 So let me start by saying Professor Bliseau and the other organizers for setting up this nice conference and giving me the opportunity to present a project which I did actually two and a half years ago with Martin Reuter. I will skip all the references in the following, since you can find them here to clear up the slides. And I decided to actually give you another introduction to the main problem and conceptual issues which arise once we study background field independence within the realm of quantum gravity. So most of my talk will actually be part of an introduction, since I decided to put all the technical details of our work aside and focus later on only on two main aspects, which are probably then the conclusion of this talk. So I hope to be in time that I really can make my point. So as we all know in quantum field theories, we are really requiring our theory to be background field independent in order to have a full prediction of the dynamical degrees of freedom. So if you violate background field independence, then you have to have somehow justified why you pick this particular background field and not any other one. And in order to avoid this, yeah, if you don't have any physical justification with this, you have to follow a background field independent approach. Now let's go to gravity and start with a classical theory actually. So GR is known to be Manifest Background Field Independent. And since we're now dealing with a metric field, it's also the same as talking about background geometry independent. This part of the geometry, which is fixed a priori, this is topology and the differential structure, which you need in order to define a well-posed mathematical differential equation. Nevertheless, we are not considering these objects to be the dynamical degrees. And as such, we really can talk about a Manifest Background Independent approach in this context. What I want to now emphasize is something which we also encounter when studying the FHE, namely when we approximate an exact equation. So for instance, we can perturb around a certain field, we call it a background field. And what we see is if we truncate now the field equation to a certain order, that the observables will depend on this background field and as such we have a clearly background dependent approach. This can always fail once we have a perturbation or a kind of approximation to exact equations, we might lose easily this nice property. Now going to what we actually want to study, namely the quantum theory of gravity. There are many different proposals for approaching this problem. What they all have in common is basically that in contrast to quantum series of matter, we are now considering space time to be the dynamical degree of freedom. As such, we are now again having the problem that background field independence goes hand in hand with background geometry independence. So let's say we take a metric field tensor field in order to define our dynamical degrees of freedom, then this one also defines our geometry and should be a full prediction by our proposal in the very end. So the problem is then that in usual approaches to quantum theories, we make heavily use of prefix or preset geometry so in order to find equal time commutators, time direction on its own or even scales in the end which we need for the RG, we have to have a notion of geometry in the very beginning. So that has led to many different proposals where you start off by putting background geometry at the very outset, so at the starting point and depart from the standard methodologies. This leads then to what we would call manifest background independent approaches of which CT, loop quantum gravity and causal sets are particular examples. Now we want to stay closer to the standard prescription and what we can do is we can make use of the background field method which we already heard and the previous talked about. So the general idea is unrelated to gravity at the beginning so what we actually want to consider is a, let's say on a formal level, that's a sketchy way how to describe the background field method anyway, but what we want to discuss is basically or compute is a functional integral over the physical sector of our fields. So we know when we study the whole set of remaining matrix that it's much, much, this is full field space so we have a lot of redundancy generated by the group of gauge transformations. So starting this object we somehow have to restrict ourselves to a certain, okay I'm not talking about curve of problems also, but to a certain global section of this bundle over there. And if we do this in a certain way then we can construct the observables, however we have to make sure in the end that our observables do not depend on the particular gauge fixing hypersurface we've chosen. And this is in principle difficult to test then, however what we can use then is we pick randomly one particular point here in field space and we call it the background field. And based on this background field we then construct gauge fixing condition and the corresponding hypersurface equations and if we do it in a consistent way then we see that the observables in fact will not depend on whether or not we move our entire construction by applying gauge transformation even also in addition to this background field. So the vertical movement along the fiber basically doesn't affect the observables. This is quite fine, however what we also have to ensure now we have introduced this additional field and our servo should not depend also on the position of the background field on the horizontal axis basically. So we need additional criteria to ensure this and here background independence adds up the missing direction basically so we require that for fixed dynamical field the observable should not depend on a particular background field chosen. So this is a general idea and with this you can then have a consistent construction on how to get a gauge invariant effective action basically. In gravity now and this is also particularly useful and since now we have a particular field in the construction itself which we can then use to define quantum series in the end. So in gravity we using this background field method we now have a reference geometry so we can define scales later on if we study FIG then we can define the scale K we can if we have this background field then we can really with a scale K also define a consistent notion what is ultraviolet and infrared it doesn't change during the dynamics. What makes this also difficult in the end and I will mention this point at several occasions is that in gravity we don't know yet what the underlying measure is. So in the FIG in fact we will later on have to reconstruct the underlying measure from the corresponding trajectory we are looking at. Okay now coming to the Wetterich equation written down for gravity using the background field method and in this case we see okay clearly we now have an effective average action depending on both the dynamical field and the background field and in particular the cutoff now depends as Jan already pointed out on the background field so it sets a scale basically how to integrate out the modes. Now solving this equation is as you know know very hard let's say we have done so then we are only interested in those trajectories or so solutions to this equation which lie within the final dimensional UV critical hypersurface of a suitable fixed point. This will then define candidates for quantum field series however as you see these are still dependent on two metrics in this context so there's some some kind of inconsistency yet or not inconsistency but some missing ingredients basically so we have to put in additional requirements and these are exactly those which we have constructed in the previous slide so we want to have constraints on the set of solutions which implement background independence and gauge invariance and for instance we could also ask ourselves okay can we restrict by do we have to restrict also further by requiring that series the underlying series should be unitary is the classical limits satisfied and so on and so on. Yeah I'm not going to in too much detail about this so John Paul already pointed out that the what identities are compatible with the functional realisation group equation that means when they are satisfied at one particular scale then they hold for any other scale for a particular solution but this is only on the exact level and once we apply approximation this might easily fail and will in fact fail then. A special limit of this water identities we are now considering only the second one since they are of interest in this talk is a limit when k goes to zero where we have our effective action and there the if the water identities are satisfied there we can then conclude that the series background field independent on in this case on also background geometry independent. Okay let's have a short look at the water identities for split symmetry as John Paul already pointed out so in general you can construct or you can decompose your dynamical field in the background field and a fluctuation field we will here just look at the linear split what also Jan already did so he split the field in the background field and the fluctuation field and there's no geometrical additional contribution of higher terms in H bar actually. So if you then look for the underlying or for the related water identities you get this equation over here and I only want to make two short comments so still so you can only address the question whether or not this action functional satisfies what water identities wants to study biometric functionals since clearly you want to measure how the different effects for the dynamical and the background coupling are different in the approach this you're measuring here on the left hand side and furthermore there's this additional piece over here this is a total action which appears under the in the measure so in a way we have to reconstruct this from a particular trajectory in the end since this equation is probably as hard as the functional realization group we will have only closer look to a very particular approximation to this one and this is the split symmetry at three level so we approximates this equation at three level meaning that we're looking for the BST invariant part of the effective action to have no background independence and we are not interested in the full or in full solution satisfying this criterion but in fact we are looking only at a limit K going to zero since there's a physical part and we okay in a way we are almost sure that on intermediate scale there should be a violation in the very end okay so that is our truncation we start off as I said biometric in nature so we have one Einstein-Hilbert term constructed purely by the background metric one by the dynamical one we have four independent couplings so the theory space or naive theory space we start off is four-dimensional and then we plug it in into the Wetterich equation we solve we try to solve for trajectories which are emanating in the ultraviolet so K going to infinity from a suitable fixer point and in the infrared we now demand that the series are background independent in the approximation I just mentioned namely on three level and it is in this for this truncation this is just corresponds to the vanishing of this first term in the limit K going to zero meaning that all dimensionful coefficients have to vanish and up or we don't know if there's at all one trajectory which satisfies this criterion since we're not working with a dimension less coupling basically so it might be that this is not compatible with this truncation it might be also that we find solutions but they are not connected to the fixer point so we have to check if there's a hint already at the level of the Einstein-Hilbert truncation so now we come to the actually results of the of the project I did with Martin the technical details are very involved so we have to solve this biometric functionalization group equation in a certain way the new techniques which we are liberated on but anyway in the end what we're interested in is now checking if there's a fixer point or not a non- Gaussian fixed point compatible with our requirements so having a classical limit and we indeed find one so in the ultraviolet we found a set of solutions namely all trajectories which have initial data with a positive Newton coupling they emanate an ultraviolet from this fixed point so so far about asymptotic safety now what about background independence therefore we have to check if there are solutions in the infrared so at the opposite side of the scale which for which the dimension full coefficient actually vanished and so what we did basically is we first noticed that the dynamical sector decoupled from the background sector we could solve these two differential equations first we plug them in in the background sector and then we look for exactly the properties we are just mentioned it turned out that the only way how we can ensure that the dimension full coefficient vanish is if we put ourselves on a very particular point in phase space which is what we call the running UV vector and this is the definition the point that comes up when we solve the now k-dependent beta functions for the background sector and set them to zero so it basically determines the entire flow of the background sector when we move on in the scale from k0 to infinity so the requirement of background independence basically fixes the background sector completely we have no choice in this so we get a two-dimensional reduction of all possible theories in the end now the only thing we have to test in order to see if both constraints are compatible here we have to see if there's a non-trivial overlap basically of initial datas and you can clearly see there is one so starting off with a four-dimensional naive theory space we have then a constraint of asymptotic safety just puts ourselves to the positive part of the Newton coupling then we look for background independence we see this this Newton coupling is also positive so it's compatible with the former condition and in the end we end up with a two-dimensional subset of now what we consider physical or relevant theories so instead of having to fix four three initial parameters we now only have to do it for two so what I let's say want to emphasize at this point is that there is quite a remarkable thing happening here namely that there's this particular point a mathematically particular point of the peterfang or differential equations which actually guarantees us on the one hand side background independence and acts on the ultraviolet as a fixed point or as a background part of the fixed point and what we would be interested in is actually is there a general mechanism underlying this topological structure which actually in this in a way implements a background independence part in the effort itself so this brings me to my conclusion so in quantum gravity we now have the problem that geometry is also part of the dynamics so this brings on us into a lot of difficulties but we can apply the background field method now this also sets up a scale for for instance defining the cutoff in our setting it's not only defined for or not only necessary for constructing agent variant defective average actions but we also then have this reference geometry as I just said the underlying what identities are as involved as the effort G itself they will test the extra background dependence and what we did actually was only considering the three level of this equation we look for solutions to the equation which are asymptotically safe and background dependence we found one with increased predictivity and what was striking us was that there's a kind of mathematical underlying object only for this truncation so far considered but it might be that it's generalized to some other that it's more deeper physical reason behind this running uv attractor so what i think is related to future tasks it's surely the holy grail of the fg e namely finding error estimates for truncations if possible at all probably we need that help from mathematics departments also but what i want to stress at this point okay we could start and develop truncations or design truncation just based on one of the exact properties we already know for the fg e for instance the water identity for split symmetry and then construct such truncation consider them or consider flow equation solve them how we cannot be sure without such a criterion or related criterion if we improve actually on the truncation itself since we have a lot of exact or we have different exact properties and improving of one on one might be on the fail of the other one another issue is surely then when we want to come study the complete water identities for gravity we have to solve the reconstruction problem in a way or approximate it in a way that would be then we can go to the next order above three level and test the water identities and surely what is probably most accessible is establish new techniques for studying actually biometric truncations makes it more simple or on the level of simplicity of single metric truncations if we have more results and actually can test if there are structures like this present in the end okay with this i like to thank you very much for your attention