 OK. So, I would like to thank the organizer for the invitation. And in this talk, I'm going to speak about the local and the functional RG. The local RG is an RG approach developed mainly by Hugh Osborne, while the functional RG is the topic of this conference. The aim of my talk is to highlight and explain the connection between the two approaches. So, let's go fast a moment through the RG theory, because at the beginning I want to give a lot of statements, which are RG statements, which do not depend on the method. And then from where we will arrive, I will start to explain you how the construction of Osborne goes. And then I will make a connection with the standard construction based on the FRG. So, as we know, the RG theory is basically the theory of the flow in theory space. Every theory consistent with the symmetries. And then, once we have an RG approach, we know there are particular theories, which are particularly important, and these are the conformal theories in many cases. They are our fixed points. And once we know them, we know that around these fixed points, we can construct all the linear theory, and usually this linear theory around the fixed point we can approach perturbatively with conformal field theories or many other methods. The local RG and the exact flows, so in this case the FRG, are the only approaches that we know that are able to construct the RG flow arbitrarily far away from a fixed point. So, in that regime that I will call is the genuinely nonperturbative regime. And in particular, when you are able to construct the RG flow between points arbitrarily away from a fixed point, you can also construct the RG flow between two fixed points. And when you can do that, you can access very important quantities like the C function and the A function, and you can use the knowledge about this function to learn about universal properties of the fixed points, which are not related to the scaling region, but to be computed, they need to be integrated along the full trajectory connecting to fixed points. So, then the local RG of Yugoslavl is basically an approach which promotes the scale, the RG scale to be field dependent, position dependent, so it promotes the scale to become a scalar field basically. And then he uses vile consistency conditions to derive some non-trivial relation relating RG quantities. And RG quantities are beta function, A function or C function, and things like zomologic of metrics, and other quantities that naturally arise and characterize the flow arbitrarily far away from a fixed point. While the exact flow, as we know, is based on implementing an exact flow, which can be constructed, for example, along the standard way where we have the Vetterich equation. So, in particular, when the kind of problem that is intrinsically non perturbative is the flow between two fixed points, and in even dimension, one can define two interesting quantities. For example, in two dimension you can define the C function, while in four dimension you can define the A functions. So, these are functions that depend only on the couplings, and they have peculiar properties. For example, if you compute the difference between the central charge, between the UV fixed point, infrared fixed point, you will find the quantity, and this quantity will always be bigger than zero, because basically the central charge, the C function, counts how many degrees of freedom we have, and when we go down with the RG flow, we are burning information, so the number of degrees of freedom will decrease. So, the integrated C theorem is a general RG statement that tells us that delta C is bigger than zero. Recently, there has been a lot of work only in trying to extend this statement to four dimension, where the quantity that we call C is called A for historical reason, but it's basically the same quantity. And then, there is also the question if the flow of this quantity has monotonic properties, and for example, in two dimension, it is possible to prove that under certain situations, the flow of the C function is monotonic, so the beta function of the C function is positive. While in four dimension this statement is still known only at the perturbative level around certain fixed points. So, what I want to do is to try to introduce the approach of Osborne and then also the FRG view on these matters, starting from looking at the effective action in the UV and the effective action in the infrared when I basically follow a flow that connects two fixed points. And it turns out, and this is a characteristic of Osborne approach, but also the approach that I will use, that many RG quantities in RG question are easier to understand if you work on curved space, because even if you are interested in flat space matter theories, if you go to curved space, then many RG quantities start to become couplings in the effective action, but they are couplings of certain operators, which involve the curvatures, so are not present when we go on flat space, but will be present only when I evaluate a certain number of functional derivatives of the effective action. So, they will show up in flat space only when I look at the correlation. For example, one quantity, again the conformal anomaly of the central charge in two dimension will show up only the two point function. Something analogous in four dimension will show up only on the three point function or four point function of the energy momentum tensor. If instead we work on curved space, all these effects will be related to a certain operator, and in general these operators are non-local, because for example the central charge is the coefficient of the polyakov action. Obviously, if I choose a background which is simple enough, like a conformally flat background, then this will automatically introduce another field which is the dilaton, and everything will be local in the dilaton. So, the starting point is to ask which one is the general form of the effective action when I do this co-variantization and go to curved space. So, if this is a conformal field theory, the effective action at the fixed point will be made by one part which is basically the, is viling variant and is the co-variantization of the conformal field theory action, which in principle I know because I know all the correlators of the conformal field theories. And then, and from now on I will work in two dimension, then there is another part which is intrinsically gravitational, so it depends on the curvature, it will be zero on flat space that encodes the conformal anomaly, and in this case it is well known that this action is the polyakov action. So, basically at a fixed point, the general effective action of matter and gravity will be of this form. So, once we know this, we can immediately obtain one important result which is the west domino relation which introduces the west domino action because if now we make a vile rescaling of the matter field and of the metric and we subtract the un rescaled effective action, then the viling variant part will simply go to zero and I just obtain the difference between the polyakov action which is rescaled and this one naturally gives me the west domino action in two dimension and the west domino relation. Something analogous I have also in every even dimension for example in four and six and so on, but for now on I will just stick to two dimension to make the connection clear and simple. So, basically at every fixed point I have this relation that must hold and this is the central charge. So, now the next step is to ask what happens if I slightly deform my effective action away from a fixed point. As soon as I go away from the fixed point all the operator that I use to deform so all the relevant perturbation that I added to flow out will start to flow by themselves and they will have k dependent couplings. But now if I make the difference, so if I just try to ask myself how the west domino relation is deformed or how the west domino relation looks like away from criticality, I understand that what is left on the right hand side is something which will be zero on flat space because on flat space this thing will be zero and it will be the west domino action away from the fixed point. So, it will be a running west domino action. And since I know that when I go back to UV fixed point or to the infrared fixed point I must first of all recover the standard west domino action. I know that one sensible expansion of this running west domino action will be something that involves a running version of the west domino action, the standard one with the running central charge and in particular there is the possibility that the two operators that define the west domino action will be splitted because there is no symmetry that will protect the two coefficients. So, in principle I have two running C function and then I can have many other terms which will be proportional to beta function in the sense that when I go to a fixed point they will be zero. And so here I have everything that vanishes at the fixed point. So, this is the general form I expect and note here that I made the rescaling also of the scale. So basically this is a way to implement the Stuckelberg trick. So, once I did this first step the first question I need to ask is what is the relation between the two central charges? So, again, since when I go back to the fixed point the two central charges must, this part of the running west domino action must recompose the standard west domino action, the difference between the two central charges must be of order of beta. So, it is natural that I parametrize their difference introducing a vector. This is a vector in the space of couplings. And in this way then I will I can rewrite my west domino action having this object here and this other object. In particular later on we will realize that the correct definition for the running C function is actually this shifted combination. So, this is one of the result which is also characteristic of Osborn approach but from a general RG point of view it emerges naturally because it comes from the fact that as soon as I go away from a fixed point the west domino action will break apart and all the coefficient will start to run in a different way. This is the reason why in four dimension we have an A and A tilde and this is the reason why there has been a lot of confusion at the beginning in four dimension when people try to prove the A theorem to really realize which is the right quantity that we should look at the scale derivative to see if the A theorem is satisfied or not. But then there are other terms which are beta terms that they are proportional to beta functions. And one way to understand how this term is or at least how is the first contribution to this beta term is just to remember that as soon as I go away from the fixed point I also have a scale anomaly because I have the beta function and I know that the trace of the energy momentum tensor will contain something proportional to the operator and proportional to basically a quantum scale anomaly and the classical scale anomaly if this is present, if the coupling is not a dimensionless coupling. So basically this shows us and give us a first hint of the general form of these beta terms. They will start with something which is proportional to the dimensionless beta function because these are the ones that vanish at the fixed point. But then there is and there is the operator that is used to the form so the matter part and then there is one power of the dilaton. So we will see this is the first contribution also to the flow of the C function when in a moment we will derive the flow equation for the C function. Then if we go on we see that already by this first couple of hints about the general form of the running waste domain action we come to the situation where we are invited if you want to perform a derivative expansion of the running waste domain action. And in particular we understand that the potential will start with the term linear in beta and proportional to the matter operators. And here is an important point. In all this discussion if we take operators which are primary so which transform in a nice way with respect to the vile transformation I will have that the matter fields enter only the potential part while the genetic part and the curvature part they will involve at the lowest order the C function and this vector which accounts for the shift between the two C functions basically. But in principle I have many other terms which are present at every in the running waste domain action. And how do I determine this? And this is what I am going to answer. I just want to say one thing before. Here I am writing every in terms of the dilaton. In every moment of my reasoning I can eliminate the dilaton by basically writing the dilaton as one over box R and so I will obtain that a covariant expression in terms of the curvature which is non-local. And this will be basically the general form of defective action away from the fixed point. So how do we determine the next terms? So the first thing is to exploit the consequence of the Stuckelbert trick. Because we have rescaled the k in terms with the field so now basically the couplings become space time dependent in a very specific way. And this fact naturally introduces beta function because if now I perform an expansion I see that here naturally I have a term proportional to the beta function of the coupling and here for example I have a term which is naturally the flow of the C function. So once I do this I simply see that my general framework obviously at the lowest order will naturally recall this scale anomaly but if I go on at the next order I see that an interesting structure will start to emerge because I see that new terms in this effective potential for the dilaton are suggested and the coefficients will be proportional to beta functions or to derivatives basically the stability matrix. So if I go on thinking like this and I do the same thing also for this vector omega and for the central charge I start to see that the coefficient of the derivative expansion start to show a nontrivial RG structure that involves a lot of RG quantity which are mixed between the various orders. And then I still have the possibility of having new objects here and this is very important because whenever I have new objects these objects are in principle present and they define new RG quantities. For example now we will and here is the connection with Osborne approach will emerge. I can simply for example the dots here must be filled by something which is of second order in the beta function and since the beta function are bosonic in this case it will be a matrix. So naturally here I will write a bilinear form which involves what we will realize it is the zomologic of metric and so on. So a lot of RG quantities are naturally encoded in this way of thinking. So now I can introduce the local RG because what they set up to now was completely general it was true for any approach because it was just RG ideas. So the idea of Osborne is this one. We have the difference of the we have the wisdom in a relation and we know that we have promoted the scale to be a field so the couplings actually becomes some fields. So the idea of Osborne is that if I have to make an ansatz for the West Zumino relation I do not do the derivative expansion that I was doing before. I do another thing because I know that at the end what I need to do is to construct all possible invariants made out of the dilaton and the curvature. But what the idea of Osborne and the idea which is underline the local RG is that since now I have promoted the couplings to be fields I have to use also the fields. And so I just start to construct all possible invariants and by doing this you see that since you can construct something like this with one coupling index free you are forced to introduce a vector which is the vector that basically we have seen from our point of view before and you are also forced to introduce something which is like proportional has two indices and it's symmetric and this is basically it's a logic of metric. The connection to the derivative expansion is very simple because now if now these couplings are in Osborne approach the couplings are real scalar fields. From the point of view I'm presenting you they become scalar fields because the scale has been promoted in that peculiar way. If I use that fact then for example the gradient of G if now I think that the gradient of G is the gradient of G and the dependence goes through e to the minus tau I see that the gradient I can expand and empowers of ingredients of tau and I find this term while for example this term will give me a contribution like this one and so you see that here we find some of the terms that we already introduced before and then we find some of those terms that I left in ellipses before so quantities which involve RG quantities. So basically the underlying idea of the local RG is this one. We just look at the the form of the West-Zumino relation and we have a natural West-Zumino action that runs and we perform not the derivative expansion but we perform an expansion using the fact that we have promoted the coupling two fields. Ok, now now that we have the general setting we need to find non-trivere relation the way in which the local RG proceed is using the fact that the vile transformation are a billion. So for example for sure at the fixed point this will imply a local an infinitesimal vile consistency relation with this kind of infinitesimal vile transformation but since the vile transformations are a billion also and this property is not related to the fact if I am or not at the fixed point I will have with a modified operator and now involves beta function and variation with respect to the couplings a modified let's say an infinitesimal vile consists of sec condition which is valid also away from the fixed point. So now Osborn idea is just the one of taking the ansatz that we wrote before and we plug it inside here and then we just do the variations and we collect all the coefficients of the operators in monomials of tau and this quantity must be equal to zero and this magically gives some non-trivia relation. I just highlight you one of this consistency condition gives that this combination this equation must be valid and if you want from this relation we realize that exactly this combination is the one that satisfies the CT or M so this is the correct candidate to be the representation for the C function away from a fixed point. This is very nice and this can be also adapted to higher dimension and so basically this idea uses exploit the symmetry of curvature invariance to obtain information about their G and also exploit together with the ability of the vile transformation. The fact is that in this approach up to now we have not explained how to compute the beta function or how to compute examologic of metric or how to compute any of this quantity because it's just a formal reasoning and so it's exact but it doesn't tell us how to construct things. So since now we work in we know the FRG we can also do a further step we can just take a scale derivative of the vestumino running action and this will be we know the right hand side because the right hand side now we can plug in the flow equation and then we know for example that the C function is the coefficient of the biliner term in tau when I go to flat space and so in this way I'm able to obtain a flow equation for the C function. So by combining the general reasoning with the FRG we obtain an explicit constructive explicit constructive relation because we know how to compute C and we know how to compute the betas and what is very important is that the flow of C is induced by matter-delator relation you see matter fields and dilatons and these are only present in the running vestumino relation. I cannot change. So now if I just draw this diagrammatically this is the flow of the C function the curly lines or the dilaton why the standard line is the matter field and then I can just it's very nice because now if I have a truncation I can just where my action runs I just know how to upgrade it to curved space by following what we said and I can compute the C function. And for example if I do this in full generality with the form of the effective of the running it's a vestumino action that we discovered before and we realize that the matter fields enter only in the potential and that the term linear in tau and of a certain power in the matter is proportional to beta function I see that this diagram gives me something which is quadratic in the beta function while I see that this diagram do not contribute because I also have to take only the term goes with p square and so this gives me back this relation exactly this relation because and what is very important it gives me also an explicit construction for the zomologic of metric okay so basically the key point is this one we have a general Rg construction and then at one point we can decide to see what we can obtain by using consistency condition and this is basically the local Rg approach but at the same time starting from exactly the same equation which is the vestumino relation away from criticality I can just make a t derivative and I can connect everything to the functional Rg and that's it so this was my main goal to show you the connection so basically at this level the functional Rg is able not only to derive the same consistency relation but to have them derived in a framework where every Rg quantity has an explicit representation where I can put inside something turn the machine and compute them I will stop here