulk has mass equal to zero but now has been - and it also leads to dimension for state which is just the stress tensor operator in the boundary theory and that's the correct dimensional stress tensor okay so I mentioned a couple of states so in any conformal field theory you will have you'll definitely have the stress tensor so it will have the graviton in the bulk okay so that's a state that will always have and then different other quantum field theories could have other states now let's analyze what other states we get from the point of view of the bulk well I mentioned that in many examples we have a string theories so string theories contain string states and among the strings States there are the massless string States which about which we talked a bit here about so we have those but in addition we have massive string States we have a mass of the order of 1 over over the string length remind you that the string length is so the tension of the string is going over the less squared so definition of LS you have a string of this tension will have such such states and it will lead to operators of dimension R over LS okay so all the massive string States will have dimensions which scale like R over LS and if you recall what we discussed here so we discussed the gravity approximation will be correct precisely whenever LS is very big so precisely when all these states have high dimensions now if you go to just the first massive string state you'll find that among those massive string States there there are stage we have spin bigger than 2 so we have for example states of screening for just the first massive string States so now if the state was very light then gravity would not be a good approximation you have sort of light states of spin to spin for you probably are forced to include all kinds of spins so they approximately demand that Delta where LS is very big is given a high mass to all the stage we have have a higher spins being bigger than two and make sure that what this remains is what remains is are all the states with spin two and therefore welcome probably the certainly inside space the only consistent directions of low energies of spin-2 particles are those of gravity so let this lead to a natural idea that that if perhaps you have a conformal field theory where all particles of spin bigger than to have very high conformal dimensions you probably record discover gravity in the infrared though this is a conjecture hasn't been proven anyway so but this is something to remember in particularly saying why if you had a white QCD for example will not will not have a nice gravity 2:11 for n equal to infinity because there we expect to have higher spin particles like the resonance resonance we see which have spin bigger than two and which are not much heavier than particles with spin less than X less or equal to two anyway so the guide approximation will not be arbitrarily good in those cases of course the string approximation will be correct now there is one important property of the spectrum that I completely I didn't I didn't mention and it's the fact that in in a weekly couple string theory or in a theory of gravity but for large radios you can separate the states according to single particle state and multi Multi particle States so in other words the states can be organized according to a list and approximate fog space where you have the state of one particle then you can add one particle will have some energy e let's say e for the one-particle states and then you can have two particle States which will have energy well the energy of the first particle you put in the inertia of the second particle plant some small Corrections so in a fear of gravity the small correction will be proportional to the Newton constant in the theory in this series we are talking when the gauge theory will be proportional the correction be proportional to one over N squared and this structure of large n gauge theories arises when you consider the the when you consider single trace operators so operators which can be written in terms of many sessions of the field versus operators which are of this form for example so you can consider an operator which contain two traces and versus these operators now if during the Larsen limit in the regime where the top coupling is fixed you can compute the anomalous dimensions of these operators and that would be a typically hard calculation and my next two lectures will be about computing the dimensions of these operators but then the dimensions of these operators will be just the sum of the dimension of this operator plus this operator plus some small correction okay and again this is the structure which matches the gravity structure and it essentially boils down to the same thing I mentioned here about planar diagrams so that darshana Series in in the planar approximation correspond to string theories again the approximation three level approximation that we have this expansion in terms of the genius of the diagrams translates into the statement that we have the expansion in terms of the genius of the world okay now now kind of weakly coupled theory ever have a gravity dual well a weekly couple Theory always contains a weekly couple gauge Theory although always contains operators of this form the plus two to the s Phi where let's say you have two operators Phi Phi could be asked I could be a scalar and I filled in the adjoint and so on and we act with derivatives on these two fields and we can make a primary operator out of this thing so something which it transforms in an irreducible representation of the conformal group and these operators will in a weekly couple theory will have a dimension which is typically small so it will be a further s so Delta will be a further s I mean and so this is not much bigger than what you could have let say s equal to three for example and you will have a small dimension so weekly couple theories cannot how well cannot be well approximated by gravity she will have really to go to a strongly couple theories where these Corrections that we get so typically we will get some Corrections to the dimension which of course depend on the spin and in theories that have gravity duels these Corrections become very very big for all the operators that have spins bigger than two so getting larger normal estimations is fundamental for getting a good gravity the world that's what I'm trying to say okay so now now that we discussed all these operators we can start computing correlation functions so in conformal field theory is the natural observable are correlation functions of local operators of these operators we've been discussing and one of the main elements of ideas if they were an element that this is used all the time is the relationship between correlation functions of local operators and a certain calculation that will do in the boundary theory so let's first start discussing an operator which has higher normalize dimension correspond to a single particle of large mass then in that case well anyway before we say that the object we are considering is this correlation functions of local operators in the conformal field theory and now let's consider for example the two-point function so of x1 or x2 now by scaling symmetry or conformal symmetry this two-point function goes like x1 2 to the power 2 Delta times some coefficient which we can set to 1 by normalizing the operator properly ok if Delta is large then we have this equal to 0 we have let's say X and we separate two points by some distance in the X direction and we have a massive particle in the interior massive particle can be it will break which well approximated by a geodesic so we have a Jew disick that goes between these two points on the boundary now this follows from what I said before though perhaps not very clearly so I discussed before that we had these coordinates in ideaa space where a TS place look like a cylinder and there was a state that was a massive particle that was propagating along the time direction in the cylinder right and we also had this state operator mapping in the boundary theory where a point here well we're where we mapped this whole cylinder to a radial evolution around around this object so we can view the say one of these points as the this was at this point here and the second point we can build us at some point at infinity and then we have this particle that propagates along the ceiling the same particle that we discussed before all that I'm doing here is making a coordinate transformation that maps those two points from zero and infinity to these two finite points here and then the trajectory is a particle corresponds to justice trajectory this is a semicircle and these are the genetics of anti-de sitter space either space or hyperbolic space here I'm taking this X to be spaces along a space like direction so everything I say here also witholding hyperbolic space okay and again that's the simplest way of getting the the geodesics in hyperbolic space is to do this mapping I just mentioned okay good so that's this is the picture now how do we get this result from this picture well we need to consider the bulk theory now in the bulk Theory if we have a particle will have some contribution to the partition function that goes like the action of the particle right and this action will be equal to e to the minus M times the length of the of that trajectory now you can come an able wait this length and so that's simple to do we as I said this this is just a semicircle so we have the semicircle we can have this angle theta now the metric is I the metric is over there so it's whatever the metric is for a circle which is just the integral D theta divided by that factor of C since we have the square root of the metric we have a factor of shuttle C of theta okay so that's the metric we need to evaluate that's the integral we need to evaluate the length is equal to this and we find that this infinite okay okay well that's typical so you always get infinitive and confirm and fill series but this infinity is due to the fact that well we're going all the way to the boundary the boundaries infinitely far away so this is just the same as the statement that of saying that the boundary is infinitely far away so I didn't mention that but in proper distance the boundary is infinitely far away so in order to do any computation you really need to we will need to put some cutoff you need to put some cutoff at Z equal to epsilon and really compute only up to Z equal to epsilon and then learn to take the thing the epsilon going to 0 limit now this is in correspondence to the fact that in field theories we often well we have to put some ultraviolet cutoff compute some quantities and then take the charge let's cut off 2 to infinity so what was an ultraviolet diversions in the field theory translate here into sort of an infrared diversions because here this is due to the fact that the volume or the length in a TS are infinite so it's an infrared story but it's an ultraviolet stirring the bond in the boundary theory and this occurs over and over again this is just the simplest incarnation ok so you can so we need to integrate only up from C bigger than epsilon and so if this custom let's say size X we have to integrate that to a theta which is a further epsilon over X and so this thing will give a log of epsilon divided by X ok where X is either this distance or half of the distance we don't care about the factors of 2 okay so what we get is for fixed epsilon we get into the minus M I forgot the factor of R here which will give us the correct dimensions times a log of epsilon over X and this log is negative so it would better get a positive sign here get that and so here you see a factor of epsilon which will get epsilon to some dimension and if we did we were to keep track of all factors of two we would get a two here and so we'll get up to Ana world factor that depends on epsilon 1 over X to the 2 1 over X to the 2 Delta which was what we wanted so the X dependence is precisely the dependence we had there the overall factor that depends on epsilon so this epsilon to the power 2 Delta is something we need to absorb in the definition of the operator and it's what's given the operator its dimensions ok very good yeah that's an exercise so yeah I didn't explain why it was a circle but that is in the exercises I guess I didn't mention the exercise but I prepared some list of exercises and you should get them probably tomorrow or today do it fast enough well one of the exercises is to show that this is a success circle now now this derivation is perfectly correct when M R is much greater than 1 if M R is close to 1 again you have to want a mechanical version of this so you'll have to sum a roll path and so and so onto the path integral and that's just simply the same as solving the wave equation so in the classical wave equation in this place and again that's another exercise so so the exercise consists of so let me well it will be printed exercise so the the wave equation for a scalar field Phi it you might find it convenient to start first with massless fields it's a little simpler and then put a regulator at the scale epsilon and you put some arbitrary boundary conditions so let's say at C equal to epsilon you put some arbitrary boundary conditions Phi 0 FX and then you can solve the wave equation and what that is supposed to give you is the following so I'm going here to take the final dictionary so if you calculate in the quantum field theory you consider the generating function of correlation functions for the operator o that is associated to the scalar field Phi this is an integral in four dimensions on the boundary theory that is supposed to well its approximated by e to the minus the classical action for the scalar field on the classical solution okay that you have found by solving the wave equation and setting those boundary conditions at the boundary ok let may be a little more explicit so in the bulk we have an action of the form grad Phi squared if we have a massive field we also of course have its mass so this is the action this is the bulk action so we are supposed to do is find the equations of motion for this action so with these boundary conditions replace it here back in the action you will get some terms that diverge with epsilon blah blah blah and you get also some terms which correspond to the non-local part of the correlation function of this operator so in other words when you do this the easiest way to do it is in Fourier space we Fourier transform this Phi of T and so you will get them well you could also do it in position space but Phi 0 well anyway you can do it in the exercise or you can ask me questions if you want now this is I put this here this approximate sign but there is an exact version of this relationship the exact version of this relationship is given by computing the full partition function in the gravity theory with the corresponding boundary conditions so the partition function in the field theory as a function of the metric the four-dimensional metric or coupling and these couplings could depend on the position and the metric also could depend on the position is equal to the full partition function of gravity with boundary conditions fixed by the metric or by those four dimensional values metric so there is for example a 5 d metric and at the boundaries epsilon is equal to that four dimensional metric that was appearing here so here there is some G for the four-dimensional metric five dimensional metric is set to this value okay and so that's that's the full dictionary and in the regime where we can neglect the gravitational back reaction and so on and interactions between these five particles this is this is correct if we had the more complicated action here which included for example some v cubed term then I will have also contributions to treatment functions for example right and in general there will be a perturbative way to evaluate the right hand side here in terms of some diagrams sort of Feynman diagrams where you start from some point in the boundary where the operators are inserted here the boundary these are also sometimes called Witten diagrams where you have some interaction Phi cube interaction in the bulk and this this lines shall simply note the propagators from the boundary to the bulk okay now okay now it's very important in the story that they are we're supposed to sum over all geometries and we can also include geometries which are different from the original geometry different topology but have the same boundary and the nicest example is the example of a black hole so black hole is a geometry which is different than the geometry of durational ad space but has the same boundary condition so you can consider a black hole inside anti-de sitter space it simpático you'll have the same anti sitter space you had without the black hole but inside you have something with a different topology now you can also have things which are called black rains so in this situation where we have the Z direction and we have let's say the X directions along here we could consider a black brain so a black hole which is extended along the three special directions of the theory so this is the stage that is translation invariant in the three spatial dimensions and there is such a solution it's given in the exercises so that's some explicit geometry which here is very similar to a TS space and it has a horizon here the position at this position so it has a horizon and sub positions is zero and the position of the horizon is related to the temperature in the boundary Theory and now what could be the relation between the position of the horizon and the temperature do you need to work any you need to do any work to find this relationship at least the scaling what determines their relationship is it for example of temperature to the three temperature to the to the minus one right is determined by scaling right so we need to understand under a dilatation symmetry these case like mass see we said sketch like length so with minus one okay okay no work that's nice okay there is a numerical coefficient which you cannot get this argument inversion now then we have the free energy which which can value obviously now that you learn how to calculate this using scaling you also know how to calculate this guy and so what is how do you calculate this one well first of all you need to make an important assumption well which is kind of obvious which is the fact that it would be proportional to the volume in the three dimensions right so it's infinite because we have infinite three dimensional volume so it's proportional to the volume in the three dimensions right and well so this is the total free energy so this is the thing that appears in the action is dimensionless and so the rest of course has to be better to minus three or TT q okay and then there is some coefficient here in front okay now what was the coefficient in front be how much calculations do you need to determine this coefficient in front so like that we would like to understand the dependence on the Newton constant of the coefficient in front so this is some calculation that will come from evaluating the action of some solution the action is just the action of the Euclidean version of this black hole that will give us the free energy and it is well the action of gravity this one over G Newton times the action I mean the usual action squared of gr + blah blah blah right so one thing we see is that well it will involve 1 over G Newton so because it's whatever action we had there none of what we said so far depended on G Newton the black hole solution doesn't depend on jinwu tone we evaluate this will get something and we have some G Newton in front because the same thing we get here is what's going to give us this bug this part right that we already determined and all we are determining now is 1 over G Newton now but G Newton is an object which is dimensionless in the 8th dimension fools in the bulk theory and it should become dimensionless by absorbing powers of R the radius of ideas now dimensions in the bulk are not related to dimensions in the boundary theory that's important so we are not going to cancel the dimensions of G Newton by some power of T that doesn't work because recall that the rescaling what it does is an isometry in ideas right it taxes isometrics in ideas they don't have any dimensions if you wish from the bulk point of view so these dimensions of this guy have to be fixed by power so far and in fact dimensions just that are cubes anyway so this is a pre factor and now if you compare this with a free field theory so in a free field theory you get something similar to this so beta F is something similar to this except that you have the affected number of degrees of freedom so right so I mentioned in the beginning of the lecture that the effective number of degrees of freedom is personal to this number and if you wish this is the derivation of that fact ok so when we consider the thermal partition function of this object which is computed the thermal partition function of the boundary Theory is equal to the a hulking Gibbons Hawking partition function for the gravity theory corresponding to this black hole and we get this answer and are there any questions now in theories that come from gauge theories this number is typically going like N squared in the of course in the free theory we can calculate it but in the bulk Theory well we have this formula but we can't necessarily match the numerical coefficient the numerical coefficient could be some function of the tossed coupling which in principle we have to compute yeah maybe I guess maybe at this point we can make this plot so as a function of lambda you can calculate that an effective over N squared and for one particular example of this duality which is the case of n equal to 4 super-yang-mills which we'll discuss in detail tomorrow we find that if we call this one here that's the value at weak coupling this is a function of the tough coupling which goes between 1 and 3/4 and we really only know these two parts of the function so we know what it is a very strong coupling we know the first string correction we know what it is a very weak coupling and we know the first correction they are good coupling now what I will discuss tomorrow I will discuss similar calculations but of anomalous dimensions of operators where we can actually compute the full function that interpolates between Witkin strong coupling okay so far we've been discussing there are one more thing to say about this black holes can erase prolly here we can calculate correlation functions in the presence of a black hole and so we have this black hole and we compute the correlation function so as we said the correlation function corresponds to the mission of a particles and then a reabsorption of the same particle and now if you emit a particles and there is a black hole what will the particle want to do yeah I will want to fall into the black hole of course so it's very unlikely that the particle would come back out right and the longer you wait so if you send it in and you would say oh wait wait wait I want to take you back then probably the fact that you have the black hole doesn't matter too much but if you wait for a long time then you experience I CLE will see the particle so in fact compute such two point functions you find that they will decay exponentially with time so this is some time T with some number that and to compute the number you really need to do some calculation but of course you can compute how it depends on the black hole temperature that's not too complicated same story that we discussed before but the precise numerical coefficient you have to do some calculation okay it's some number actual number divided by beta this is the same as what this discussed before and that's how they will decay so they decay because the particle wants to fall into the black hole and in the language of black holes what happens is that this long time tail so this number here it's related to the so called quasi normal frequencies of the black hole so here you can consider excitations around the black hole and they're not quite energy eigenstates because they are falling into the black hole so the frequencies of those eigenstates are have a real part and they also have an imaginary part and this imaginary part tells you about this long time tails okay good now in the particular case that you choose this operator to be the stress tensor this correlation function tells you something about how how the black hole responds to small pushes so you push the horizon a little bit in one direction and then the fluid this thermal fluid in the boundary Theory will respond in some way or the horizon itself will deform in some way and it will lead to different expectation values for the stress tensor later times so you can view these two insertions of stress tensor as saying well you deform the metric a little bit to give a little push to the fluid and with this one we just measure how the fluid is moving well I guess I'm probably going a little fast but I probably should have mentioned that of course this finite temperature configuration is breaking the boost symmetry of the problem right so you're you have your four dimensional conformal field theory you raced it to finite temperature now it's filled with some thermal gas or fluid it's not really a weakly interacting gas I mean it's something that where the particles are interacting very strongly and now this fluid will have properties with us any good fluid it will obey the navier-stokes equations and so on and you can calculate the coefficients opinion this navier-stokes equations like for example the viscosity by doing such calculations so the viscosity tells you how you're dissipating energy and these exponential tails are related to that so they are related to how if you shake this a little bit how quickly that the information about that shaking dice out and that that's related to dissipation now in general the the slogan is that falling into a black hole is the same as dissipation in the in the under theories and the bond theory you have this thermal gas and you start with some excitation localized excitation this excitation load slowly thermolysis and becomes equilibrated with the rest of the gas and that process is described in the dual gravity theory by a particle falling into a black hole okay so you can compute this viscosity the shear viscosity which is the only viscosity you have in a conformal field theory and it's related to the absorption of absorption of gravitons and absorption of gravitons by a black hole our report they are proportional to the area but now you remember that the entropy is also related to the area so you think that it might be a good idea to divide the this viscosity by the entropy and indeed you get some number which this turns out to be a pure number which is 1 over 4 pi and this is a general result in in any theory that has a gravity dual and it's correct only in the gravity approximation so once you start including corrections higher derivative corrections to gravity this gets corrected and could get corrected with plus or minus okay what time I'm supposed to stop now or okay as I wish okay well the one thing I wanted to mention is go back to the original diagram for the warp factor that I drew and and try to say what it corresponds to so in the beginning we threw a word factor which had some minimum here right now this is the typical situation you will get when you consider confining theories so some theories where there is no masks up I mean such a word factor does not have the conformal symmetry anymore so these theories will have a mask up now here the word factor is not that goes to infinity again but in the typical situations what happens is that the space ends here and there are different ways in which space can end typically in the examples there ISM there are some extra dimensions let's say an extra circle which has some fixed size here and infinity and then when you come here it's shrinks in a smooth way together with the radial direction in such a way that gives you smooth geometry that's one way you can work I mean in there but basically the space ends here and then if you consider a particle the minimum energy for a particle of mass M will be achieved here at the minimum and that minimum energy is nonzero so I need any excitation you typically want to put here even a massless particle like an Atlas graviton will be forced to have some wave function which is localized in this direction for the same reasons that we saw in global ideas so this energy will be nonzero and from the point of view of the four-dimensional Theory you have a non zero mass for the particle so in this situation you typically have a mask up now sometimes in some very very specific examples there might be massive degrees of freedom that have to do with flood directions of the field theory and so on but the typical situation without supersymmetry and so on is to have a mascot unless well typically if you don't have a mask up there is some good reason either supersymmetry is protecting it or maybe there is some broken spontaneously broken symmetry something like that now I mentioned that the amass described five dimensional graviton gives rise to a massive spin-2 particle in four dimensions and you can do the counting of degrees of freedom to check that indeed this is consistent the number the numbers match between five and four dimensions now okay so well a few more words about the radial direction I I think I can't emphasize enough perhaps the importance of understanding this connection between positions in the radial direction and the sizes and so on on the boundary Theory so let me emphasize it once more we mentioned that we have this metric this is the metric of idea and all the computations that we do usually in what we do when we match two sincere quantities we are typically looking at something and approaching the boundary at fixed distance X so we will approach the boundary at fixed X distance this is sometimes called comoving distance or and so on I mean it's called like that if you think in terms of the sitter space so this metric is very similar to the metric of the sitter space so d squared is ideas the sitter space is the same thing except that this is a minus sign so the S here and so now we typically Strawn like this ADA here is less than zero is conformal time again an infinite time to the boundary the boundary here in the sitter's is at infinite future this is just described in an exponentially expanding universe this exponential once you here go to the log of Etta so it's exponential in the log of it it's a power here and we have this is the end of the sitter and here is the very far plus and again this fixed fixed x-coordinates correspond to what's called fixed common distances so you have two observers who are just following judy six they would follow this fixed x distances okay now you are probably familiar with the fact that in the sitter this distance is called infinity in the far future and the proper distance goes to infinity and here again the proper distance is go to infinity now in the case of the sitter we know we have horizons and that anything we and this whole region becomes causally disconnected from the observer that is sitting here and so any perturbation that we that we do here will not affect the observer that is over here so these two universes will one stay with which a certain time they will evolve sort of separately this is a separate universe argument now here we have exactly the same story governing this behavior as a function of the cutoff so we do some calculation at fixed x and then when we start moving out the same mathematical reasons that here lead to the two this separated universe blah blah blah imply that devolution as we change epsilon is going to locally effect each of these two guys separately okay and that's why we we can extract so we are computing for example correlation functions we can absorb this whole evolution in epsilon as some local renormalization of the operator and so on and it's also related to the fact that we get the local since here in the boundary so we have the possibility of changing arbitrarily the metric here it will not affect how the metric is approached how the limit is approached here I mean it will affect the non-local correlations but not the local behavior of the theory so now there are some limits so there are many similarities between the city and anti-de sitter and many of the perturbative calculations you can do are identical you can translate very simply between one and the other now there are some differences and that some some of the differences are that well we can choose examples where anti-de sitter is completely stable will not decay but the sitter in all the examples we know string theory will be k and there good reasons for thinking it will always be k and and well also in the case of fantasy there are examples where we know that the well conformal field theory in a sitter we know the dual conformal filter in the sitter we don't know whether any conformal theory will exists or not but one thing i can resist mentioning is the fact that the quantity that appears there see gravity with fixed boundary conditions is just nothing else than what you would call the wave function of the universe so if you let me draw this picture this way now we have now a Euclidean space let's say hyperbolic space some or some locally hyperbolic space it could be globally different and we could have perturbations level de and that quantity there should be viewed as the wave function with some fixed metric here in the boundary right so that's another way of thinking about that object see gravity of course it's the wave function of the universe in some Euclidian regime the one that appears in the theater is the wave function in the Lorentzian regime more well real times where things happened okay and so now I like to mention one more thing one more general thing and that's about global symmetries so we often encounter field theories quantum field theories with global symmetries now imagine you have a quantum field theory with a global symmetry then there is the corresponding current conserved current and it turns out that in the dual IDs space you have the current is associated to gauge field mu for this this mu now has five indices rather than four so we have two corresponding gauge field the global symmetries and the boundary corresponds to gauge symmetries in the interior and that's good because global symmetries are exactly conserved right they cannot be violated in the Boundary Theory but we know that if we had a global just purely a simple global symmetry in the bulk we black holes could violate this global sim at the formation and evaporation for black holes could in principle violate this global symmetries but the formation and evaporation of black holes cannot violate gauge symmetries and so that's at least one necessary consistency condition for this to work is that these global symmetries are carried by gauge fields which black holes can also carry so they can also carry global charges and while similarly if you have a super symmetry you'll have super gravity and so on okay I think I'll stop here and continue with more detail so next the next two lectures will be about n equal to 4 super-yang-mills and we'll discuss that case in great detail ok thank you

Channel: Institute for Advanced Study

Views: 62,609

Keywords:

Id: sYqtGXN59uE

Length: 83min 35sec (5015 seconds)

Published: Mon Jul 10 2017