 Ok. I want to thank the organizers for having asked me to talk here. In, ok. In practice, ok. I have been asking to give a small introduction, but, ok, I will try to be shorter than what they hope to me. Ok, so, we know that the background fields have to be introduced in several theories, and one can introduce even where they are not needed, but so, we know that there are theories, essentially, like gauge theories where we usually want to have a computation with effective action using the background field method, and this in order to deal in some way with gauge invariance. And gravity is one of these theories, but the background is needed even for other reasons. For example, we need an ocean of scale, and Jan also has described this. And also in some, let's say, in cases where you have, let's say, no linear theories, it may be important to introduce a background. For example, if you want to have a geometric approach where, since your fields live in some manifold. And then, ok, for gauge theories, again a short, doesn't work, ok, a short comment, again on the problems, ok, we have a background field method, so essentially we will have the presence of two gauge fields, and in this case, we have, as Jan has also said, some slavno-telo identities, and they are modified in the functional energy approach because we have an infrared regulator, we have the presence of the background, and the dependence in the background is described by the modified splitting-ward identities. And these are indeed very complicated, and the gauge theories, it's very difficult to deal with them. And actually they are related, these identities in some ways, ok, because in some sense the degrees of freedom in gauge theories, since some of them are spurs in some sense, are not physical, could be disentangled with some non-local field parametrization, and this means that there are some relations. For gravity, we know that it is a gauge theory, we have all these problems, and what we really be concerned is that physics should be independent on the background. So, whatever background we use, we should get the same result, and this is a very hard point in this game. And also the double field dependence is technically giving an enormous problem because in practice, when we want to study an effective action and we plug in the vector equation to study the flow and the fixed point structure, eventually we need to make answers on the truncations, and there are, with two fields, many possibilities, and a lot of operators that can appear in the truncation, which may be relevant, and they are constrained by these identities that we are talking about, and it is very difficult in this case today with this. So, we know indeed that what one is doing is trying to separate a first piece from this effective action where the full metric g is essentially coincident with the background plus corrections, and these corrections, let's say, all the real structure of this action depending on the two fields is really containing a lot of information which we should not miss, essentially. So, we have, let's say, here, for example, as Jan was showing, all the vertex expansion is hidden inside here. We know that even if we try only with this approximation or with the studies about the flow of the vertices, which give useful information, we know that we get the same picture that asymptotic safety seems to be there, but there are a lot of disagreements, and these approximations clearly show that we are not keeping them under control. The other interesting field of application that I will say was the nonlinear sigma model. Here, let's say, we have normally a non-trivial manifold, so, let's say, there are different possibilities. One as an effective theory computes elements, and then one does not need anything special. One can introduce any background, but the fluctuations around any background are enough to compute, let's say, some physics. Otherwise, if you want to work with an effective action, which is an official quantity, one should be careful, of course, this, in some sense, depends on the parametrization heavily, and if you introduce a background, then you have a double-filled dependence, of course, and geometrically you implement a double-filled description in term of a base point in the manifold plus some vector, which permit to develop, for example, a geodesic along this manifold, and so the point is, again, we have here a problem of two-filled dependence. OK, then, after this small introduction, let's say that what I will present here, it's a much simpler attempt to solve a very small problem, which is related to theories, which are not gauge theories, and also we concentrate on a special class of splitings, which are not the most general, that one can encounter, for example, engage theories, which essentially mix in a very non-local way the different fields. This is the point. So we will try to stick with some background fluctuation splitting, which are local. OK, so this is the starting point. Here there is also a small view of the functional approach. I can define the effective average action in this way. So the red part here is essentially the regulator. If I move it, I have the standard approach in one-to-field theory, and the regulator is here. And zbar is the average fluctuation, so it is essentially what is called the classical field. And one can consider that this field is generically splitted in a background and a fluctuation. And we consider that if this fluctuation should be a vector in order to have a covariant description. And so if one has this splitting, automatically one can only take a derivative with respect to the background in this equation, get this modified splitting. It is here written. Essentially this part, which depends on the regulator, is very easily understood, because when one takes a derivative with respect to the background, this goes inside integration with the derivative and clearly corresponds to a contribution with a quantum average. And then, of course, one taking a derivative with respect to the scale can get the better equation, and these modified splitting word identities are compatible with the flow. So essentially if this object here is zero, also its variation will be zero and maintained along the flow. It is a consistent relation. OK, so let's consider first a case where we consider only the standard quantum field theory with no infrared regulator, so we don't consider RG flow here. So the splitting word identities were written in this way. And one can ask which kind of solution are here, possible to be derived, and it's very difficult. One can see immediately that if the variation of the fluctuation with respect to the background in order to keep the total field fixed is done, one can expect locally an expression of this kind, and this means that when you take the quantum average, you will have a complicated function of the action, so it will be a very complicated non-local structure. There is one case, instead when this is possible to be solved, it's just when one restricts to splittings containing only these two terms, and so if we take this kind of structure, the most generic structure for this multiplet, then one can see that plug in this inside here, one gets this constraint for the effective action, and then this can have a solution if the integrability conditions of Frobenius are satisfied. Essentially it's a PDE to be solved, and the solution can be obtained solving these two equations, where here in a compact notation we have introduced some tensor forms, one form. So the solution is given by these objects, and so the components are here. And from these solutions, one can immediately obtain the dependence of the total field on the fluctuations and on the background. Indeed, one can plug inside this equation, the object, and get this relation, which can be trivially integrated to be this. So essentially the total field is a function, arbitrary function, and some of the functions of the background plus a linear transformation of the vector given by this matrix. And in particular if, let's say, f is not singular, we can always identify g with f and get this expression. So for the case where one redefines g to make u become this, so the derivative with respect to the background of this function f, this becomes an exponential map. Essentially it can be written in this way, and these gamma are the covariant derivatives of the connection with respect to the lower indices, and essentially the connection is this. So it's really depending on this function which defines the splitting essentially. And in this way we can write a very special solution, which is single field dependent for an arbitrary, let's say, splitting defined by a function f in that way. One can also plug back inside the definition of the effective action, this, and one can write all in this way. So one can see that indeed this effective action is a function of a single field. And one can see that if you reparameterize the background here with an arbitrary function h, reparameterize the quantum fluctuations accordingly with the gradient of h, and redefine f in this way and u in this way, the relation which was given here essentially is corresponding to the transformation of h. So this is clearly a covariant approach. And one can check that, for example, at one loop, which is very similar to what we will need after for the functional RG, one can compute essentially with the standard trace log formula the one loop effective action correction and one needs the second functional derivative in the fluctuations here of the action. And so taking these expressions, one at the end can write the second functional derivative in the quantum fluctuations as the second covariant derivative in the background times these functions. And essentially if one is computing the trace log part, one can get rid because of the measure in the path integral of the other factors and essentially get this formula, which is what is well known in indeed also in the Wilkowiske approach in some sense. And what is important here? So since it is single field dependent, I can compute, for example, with a background and after introduce automatically the classical field dependence and I have it at all orders automatically, let's say, of course. Ok, and for the symmetries ok, there is something to be said, of course, because here we have, let's say, constructed a splitting based on some connection which is flat, which was defined. And so this means that indeed one can make a transformation to a reference system where this connection becomes zero, of course. And, ok, let's say, if we have an action and it's very complicated described by many times inside, let's say that when we make a transformation of the fields, we transform covariantly the action and if we want that this transformation is a symmetry, the action should become the same. So we need this identification and also that the extra dependence on the connection is not there. So also the connection should not be modified. And this means that if I go to the reference frame where this is zero, that the transformation of this connection which is given by this term should be zero and this means that in that frame one should have a transformation of first order in the fields which also include the linear transformations. And so this means that if we have some symmetries in the action, this symmetry should be linearizable. That means that there exists some reference frame in which I have a linear representation for this symmetries. And this symmetries how, let's say, when can be linearizable. OK, there is already a well-known lem due to a column in versenzumino which tells that if the action is generated by the action of some group acting on the manifold of the fields we need essentially one should check if there is a fixed point in this manifold and this means that if there is a fixed point around this point I can find a system of coordinates where the action is linear. And what is interesting is that this fixed point in principle can be also outside the manifold. If we are interested only in the local properties of transformations we can have in another coordinate system not the one where the linear action is let's say, realized. And so in some sense we have a possibility that one can, for example, having a manifold extend it in order to include the fixed point which would be under the action of h and then prove the existence of a linearizable symmetry or otherwise in a case where this is already the case one can remove it. And so there are cases where this can happen and one of them, for example, is a nonlinear sigma model where, for example, with the addition of one extra field going to the cylindrical manifold one can see that there is a fixed point that can be realized at infinity. And this means that, for example, one can study models like the battery is over. One can study models as this one and, for example, setting the function f of h in front of the matrix of the nonlinear sigma model to a constant. This means that this theory in reality as h and the nonlinear sigma model completely decoupled and then the physics of this model contains in a separate sector the physics of the nonlinear sigma model and then I can study this. A little bit. You mean that it was the finger? Ah, OK, sorry. OK, so this is the example that I wanted to show. And then for the effective average action in some sense one very simple possibility is to consider a solution where we take advantage of the solution given before and we search for solutions which make the modified splitting or identity to be satisfied in the presence of the regulator set into two zero also the other contribution. And so this is what doesn't mean in practice that I would have a a parametrization invariant effective action at any scale there is a flow and the parametrization is scale independent and so this single field dependence is maintained along the flow. And how this is done? OK, it is possible to transform this part to this adding in both terms and subtracting a connection term this make this relation completely covariant then if I use the solution the quantum field level without regulator I see that this covariant derivative is constant so this term becomes zero and then we need only to set this to zero and this is constructed in this way so it is possible to find a regulator which is background dependent such that this term which is the covariant derivative of the regulator is zero and with this essentially and I stop here one can construct a flow based on the second fluctuation derivative which is given here and where this these new terms are cancelled in the ratio inside the trace and one can even redistribute these factors f minus one which are dependent on a total field inside a new cut-off term let's say and write in a manifest way this flow as a single field dependent and for the symmetry all the things are the same so in practice if I use this equation to study the flow I can see immediately that I can have a single field dependent so this means that I can only use the background field approximation to compute the action let's say rise to a full field that depends including the classical field and I can see that for example if I have a solution which I consider in LPA a fixed point of the flow of course by reprametrization this will appear as some second order in the derivative expansion object because of course for example if in one frame let's say if I fix jk to one in the LPA and I make a reprametrization this j will not be anymore one so this will show that the equation for the flow are covalently transforming and the solutions can be found at the same level of approximation in any frame of the fields and only to show you a few examples here there is an exponential parameterization and one can study with this connection if I have a polar coordinate system for an ORN model in this case O2 I can use these splits in the expansion I have of course the connection terms here and one can deal with all these models in a nice way unfortunately these are very simple models in some sense the only interesting thing is this possible application to non-linear sememodes but gauge theories and hand gravity are instead a very hard problem and ok, so there are proposals we will see Thank you very much