 Hello, everybody. First and foremost, I want to thank the organizers, particularly Roberto, giving me the opportunity to speak here today. Actually, Tim Morris was supposed to speak, but cannot be here unfortunately today because of personal reasons. And I would like to present a series of three papers of Tim in collaboration with you indeed, and Zoe Slate is also here, which all deals with the question of what is the influence of demanding background independence to RG properties, particularly the notion and the existence of fixed points in the quantum field theory of gravity. And as a somehow simpler time model, we've been looking at conformally reduced gravity, and I will focus in particular on the second paper in the series. Yeah, so we've seen very good introductions, particularly on, and that's why I have only one slide, but let me just, that's probably the compensation of what you should take home, that first of all, whenever you choose to apply the background field method in quantum field theory to be able to carry out a particular calculation, you will split your full field in some kind of background plus remain the fluctuation. This does not have to be in this linear fashion, but it will be in some way. And whatever you get, the results of your calculation, they should not depend on the particular way, the artificial way and how you do the split, or the physical quantities or only a function of the full field. So this is true also, for instance, in Young-Mills theory, in any quantum field theory, but might be trivial. It's usually introduced through the gauge fixing sector and resulting in the ghost sector. But additionally, if you want to study quantum field theories of gravity, you have additionally the problem that when you want to use the functional RG equation, you have to define a notion of scale, and this is usually done by the spectrum of a similar plus operator, which you defined with respect to the background metric. So in a way, how you classify the fluctuation of your gravitational field depends on the field itself, namely the background part. So through this dependence, which will show up in the cut-off function, which depends on the plus operator, background dependence will be lost at least in intermediate scales k, and you will be able to restore this, however, by demanding that this modified split water density will be at least in the limit on k going to zero. So let me just clarify a bit of our notation. So we're interested in conformally reduced gravity, where the metric, the full metric with the twiddle, is expressed as some conformal factor, f, and some fiducial background metric, which we actually fixed to be a Euclidean flat. So the conformal factor field phi twiddle, which we split in the background, kai and the fluctuation's var-phi twiddle, which then, its classical counterpart will be denoted by just this var-phi. The background metric we denote with a bar on top. I note that the conformal factor is kept arbitrary in terms of the parametrization of the conformal factor field. Also note that this function f itself does not depend on the scale, and in this theory, no need any gauge fixing nor ghosts. So the first step is obviously to derive the functional RG equation, the Wetterich equation, which we have here. Note that the derivatives of the inverse propagator are taken with respect to the fluctuation field, and then the cut-off regulator will depend on the background field, and the RG time is just a lack of decay. So the demanding background independence is done in equivalent to imposing, in the case of the linear split, this split symmetry, where you just shift the fluctuation field with some field epsilon and the background in the other direction. And then you can derive from average action what we will refer to as the modified split water density, and this identity encodes to some extent how much this split symmetry is violated by the effect of action. Note also that the structure is quite similar to the flow equation. But where the time derivative acting on the cut-off is replaced somehow by the dependence on the background field, acting on R directly or on this inverse square root of the determinant, which includes this confirm factor. So the first thing you may want to ask, and also that has been said before, if the system is compatible, if the water density is compatible with the flow and what I mean with that I repeat it again, is given that the water density is satisfied at some scale and given that the effect of average action evolves according to the flow equation, the water entity should be satisfied in each and every scale. So rewriting the MSWIs we've just seen as left-hand side minus right-hand side is equal to zero, you can derive the flow of this water entity and what you get is this, and in particular it will be proportional to the water entity itself, so this is going to be zero, and then the water entity is a constant which is zero. This is somehow trivial at the functional level because you just do simple manipulation to the same partition function. But however this is not at all trivial anymore if you truncate your system in any kind of practical calculation and it's also not clear at all what is however I call the overlap of information, so it's not clear if either one equation contains the information of the other and in particular we see that that will depend at least in our truncated example will depend a lot on whether compatibility is satisfied or not. So then we wanted to study the system in the derivative expansion up to second order in the derivatives and assuming a slowly varying background field chi we can terminate the kinetic term effectively at the LPA and this is the most general action you can write down there. And then from the full functional flow equation water entity you get two equations at each and every order in the derivative expansion and I will just show the two equations we get for the potential here where q is the propagator to this order in the expansion and we have this very similar form as before so here you will have the flow you will have the time derivative acting on the cut-off here you have the background field acting on the... The first thing you can study in this truncated system again is compatibility and you can calculate again the flow of both water entities you have now here I just show you the flow of the water entity for the potential v and you see it gets quite a bit more involved so you will have something that lurks a lot as the identity we've seen before in the functional case and this is actually going to vanish assuming this is satisfied so the water entities are satisfied themselves but then you have this double integral which involves this commutator like terms of the regulator where for future convenience we abbreviate it with gamma this thing which is basically the variation of the conformity factor with respect to the background field Now one way to have the flow of the water entity to vanish is to demand that this commutator like term which is written out here is actually zero and this integral will obviously be zero and you can integrate this equation just noting that if you write it in terms of quotients this will actually be independent of the momenta and then you will see that this has to be linearly related to the time derivative by some function that is dependent on the background and maybe dependent on the scale as well The flow of the water entity for the kinetic term is much more involved, much more complicated and you will get additional commutator like terms as so with higher derivatives in the mentor but again it turns out that they vanish if this condition is satisfied so it is a necessary and sufficient condition to ensure compatibility is actually satisfied at each and every order this however I should mention is not necessary so they might not vanish but then this equation and all the others will give you additional constraints that are non-trivial and you have to satisfy so in a sense they are secondary constraints but then if this is not trivially satisfied you will also have to look at the flow of this object and so on so all the derivatives of the water entity itself and you have to demand that they are zero this will give you a tower of secondary constraints which you presumably cannot satisfy simultaneously so this compatibility condition can be satisfied it turns out that this is satisfied either if the anomalous dimension of the graviton is zero and then it is trivially satisfied by this being a constant or if this is not zero you get the differential equation like so with some constant and this can be solved by choosing the cutoff profile in the dimensionless variable with rescape of that variable p hat as so being power law okay now I want in a couple of five remaining minutes I just show you three results we obtained in the LPA and first of all you see if you rewrite the flow equation and the water entity in terms of dimensionless variables that you get a non-autonomous system in t so actually demanding you are at the fixed point where the time derivatives of the dimensionless potential vanishes then this equation is going to be independent of drg time t this however by the change of variables to dimensionless variables sorry you introduce a time dependence in the dimensionless conform factor which will show up through the gamma here you cannot eliminate and you can only and this will in general not allow it to have fixed points unless again you set either 8 to 0 because then this is going to be 1 or you have the parameterization the conform factor now to be power law not the cutoff but then again as I said if the equations are not compatible not everything is lost a priori but you can try to solve the system still but what we did is without including the additional constraints we just tried to solve in a particular case in what I mentioned with the optimized cutoff but then resulting incompatible equations and it turns out you can combine them and separate the scale dependence as so by the method of characteristics where you have this function v hat is then only dependent on some initial data and plugging that back into the flow equation which you have in addition to this combined equation you get then this form of equation and again this will depend explicitly on t unless you again set eta so you can only obtain solutions if eta is equal to 0 but then again the system was compatible in the first place third thing is what actually happens if we have this so what happens if eta is actually 0 and then again it turns out you can combine the flow and the watered entity and after some redefinition of the scale which will depend on the parameterization you end up with some flow equation that is manifestly independent of both the background field chi as well as the parameterization and it also turns out that the fixed points so there's a line of fixed points here I don't want to go into this too much here coincides with the fixed points you would have obtained by just studying the system without the watered entity but you have to demand this condition and it's very interesting that this under certain conditions is possible to formulate the flow combined with the watered entity in an absolutely background independent way alright let me conclude then so we investigated potential conflicts that can exist between a notion of fixed points and background independence in a quantum field theory of gravity we saw the compatibility of the flow equation and the watered entity is always satisfied at least in conformally reduced gravity at the exact level but not in the derivative expansion but however can be guaranteed when either eta equal to 0 or the cutoff is chosen to be power law we claim there are no solutions if those two equations are incompatible and confirm this with some example we also see that even if they are compatible fixed points, main general still be forbidden but however there seems to be well I don't know if always but at least in our case there is a way to actually combine the flow and this modified split watered entity to uncover some background independent description thank you