 Okay, good afternoon everybody. Okay. Good afternoon. So welcome to today's colloquium I'm very happy to have a professor on as a BC as the speaker today So I'm as a joined the high energy cosmology and astroparticle physics section of ICTP very recently just four months ago and She will be talking about It's a very interesting title the conformal bootstrap from boiling water to quantum gravity So an essay before coming to ICTP. She was an associate professor of theoretical physics at Uppsala University and She did her PhD from Niels Bohr Institute, and then she was a postdoc at Oxford and also at Harvard And now we are very happy that she has joined ICTP ICTP has been actively pursuing the policy to increase the number of women scientists on our faculty and We were happy very happy to get somebody with her accomplishments to join us. I know her She is close to she works in areas very close to my own research So I'm familiar with some of the things that she has done So as you know conformal field theory has been an extremely important tool For analyzing all kinds of physical phenomena, especially for example even in condensed matter physics The boiling water as she said at some way high pressure at some second order phase transition critical point is described by a conformal field theory but one of the really amazing surprises of the Maybe last 20 years is that conformal field theories of a very special kind are Have at exactly equivalent to a theory of gravity and this is really a big surprise because Conformal field is a normal field theory Things don't fall in a conformal field theory and it is equivalent to something where things fall and this connection Anise wrote an important paper one of one of our important contributions is to really extract properties of gravity at At the quantum level by analyzing the conformal field theory. So it's a very interesting way to learn about gravity By analyzing conformal field theory. So I hope it's probably one of the topics that she will come through towards Maybe end of her talk So, so I'm very happy and there's a floor is yours. Thank you very much. Thanks Thanks for So thanks for the nice introduction and for giving me the chance to talk about my work and what I'm Working on since the last 10 years almost So actually a tissue spoiled all my talk But I'm very happy that at least we are all on the same page. So I would like to start this talk by something that you might be all familiar about. So I Put here the face diagram of water. So Here you see the pressure here. You see the temperature. So in this part Here the water is like solid these two lines in between these two lines water is liquid and all in this part here the water is on his vapor face and Actually, you see that these lines they go up here But the face diagram here stops and this is not a type or a problem of my drawing here actually the Face diagram really stops and this red point is very important and it is called Critical point So let's try to understand why this point is important and essentially the next 10 or 15 minutes are devoted to understand why this point is important and what are the property of this point here so This is this critical point is the end of the face equilibrium curve and in the neighboring region of the critical point the physical property of the Liquid and vapor which are the two regions which are close by the critical point to change dramatically and The faces so the liquid and the vapor they become very very similar to each other so the idea is to understand again why this happens and why And how we can describe this critical point more in detail. So let's try to understand again a bit better that So in order to see that Oh, sorry, we would like to To study some fluctuation of the fluid density which I will denote by this DRO here. So you see that Here we are studying this fluctuation at different positions and we see that actually if we go to longer and longer distances which are Measured by this parameter psi, we see that these fluctuations behave in two very different ways So in particular, we see that there is some Exponentially suppressed behavior here where when the distance is much larger than this parameter While there is this power low behavior when the distances are much smaller than this parameter psi this parameter psi actually changes As this very interesting behavior when the temperature goes to the temperature of this critical point So this TC labels the temperature of the critical point here. So this is the Coordinate of this point TC here now when T goes to TC this Correlation length become infinite So what does that mean? It means that near the critical point and at fixed pressure They correlation length diverges. So it means that all the scales of our problem drop out now From this kind of Relation, it seems all a bit abstract. So in the next slide, I would like to show you a Simulation that so this is a real simulation that has been made by this person here I'm not sure if this slide will be available But you can click here and go to the person who made the simulation and here we can see that there are three Blocks, so this is again the critical temperature and we see how if we zoom in or zoom out from the system So this is exactly the same system that I had the critical Sorry the face diagram before and you see on the left. We are a bit below the critical temperature on the right We are a bit above the critical temperature. So let's see What happens? So here we are essentially you see this B is denoting how much we zoom out So we go we go we go and we are evolving to some situation in which We have this kind of equilibrium. So we go a bit above So you see we are not very far from the critical temperature and you see that now we go into another phase So we have to think about the white dot to some Vapor phase and to the black dot as the liquid phase so instead of the critical temperature You see that nothing happens if we go it seems that we are repeating are repeating the same pattern So this is very interesting Maybe I didn't put the video until then this is very interesting because we see that Exactly you have to be exactly at the critical temperature that something happens. So there's something that happened It's essentially that the if we zoom in or out nothing changes So we see maybe we can go back for a second. So we see that Here while here we go to a some very order phase at the end of this zooming in and out and on the other side here If you remember we had some like all black Configuration so here we go to a very disorder phase When we are exactly at the critical temperature, we keep seeing exactly the same object This goes again in the same description that I was saying before that essentially when we are exactly at the critical temperature Lord the scales drops So if we are very very close to the system and very very far to the system We exactly see the same patterns and this is what this simulation is showing now the Another important feature of this kind of point is that Critical exponent all the critical exponent that we see before we saw psi we saw new They describe the behavior of physical quantity near continuous phase transitions are all divergent So all the critical exponent s t goes to tc. They are all divergent And also there is something else that if we change the kind of substance that we look so in If we take different liquids for instance or superconductors While the phase diagram that I show you before is different We see that the critical exponents are the same so the PC and tc may be very different But this exponent psi that I show you before eta that I show you before they are all the same This goes under the name of Universality which makes this critical point even more important Because we can study this critical point only the critical point to kind of Discuss all this correlation length that I show you before for a lot of systems so What I would like to to discuss in the next few minutes is to understand why there is this Universality the specter fact that we are talking about very different systems So Okay, this is what I said So for instance if we consider water this this is what we show before This beat has been measured to be this number here but if we go to the spontaneous magnetization of uniaxial magnets while these TC is different from the one of water we see that the behavior of these Mt which is the magnetization is of the same type and in particular this beta is exactly the same number So as soon as we find a way to compute this beta We are able to describe this d row t But we are also able to describe the magnetization of uniaxial magnets So with a number we can describe a full range of systems So why this is the case? Why can we describe so different systems with the same class of numbers? so the reason for that is that those here I Put this drawing of boiling water here. I put the magnets they belong to the same universality class So this seems a triviality, but what does it was that what does it mean? It means that while the models are so in particular in this case Water and the magnets at finite scales are very different So you see it already from this picture that these two pictures are very different in the vicinity of the point they the In the vicinity story of the critical points the asymptotic phenomena which are described and modeled by the critical exponents are the same and In particular again in all models falling in the same universality class We have that the critical exponents are the same So when this happens actually this always signals some symmetry which is emergent and in this particular case The symmetry which is emerging is a scaling variance You could have thought already about that from this small video that I show you So you saw that when you were very very close to the system of very very far as before like when we were going We are zooming in on zooming out to the system before we had that in the middle Panel here where we were seeing the same object we had some scaling variance so because nothing will depend on the scale any longer and Actually, this the scaling variance in this invariance under rescaling which are dilation of all coordinates by a uniform factor So it's essentially scales all the coordinates by us the same factor, which is this lambda here Okay, so a class of transformation which are scaling variant Are called conformal transformations which are the transformation that I will be interested in and those kind of transformation are pretty interesting because while they do not preserve distances they do preserve angles and The way to think about this transformation I will see I will show you another way of thinking about that a bit later, but essentially at each point to locally they are Described by a rotation plus a dilation with a rescaling factor, which depends on the axis So is what we saw before but they rescaling this lambda depends on the points where we sit and in particular in 1970 Polyakov conjecture that the scaling variant theories that describe the critical points are actually conformal invariant which is something that we are going to discuss and Again now we see that the description of Fixed points so the fixed points that we show before which are scale invariant boil down to Classifying all the conformal field theories and what I'm going to do in the next part of the talk is to understand how exactly to compute these fixed points and So here to Let give you a maybe a better idea of how this conformal transformation work I put this This map so this is a map that comes from the 16th century by this guy here, which actually I Discovered while preverting stock that this is not his name. I don't know why it has been Latinized so he's a Flemish Cartographer which was called the Kramer but now we know him as Gerardus Mercator and This guy was the first person who did the conformal transformation in order to find Maps and it was the first person who Gave the name of the Atlas to what we call Atlas now. So you see here that So the objective of this guy was to make maps which were very useful to sail So he wanted to have the maps useful to sail So what is very important if you want to sail is to understand the what are the angles that you need to to Maintain so he decided to have this 2d map in such a way that the angle were preserved But as you can see shapes are not really preserved because there is some distortion and the distortion is bigger and bigger As we go to the south and north pole. So you see that the south pole is much bigger than we expect to be so this is an example of a map which is conformal and There are also still now. It's not something that ends in the 60th century. We use this kind of map to sail So in particular Yeah, you can see how things are are changing from the equator north and south Okay, this was just to give an idea of what the why this conformal transformation can also be visualized a bit Again, so let's go to the second part of my talk, which is how can we find this critical exponents? So now that we understand why they are important. How can we find them? So the idea that I'm going to describe is an idea which is based Mostly on the symmetry. So the idea is to use the symmetry of our of our system in order to Understand how to fix this critical exponent. So in particular, I'm going to use strongly this Universality, which means that we I we don't think we don't need to think about the specific realization of the System that we talk. So in particular, I'm not going to discuss water or magnets. I'm going to discuss a specific I mean a general Theory that has the same symmetry So in particular, we are going to study conformal theories Using an approach which is based on symmetry and with little or no input at all about the Microscopical description of the theory. So studying conformal theories means that we need to classify all the possible conformal theories that we are going to discuss. So this is just a scheme. I'm going to Say later what that means and from here we can sorry we can find all Or at least some of the critical exponents and with the symmetry We are using the fact that our theory is conformal invariant and that exists a special Object, which is the operator product expansion that I'm going to discuss later So on one hand we would like to find the critical exponents which will Characterize all our system and on the other hand we are using the symmetry So on the right side of this slide you see the symmetry that we are going to use on the left hand side We see what we are going to get Okay, so let me start by Specifying a bit better. What are the observables? So what we are going to study in these conformal theories and in particular The conformal group, which is so d comma 2 where d is the number of dimension is generated by Translations so the usual translation that you know Lawrence transformation Which are essentially this Related to the spin or so to the rotation dilation which are the transformation that I discussed before so which are Where you go from a point and you dilate with a factor your position And then some special conformal transformations which are the one that I was trying to discuss later So those transformation are essentially an inversion of again the coordinates Combine with a translation and then inversion back So this is somehow the way to think about special conformal transformation. So This is the symmetry so those are the generators of my transformation, but now how can we describe Objects so the observables in these theories. So we have operators So the operators are labeled by two number these of these delta here and the li what are these objects? So delta is related to the dilation. So it tells us how How these operator changes under dilation and this li which are is that they're related to the Lawrence transformations in addition to that there are What we can say interactions and those are related to this Operator product expansion that I was telling before so we can take two operators and Understand how what happens if we bring the two operators close to each other. So in the case of our conformal theories so we can take these Two operators one is labeled by i and j that essentially encode both these two numbers here And we can expand these two objects in terms of another operator here This okay, and then here we have a number a set of number which depends on the two operator that we are bringing close to each other and of the third operator that we have on the right-hand side and This function here this f which depends on the position where these o seats and this o j seats and Okay sits and the dimension and the spin of all these three objects Sorry forgot to write that it depends also on the spin But the most important point you don't need to remember anything about that Just the fact that this function is completely fixed by the symmetry structure So this c is not fixed by the symmetry but this function f is fully fixed by the symmetry and Pictorially you can see that in this way. So you have this o i o j then you bring it close to each other This point is this c i j k and this ok is the other operator that you get so you can think about Having like writing this object As an infinite sum over all the possible operator that appear here and then instead of having two operator You essentially have an infinite number of other operator with this weighted by this number here c i j k and Again this object here is completely Fixed by the symmetry notice that you may know Feynman diagrams Maybe and these object is not a Feynman diagram. So it's just the same way of Writing it, but it doesn't have anything to do with Feynman diagrams. Okay, so this is the operator for that this function that He's acronym is OPE Okay, now we would like to construct again the correlators which are really the observables of this theory and actually the conformal symmetry is strongly constraining two and three point Correlators, so in particular you see that two point correlator is essentially fully fixed in meaning that is completely Kinematical so if we it depends essentially only on the on the coordinates and Of this delta o this delta o is the number that I was showing you before Which is called the conformal dimension which essentially is a property of the operator that we have in our correlator So two-point function is essentially fixed that three-point function are all also fixed up to this number here Which is called the three-point function coefficient now you can say okay, but we have already seen this number Why do you want to use the same? Same letter to denote two different sets of number, but actually that's not a typo those numbers are exactly that OPE coefficient Coefficient at the tenth are saying to this expansion here So the three-point function you can clearly see maybe from here So if you put another object another operator here We you get exactly the CIJK and this three-point function coefficient are exactly the same of this object here so again two and three-point function are completely fixed as soon as we specify the operator and the OPE Which is again completely dictated by the symmetry of the theory for instance. I wanted to show now how this Delta sensei are related to critical exponents because remember again that we said that we wanted to understand The critical exponents that we had before so in the case of the icing model Which is the one that I show you at the beginning like the one of water and the one that we saw there the small video the dimension of the of the first Operator that we have is one half plus it over two so eta which was Which we saw in the very first slide is exactly given given by a dimension of one operator that we appear So this is the way of from the dimension reading off The critical exponents so there is a map between the dimension of our operator So we can compute the two-point Functions which are again observables and then we can read off for the critical exponents and Again those critical exponents are the critical exponents of all the universality class of the systems that belong to That include the icing model Okay, now let's go to the next point. So what happens to four-point function So we saw here in this slide that again this space-time The space-time part of the two and three-point function is completely fixed what happens to four-point So conformal symmetry is not strong enough to fix completely the correlator of four points. So now here to avoid cluttering the formula I completely removed the Letter down I jk. We assume that all these objects are identical and the only conformal symmetry what it does it constrain this Correlator to be a function of some UMV and UMV are just this combination of distances over this Factor and again it depends on the conformal dimension of these objects here and UMV are at this form because again this is dictated by the conformal symmetry and The Four-point function is the first one which contains some dynamical information which is encoded in this function g of UMV now we can say okay, but why don't I use the Information that I said before meaning why can't I think about expanding using the operator product expansion of these two Operator into an infinite sum of one operator and the same thing On the other two operator and this we can do that. So let's try to do that together So we can take these two operator bring them again close to each other and expand it and then we get this This kind of diagram that we saw before We can take also this other two operator and do exactly the same thing. So we gain again two factor of C's and We gain also an infinite sum So now the way that I'm labeling this object is again through the dimension and the spin of this operator that Appears inside the operator product expansion. So we can rewrite a four-point function as an infinite sum of coefficients of this OP coefficients or three-point function square so we get one factor of C each time that we do an operator product expansion and Another function this function. So this we get it from the definition of this function G that I show you before But this function G delta L U and V It essentially encodes these two functions F that I had in the operator product expansion But again, you don't need to remember anything of that The only important point is that this function is completely fixed again by the symmetry So as soon as we specify the symmetry of our problem this G is completely fixed So the only information that we don't know is this C square and this delta L Which depends on the kind of operator that we have inside So this function of this object depends only on the theory that we considered and the number of the space-time dimension That we are in and on the kind of operator that is inside here So we manage essentially to rewrite the four-point correlator in terms of an infinite sum of three-point function and two-point function Because remember that this dimension and L here can be written in terms of Can be rid of essentially from the two-point function So we reduced a four-point function into an infinite sum of three-point functions and two-point functions Okay, so now why am I doing a lot of fuss about that? Because we could have done also another thing. So the operator product expansion is associative So what can we do? This is not very hard to do But what can we do? We can take this OP here as I showed you earlier, but I can also so which essentially does Can be like pictorially depicted in this way So we are taking this X1 and X2 is denoting the fact that we are taking the OP between these two operator and X4 And X3 are these two other so we can we take this OP here But we can also take another OP Which means that we consider we take X2 and X3 and X1 and X4 So like this one here and these two expansion here That should be identical because again the operator product expansion is associative and on and on the other hand These two four-point correlators should give the exactly the same result So these two kind of expansion that should be completely identical So let's try to understand if this equality brings us some kind of Way again Our objective is to find this delta and L and the three-point function coefficients so Sorry another way to write this object is to to write it in this way So remember this left-hand side that is the way So the left-hand side is this object here if you remember this object We can be written as an infinite sum over delta and L of some coefficients this C delta and L times this function g uv and The right-hand side is exactly the same object where Here we have just the exchange of v with u because this is essentially a symmetry when we take When we take the different OPs, this is just a kinematics It's very simple to see but I didn't want to write it here to clutter again the relation But this is just a relation in formula of this picture that I have before and This u and v to the delta O comes let me go a bit back maybe So this comes because if you remember the four-point function is also this term here So now you and we are this object. So if we swap The label 1 with 3 we swap u with v and we also get this extra factor Which is essentially the u over v to the delta O that we have in that relation there But it's not it's not super important for what I'm going to tell Okay, so this is essentially a consistency condition on this Object here so on the c squared and L and on this delta and L here notice again that these functions G of u and v and v and ur are completely fixed by conformal symmetry So those are fixed that the only thing which is not fixed is the c and the deltas This fun this relation seems very simple So it can be it can seem that is completely useless that we don't Really know what to do with that. But actually that's not Very true. So in particular, let's look better at this equation So the first thing is that on the left-hand side and on the right-hand side there are infinitely many equations So you see that there is this sum here sum over delta and L and This sum runs Over delta and L which goes from some value to up to infinity So there are infinitely many equation and also they conformal dimension. So this number delta is a real number so This sum is infinite and over real numbers. So this is the first complication of this equation here The second complication is that okay, you can say I can truncate this sum here up to a certain point Let's say I cut it Until 10 and here I cut it until 10 and this should be satisfied. So I don't really I can simply Find some c squared and some deltas from this consistency condition But that's not actually true because one term on the left-hand side Is not mapped into one term on the left-hand left-hand right-hand side Sorry because remember that here this function is different on the left-hand on the right-hand side So you have U, V and V and you actually One term on the left-hand side is mapped to a infinite infinitely many terms on the right-hand side So this property is Is makes these relations much more complicated to solve but actually this is very interesting and we will see later why this is very interesting and The third thing that I want to mention at this at this point here is that This relation here does not depend on the microscopic description of the theory what that means It means that I don't need to specify again if we are talking about the water or the Magnets or any kind of system that I'm talking about. I just need to talk to say which is my Symmetry group and that's it essentially just to specify what is the number of space-time dimension that I'm working in and Then I don't need to add any extra input about the specific theory In particular in general conformal field theories can be strongly coupled So I don't even need to I don't need to be in some perturbative regime or anything like that so those are valid that any value of Of the coupling constant or in general for any particular microcovic description Okay, so now let's try to understand how to use this relation. So those relations are pretty old. So they are maybe I Don't know how much about 50 50 years old and So for 50 years We don't know really a way of completely solving them in our In general form So we don't know how to find c square of delta L and delta from this relation here So we don't know really how to solve them in full generality But we have two ways that is what I'm going to discuss in the next ten minutes to Find some solutions to this equation Okay, so again, I want to stress that the unknowns of this relation are the c-squares and the delta NL So there are some particular theories some particular case So which are where the when the theories are unitary for which this c-square delta NL Are positive which means that the c are real numbers and for this particular case we can think about This this problem in a slightly different way So I don't want to enter much into the details But I just would like to let you understand the procedure that one can use in order to find some Information about this delta NL here. The only thing that again I'm assuming in the next few slides. It's the fact that this c-square delta NL is a positive number Okay, so the machinery is called again conformal bootstrap I'm going to discuss a bit later what it means about the idea is that we would like to find some Solution to this equation here. So at the end of the day, we would like to understand What are the delta NL by assuming that the c-square delta NL are positive that fulfill this equation here Okay, so the idea is that you can hand Some a box where all these balls are dimension and Spin and Lawrence Quantum number of a particular theory that you like you just put them in a box Then you take one of these blue Ball which has associated to eat a delta one and an L one So those are the dimension and the spin of this object. Then you put your ball To you give it to some scientist and you ask is this Operator which is denoted by a delta one and L one part of a CFT or not. So is this Operator does this operator belong to a CFT or not then what the Scientist do they put it into some computer. So you run through another an optimization problem And the kind of answer that you can get is yes, sorry, maybe it does No, it doesn't so that's the kind of Procedure that you have in order to to use this consistency condition notice that this is not So here I didn't put maybe because I felt pessimistic, but just because this is what he can give so he can say that those kind of relations are Do not allow to say if For sure they come from a conformal field theory They can only say that maybe they can come from a conformal field theory But they can ensure you that they cannot be part of this. They cannot be consistent with this relation here So if you get from the computer Sorry a pink Ball Then you are sure that the ball that you were that was coming from the box here Cannot come from a CFT so that you can iterate here in this slide I just show you that for One of these balls, but you can keep running the computer until you want and you will get some maybe no maybe no and so on so the kind of Usage that you can do of this machinery is Into it can go into this Into these plots so these plots has been made by so that this numerical way This optimization procedure has been revived by this group here Eric may be seen the room, but surely see interest so it's Here is a local person and The kind of plot that you can get so is here you have some dimension delta sigma Which is related to this delta sigma that sorry this is delta epsilon This is delta sigma and the way to think about this diagram is to say that Sorry, let me not spoil so that this purple regions are filled my babies and The white regions is from no what I mean is that if you have if you want to describe Dimensions of operator that belong to a conformal field theory. They cannot have a dimension which is smaller than 1.2 in epsilon by fixing delta sigma to be 0.58 while if they are here Then they can come from a conformal field theory now the point is Do we know any theory by say by assuming some other information that are In this plot here and the answer is yes So the 3d icing which is essentially the three-dimensional counterpart of the model that I was showing before is exactly in this region here So we can use despite the fact that this is just a maybe no plot So we can use this information to zoom in into this cross here to read off the critical exponents of the 3d icing so Though those are this is the way of to think about that before so you can run this procedure you can get all this Green and pink blobs Okay, so this is very nice because again by just we didn't assume essentially anything and we were able to find At least the regions of possible critical exponents again Those are numerical ways of finding this this regions another way is to use analytic methods again The the problem in this particular case is that there are these infinite sums that I show you before and That one term on the left-hand side of the relation that I show you before is not mapped into another term on the right-hand side But the idea is that can we overcome somehow this problem? The answer is yes We need to focus on a specific kinematic regime by kinematic regime is a specific Regime of you and me where those sums simplify So this is a very technical topic. I don't want to go into this Technical topic, but I wanted just to show you one thing which I think is very important and very nice so By focusing on a specific regime you can focus on the contribution for instance on the left-hand side of this equation of The identity operator so inside this operator product expansion There is also the identity operator at actually in particular the identity operator and this counts as One term on the left-hand side and we see one can actually compute that by assuming the fact that you have the identity operator in one of these Operator product expansion we see that we need to have an infinite sum of operators on the right-hand side which have Large spin meaning a spin that cannot be finite so the presence of the identity operator which always appear in this kind of operator product expansion is Ensuring us the fact actually is forcing us to have infinitely many operators which has to which need to have a Spin which has to be very very large. It cannot be finite Okay, this is This may seem like a very simple exercise to do, but it took again from this 73 to 2013 to be established completely The same The same machinery can be run for a lot of systems and in particular He can it is possible to reconstruct in some cases the full Correlator in some other cases part of the correlator and this works Sorry, this works much better in the presence of some perturbative parameter meaning that for instance you can take Some parameter in your theory, which can be whatever then this Perturbative parameter you can tune it and you can see that in If you have a perturbation expansion you can reconstruct the One loop to loop expansion and so on by looking at previous order to using again the same method This is a super technical issue that I didn't want to discuss in this place But if you are interested I recently with some collaborator wrote a review about that so this just to say that it is possible to do this kind of procedure and There is a lot of topics that have been approached by this method here But now I want to quickly go to gravity because now you can be very confused actually a tissue ready It's pulled a bit, but you can be confused why we are talking about water now You want to also talk about gravity? Let me just mention in a couple of slides Why gravity and why we really care about Quantum gravity in general also general activities are classical theory and in particular it describes How gravity work and in particular gravity is due to the shape of space and time So we can describe how Heavy objects move actually all objects, but in particular heavy objects move into Into space time and the the essence of gravity is related to the shape of the space time On the other hand we have quantum fish theories which are based on quantum mechanics And they describe how matter and strong weak electromagnetic interactions work So in particular here we focus on very big objects and massive objects and here we focus on very small object In particular, we go to very very small scales But there are particular case in which there are situations in which quantum fluctuation of the geometry Are important so for instance at the beginning of the Universe or in the vicinity of black holes where we would need to have the two description together So in particular we have very massive objects, but in a very tiny region of space so in particular the two Description need to merge but to do this merging dismerging is what has been dubbed by the name of quantum gravity because we would need to go into the Quantum regime of gravity is very hard to achieve in particular if you want to use some naive Method of taking into account the quantum correction to the Einstein gravity Which was the general relativity that I depicted here. You don't get any consistent result In particular all the object that you would like to compute are affected by infinities So if you it's impossible essentially to compute anything that you want to compute But actually there is a way that is what Tisha was mentioning before which is essentially a definition of what Quantum gravity is and the idea is that there is a story. There is an equivalence between a theory of quantum gravity which leaves on Curved space time which is under the sitter space to a conformal field theory Which lives on the boundary of this space here and those two theories are essentially the same theory and in particular which is the most important part of this slide here the observables that you can compute in the theory of gravity and the Observable that you can compute In the conformal field theory are exactly identical. So this provides us an operative definition of what Quantum gravity is. In particular since I already talked for 50 minutes about conformal field theory now we know how we can construct the Observables on the conformal field theory side. So by using this information we can read off the observable on the theory of quantum gravity. In particular, I mean I'm not talking about Very exotic things. These things have been computed for ages in some regimes. Now, there is one regime that where those these correspondences been very little explored and is This particular regime is very important in order to again use it as a Definition of quantum gravity. So let me just briefly sketch and then I will conclude how to do this kind of Computation and how can we think about this object? So we can again think about Space this is just an illustration So we have a gravity inside here and we have the conformal field theory at the boundary So what we can do we can use Again this analytic and numerical techniques that I was showing you before to the Conformal field theories that lives at the boundary of this object to infer information About the observables of the theory of quantum gravity here. So in particular we are able to compute This kind of a four-point amplitude. So I don't have a lot of time to discuss what they are But those amplitudes are observables in this theory of quantum gravity here so again These two theories are completely equivalent. So if we are able to compute the correlator At the boundary of this theory. We are able to infer information about quantum gravity again The techniques that one can use are Exactly identical to the one that I showed you before at least I sketched from the analytic side and I didn't write any References to this page because otherwise it would have been filled of name instead of this picture But this has already been carried out. There is still a lot of things to be done So computing this object. It was completely unattainable until a few years ago and Now with this technique by looking essentially at the symmetry of this theory So these theories in particular that sometimes You can study them also in the presence of another symmetry, which is called the super symmetry Which is even more constraining and this power Gives this extra symmetry gives us a lot of power and we can completely reconstruct This first box here. We can completely reconstruct the say Not completely we can reconstruct a part of this object and we have a way also of attacking all the other Objects. So this blob are essentially we can have a perturbation Theory along this line here. So this is the simple thing. This is the next to simple This is the next the next to simple and so on so and the idea is that we can completely reconstruct this object By knowing this one we can completely reconstruct this object if you would know this and this and so on and so forth And this is again something that is completely unattainable with any other technique that we know now Okay, so I don't have enough time and also I would need to have more technical ground in order to discuss how to do that in practice, but this Come let me to come to my conclusions. So what I hope I have Told you is how To essentially use this machinery of the conformal bootstrap despite the fact that I didn't really discuss how to do that In practice, but I hope I gave you an idea of the fact that there is a machinery that Allows you to find the critical exponents and to find at least in the numerical side Some constraints which are completely rigid and robust so they don't depend on any perturbative expansion And also one can find some analytic In analytic way these Critical exponents for a lot of class of theories so here I show you the magnets the boiling water This is supposed to be some superconductor. I just Google it. I don't know if this is correct And also for for instance black holes and those systems of quantum gravity. So thank you very much Thanks. Thanks and easy for this nice talk Okay, the floor is open for questions. So thank you Given I was wondering what if there is a Understood comparison between the applicability of the analytical versus the computational methods So given a theory and a symmetric content Is it clear a priori which route will be more successful in providing the more meaningful constraints or one has to try? You mean at the difference between numerical and analytic approach It depends on if there is so a priori I would say that The numerical ones are surely efficient to give you an idea of where there are Theories or not like you there can also be that I don't know There is only one theory into this realm for instance. There can be like a purple only a purple point in your in your Numerical side like in your numerical scan and then you can focus on that particular Articular theory by using the analytic approach. So you can use the numerical Techniques to scan to all the possible theories that can that you can have and then you just focus on a specific theory This is just more ideal way of behaving the other The more realistic ways that if you have some description for instance You have a theory which you have a large parameter which can be for instance You have an extra symmetry like an extra global symmetry There is the rank of the symmetry group But then you take this rank of the symmetry group and you do it very small rank or very large rank So you can take a perturbed of expansion around this parameter And then this is much more efficient to do it analytically because you have some parameter that you can control But if you don't have this parameter usually applying analytic results is very complicated. So I think that's a bit the the the way to think about Here Yeah, I mean I got missing this numerical part because you show this maybe So how you really know that you get the a conformal field theory because otherwise Then how you connect with the gravity side? Yeah, so I don't know. That's why I wrote maybe It's really a real maybe like nobody knows You need to have extra inputs to say if it's a yes or a no in the maybe Actually, no, if it's a yes or a maybe again So the way for instance for the quantum gravity how it works is that you can compute so in general there is this symmetry which is called a supersymmetry that fix some other Critical exponents or conformal dimension so you can take these numbers that has been already computed using other methods And this fix the point the region of this space where is Where they maybe becomes yes But you are not sure that in the region where it is maybe is yes or No in general. So there is not a priority. There is no way in which you can access this point Sorry, I didn't hear Yes, if there are other symmetry you can think about that But if there are no other symmetry, you don't know if there can be region So another way of thinking is that here I show you the consistency condition only using one correlator So the operators are all identical You can take correlators where the operator that you have are different So you can take a one or two or three or four and then you can study all these correlators together Usually what this study does so the consistency of all the possible correlators that you have in your theory is much more Constraining than one single correlator. So this region they just shrink Into smaller regions in general the problem is that this computational is much more Heavy so this requires Lots lot of time and other things but another way of shrinking the region is is that So and this you can see for instance if there are other symmetry you can even shrink the region more So that's another way You showed in the beginning an exponent that was zero point three two five something and it seemed come to come out of a measurement But from your talk, I assume that you are able to calculate this from some So did I miss this did you calculate this number using this theory or did you confirm this number? Yeah, so This is if you if you go it depends in the case of water Yes, you can do that the point is that the Number that I show you it has been Calculated using some Monte Carlo or lattice simulation that you can find and indeed that you can confirm that actually you can Find it even with more accuracy that the previous that the previous Computation using Monte Carlo a lot is because again you can use this multiple Correlators or not to really with the meter that I was showing but if you can use more Correlators but again using the same idea you can you can contrast it with the result that was known. Yes There's a question here So I don't work on this at all When you We're showing the consistency conditions. He said it comes from the associativity of the operators So if the operators commute does that give you more consistency conditions and a more rigid theory And then I guess on the opposite side if the operators aren't associative then what where do you where do you look? okay, so In all conformal field theories that they open the piece associative So the the only problem that I didn't discuss because it's a bit technical is the region of Where this expansion is defined? So this is not true for all the values of x you need again to be this expansion is true when the two The two points are close to each other So there is a regional convergence of this of this OPE. So this is an asymptotic expansion. So you need to be in a specific region so the Issue that I didn't discuss is the fact that those consistency conditions are true if both the OPE are convergent at the same time and What it has been proven is that there is a regional convergence of all the OPEs that are in place So there is a regime in which one can use the associativity of both these operator product sponge this I didn't discuss it's a very important issue and But it's pretty technical. So I didn't want to enter into this thing But there is a regional convergence of the two OPEs and the OPE is associative for this kind of theories here Can we we can take a last question the director is No Okay I have a very nice question like if you if I give you like Say theories A and B would you be able to tell me like whether or not they would lie in the same universality class? Like for example from some microscopic You need to see how they scale where the T goes to TC you need to go to the very first slide that I show you and You have to see how how the correlators behave essentially and then if you you can find that that they have the same The same coefficient then they belong to the same universality class This is one answer The other answer is that if now that we understood that you need to look at the symmetry of your theory So if the symmetry group is the same Then they are the same theory essentially they fall in the same universality class So you can look at the conformal symmetry that That Describes the critical point in order to say if they belong or not to the same universality class But you could also find that the correlators One last question No, I just wanted to ask you what is the current status of recovering Loop corrections to gravity. I mean, where is where is this program stand? So far in the picture, maybe I can go back here So far this has been I would say computed this is The last paper that appeared two months ago Essentially Tells a recipe on how to compute completely this but there are some coefficients that cannot so this requires an infinite set of stringy corrections and those are not all Computed but there is a recipe to fix them using integrability and localization So you need to use other symmetries, but you can essentially compute that these Only few bits of this are known Okay, and so it is a tradition of ICTP that after the I mean I tell for the students but also for an easy that so the others Leave but the diploma students stay around that you can ask more Private questions to But the food that will stay for you outside so and let's thank and yes for You