 So let me start by thanking the organizers for putting together such a great and diverse program and Running such a smooth conference. So it was really a pleasure to be here and maybe at the end of the session Things I can say this So what I want to report in this talk is actually I want to report on two works So one work the first work was done in collaboration with the Jena all-stars collaboration So there were two students driving the project Benjamin Knorr and Stefan Lipphold And then I also want to explain something that I did with my students in Nijmegen Which brings to the title of this talk so since you are probably not all quantum gravity persons and I'm the only asymptotic safety quantum gravity talk in the plenary here Let's explain what our goal is So what we want to do is we want to derive a consistent and predictive description of the gravitational force valid on all length scales so we want to start at the Planck scale or at sub-Planckian energy scales where Space time might be widely fluctuating and then we want to connect this quantum gravity phase down to large Large distances. So here are typical distances that we are interested in So at this distance we form the cosmic microwave background Particle physics typically happens at the order of 10 to the mile minus 19 meters If we talk about solar system tests of gravity We are talking at distances of 10 to the 12 meters and down here in this plot This should be an illustration of the visible universe. So then we are talking about the largest possible length scales that we can possibly observe and Looking at the distance scales you see why you want to do Rg So you want to extrapolate physics over 60 orders of Magnitudes and this is something that will be very hard to do for example in a Monte Carlo So Rg methods are predestined to explore these connections So what should be the ingredients of our program? So we want to approach this problem of having gravity at all forces from a bisonian perspective and The first thing we need to do this is we want to have a fixed point of the random realization group flow Why is this important? So first we want to have this fixed point to control the ultraviolet behavior of our theory So in this case we can control the UV divergencies the short distance behavior and If the dimensionless coupling constants associated with this fixed point is finite This should ensure that also scattering amplitudes don't have unphysical divergencies We want the construction to be predictive. So this translates into a property of the fixed point So essentially we only want to have a finite number of relevant deformations So the Rg flow that is tracked into the fixed point at high energies dispense the UV critical surface and What asymptotically? Conjectures is that our world actually lives on such UV critical surface Then our goal is to pinpoint where we are on this Critical surface and this is typically done by measuring relevant parameters of our theory So by ensuring that the surface has finite dimension we ensure that we can locate us with a finite number of measurements Finally we want to be able to connect to our real world So we demand that the Rg flow emanating from our fixed point approaches a regime where Classical general relativity is actually a good approximation So in this regime we can then recover all the successful tests of general relativity like the classical tests of perihelion shift Gravitation lensing results nuclear synthesis and many many more So these are the three basic requirements that we demand for this program to work So the question why are we not there? So we know how to quantize theories for a long time and we have a classical description of the gravitational force so here We start with general relativity at the level of the action. This is encoded in the Einstein-Hilbert action We have a Newton's constant. We have a cosmological constant in the game and What makes this action very different from what we write down in the standard model is that Newton's constant actually comes with a negative mass dimension Then we can analyze the consequences of this finding using a Wilsonian approach So we want to compute the flow of this coupling constant. Okay, canonical way. We introduce dimension less quantities So here I introduce the Rg scale k and then we write down the beta function Whatever this beta function will be due to the canonical mass dimension There is a part of the beta function that just is given by the mass dimension and quantum corrections then come perturbatively at the order of G Then you notice from the structure of the beta function That the Gaussian fixed point where this coupling constant vanishes is actually a subtle point As a consequence if you start with a theory with a positive Newton's constant You will not end up in this free fixed point, but you will be driven away from it So this is the statement that general relativity is not asymptotically free So this is not how people started out within this program So people were interested in actually Computing counter terms So this is the famous work by Gore-Rhoff and Sanyotti And they started from the Einstein Hilbert action leave out the cosmological constant Then you find that at one loop order There's no counter term on shell counter term that you can write down So you have to push your computation to two loops and then you see there's indeed a non trivial counter term That you can write down this contains three powers of the wild tensor contracted here And it comes with a non zero coefficient So then you notice that this type of counter term is not of the form of your initial action So it cannot be reabsorbed by redefinition of Newton's constant as you would do in a standard perturbatively renormalizable field theory So then you would need to add this new piece to the action in order to get rid of your divergencies that will introduce a new free parameter and You expect that this will go on at three loops and in the end you will end up by an infinite number of free parameters So in the perturbative language, that's the same as we have seen before The statement is if we start from this action, we end up with a perturbatively non-renormalizable theory Okay, so maybe this is not the good fixed point Let's see what fixed points are on the market that we can use instead So the Gaussian fixed point here I described what goes wrong with this fixed point So if you start with the Einstein-Hilbert action you perturbatively quantize in G this doesn't work to describe gravity Then people thought about okay, we can add higher derivative couplings to our action That cures the problem of randomizability But it raises a new problem We get negative norm states and perturbatively you don't want them So this is also not the theory that you are looking for Then there is the proposal by Weinberg that there could be a non-gaussian fixed point sitting on the theory space of gravity This corresponds to an interacting field theory and here I should stress that we are essentially working at a relativistic quantum field theory at this stage So if this fixed point persists then gravity could fall into the class of non-perturbed if we renormalize with field theories Recently there was a second proposal so this playing a variant of a gaussian fixed point and Instead of working with a relativistic theory you break Lorentz symmetry at the fundamental level So that leads to an interesting class of unisotropic fixed points where you can combine the features that we have above Here so you get higher derivative terms, but they come without the price of having negative norm states So in essence this could also be a feasible option for UV completion of gravity But for the purpose of the talk I will just focus of this idea that we have a relativistic fixed point So focus on asymptotic safety, but this is the outline of the talk introduction. We have just done then I Should quickly summarize the activity that has been going on in the field in the last couple of months and Then I will focus on three particular questions So what is the role of the two-loop counter term in asymptotic safety? How can we go towards real time computations in our framework? And I will briefly comment on the question of unitarity from our perspective. Okay What has been going on and in the last So the primary tool for investigating the asymptotic safety hypothesis So it's based on the Wetterich type flow equation We see we use a background formalism to set this up So G bar here is a background metric and we look at fluctuation around the background Related to the discussion that we have in the in the auditorium There are three remarks I want to make here about this equation So first it's the role of the background formalism if we set up this equation for gravity This is not a choice, but it's a necessity So if we do standard in quantum field theory, we get precisely such a background for free That's either our Minkowski metric or Euclidean metric. So in quantum gravity the background metric essentially has to fulfill the role of also Providing us the background that in a standard quantum field theory comes for free So it will be hard to actually get rid of this point where you are standing on Then the formulation itself so we don't need to specify the initial action that we want to quantize so we can search for this initial action Dynamically on our theory space. So since we don't know what the action associated with our fixed point actually is This is a highly desirable feature and certainly it brings in plenty of approximation schemes Which we can use to construct RG trajectories of this type Okay, so over the years this has been used at the background level There were many many truncations of the gravitational action that have been tested so the story started out with Einstein Hilbert now polynomial are Approximations including R to the power of 35 have been studied and people also included tensor structures There is a new perpendicular direction to this diagram coming up So this corresponds to vertex expansions, which are made on top of Specifying the ansatz for the effective action. So this is not shown in this diagram here So here is a summary of recent works and developments so The brief summary in many of the studies that we have seen all essentially show the existence of a suitable non Gaussian fixed point Both at the background level and at the level of the vertex expansion We understand that this fixed point has a high predictive power So the number of relevant directions could be low and as low as three For my personal perspective This here is an important result So if we study the field theory that is associated with the non Gaussian fixed point in two dimensions This can be linked to a unitary quantum field or conform a field theory So this is the first link which could provide an intrinsic definition of our fixed point And then there has been a lot of structural analysis going on Understanding the gauge dependent the background dependence of our assault people were interested in deriving C theorems Deriving closures of the RG equations. So there's a lot to be done So this refers to the case of pure gravity Branch that came off. So instead of gravity by now. There's also a lot of developments studying Gravity coupled to meta systems. So the central results here in this area is that The non Gaussian fixed point that we see in pure gravity has the analogous Fixed point if we include the meta content of the standard model There was the famous prediction of the heat Higgs mass based on asymptotic safety arguments There are studies of Higgs-Yukawa systems on the way and a lot has been going on in terms of studying cosmological implications of asymptotic safety so a lot of activity on the other hand The discussion and this conference showed that there are lots of open questions Which are still discussed among the people in the field. So here are my favorite ones So non Gaussian fixed point are the critical component Exponents really complex What should we do with the background field? Well, how can we generalize our finite dimensional truncations to an infinite number of couplings or? Does asymptotic safety manifest itself in? phenomenological signatures So these questions indicate there's still a lot that needs to be done in this program So let's focus on the questions that have been around So this is a rather old question. It already started at the beginning of the program So given that we have this asymptotically safe fixed point What happens if we study the effect of the go off uncertainty counter term will our fixed point be destroyed by Adding this particular term to our truncation So the setup in which we studied this question consists of the Einstein-Hilbert action down here And here this is the two-loop counter term That was found in the perturbative studies Before actually Understanding what the effect of this term is let's recap what's going on in the sector down here So this is our ansatz for the effective average action We have two flowing coupling constants g and lambda we can compute the beta function and From the beta functions we read off that there are two fixed points We have the Gaussian fixed point and we have a non Gaussian fixed point At positive Newton's constant and positive cosmological constant So this would be the good candidate for UV completion of gravity So it's UV attractive in both g and lambda and we see it has the potential of Attracting the RG flow from the classical regime and provide the UV at completion of our trajectories at high energies Okay, how does this picture change one this once this operator is included so following the RG Perspective we added precisely this term to our truncation ansatz and we equipped it with a running coupling constant Then you can again introduce dimensionless quantities and start computing the beta functions In terms of structures, there are two surprises coming up actually So the first surprise was that once we include precisely this term It does not feed back into the flow of the Newton's constant and the cosmological constant So the beta functions for g and lambda are precisely the ones that we had before Why is this the case? It has to do with the geometric structure of the counter term and the way that we construct our beta function So this is the term that we added on the right hand side our Flow equation we have the second derivative of this term with respect to the fluctuation fields And if we work on background level we then evaluated this flow Setting the fluctuation fields to zero that means whatever I can do here in terms of taking two variations of the metric The result will always be proportional to one power of the wire curvature then it's By construction whatever index here. I construct the contraction of the wire curvature term vanishes So if I have such a term it cannot contribute to a structure that is linear in the Ritchie scalar So that tells me that the coupling constant sigma cannot enter into the beta functions in the Einstein-Hilbert sector Okay The second surprise is that we can write down the structure of the beta function For this new coupling constant It will be cubic in the new coupling sigma with the coefficients C0 to C3 being functions of g and lambda Again the argument follows from taking two variations of this. So again, we have the equation that There is at least one power of the background while tensor Coming and if I expand my flow equations in power of the curvature then I get vertices I take the vertex to the nth power and This expansion in terms of curvature will give rise to a term e to the power n multiplied by the nth power of my coupling constant and Down here. I get additional terms Which are lower power in the coupling constant Since I'm only interested in terms which are cubic in C bar I see that if I put this to 3 the series will start at Finish at sigma to the power of 3. So that explains this highest power here And from this lower power here as we get the terms in sigma squared sigma and the constant terms Okay, these are the structural insights now in order to understand what's going on with fixed points We have to understand what these coefficients C0 to C3 are And then Computer power came into the game. So we put the system onto a computer I have to thank Andrea Swift at this stage for providing the computational power for doing this computation and in the end The result took a rather simple form So these are the coefficients and the important takeaway message is that this coefficient C3 Is always is found to be non-zero then you see here you have a cubic equation in sigma and If you have a cubic equation this equation always have a has a real root So that means that as long as we have this coefficient C3 Bigger than zero Then there is an extension of the non-gaussian fixed point to the level of the go off and so not the counter So here are the numbers that we found so if we compute Analyze the beta functions. We have the Gaussian fixed point with canonical scaling dimensions And here's the position of the non-gaussian fixed point and it turns out that the new direction that is added is Actually irrelevant. So if we have This additional coupling added to the game that means we don't get an additional relevant parameter And this is in contrast to what you see in perturbation theory so The counter term does not increase the number of free couplings in our theory Then you can integrate the flow diagram that is depicted here And you can also check that the crossover from the non-gaussian fixed point to the Gaussian fixed point is actually intact So our go off and so not the extended fixed point still connects to a classical regime down here So this settles the question. I think in favor of asymptotic safety Okay new results So the motivation for looking into the effect of a causal structure is the following So so far we have done all our computations in the Euclidean setting However from experience, we don't live in a Euclidean world. So we experience that time passes also in my talk So how do we can can we include the time direction in our framework? and the basic idea is we don't want to encode our Decrease gravitational of decrease of freedom in terms of a covariant metric We want to resort to the ADM formalism So in this case what we do we take our space time and we put a fallation on our space time So there are spatial slices which come at equal time and space time is built up by a stack of these spatial slices Effective you have a preferred direction and you associate this preferred direction with time The decomposition of our metric to ADM variables that is well known So here we have the line element This can be written in terms of the covariant metric G mu nu or in terms of these ADM fields Which tell you how to go from one time slice to another and how you measure distance on the spatial slices and here is the Decomposition so instead of having one field we have a lapse shift vector and metric on a spatial slice and This decomposition is non-linear Okay, you decide for this set of variables Then the Einstein-Hilbert action that we had before Assumes is likely different form. So we still have Newton's constant cosmological constant But the curvature term splits up into an extrinsic and an intrinsic curvature Then you also notice that not all of your fields are dynamical so the lapse function and the shift vector hidden in here are actually Lagrange multipliers which Do not propagate a priori Okay for Computing the beta functions. It's convenient to specify a background And the idea is you can use a Friedmann-Roberts and Walker spacetime Then there's a canonical parametrization of your fluctuation fields So this is precisely the parametrization that is used by cosmologists If they want to study effects in the cosmic microwave background. So this is what we borrowed from them And then it turns out that if you start from this action There's Unique and canonical gauge fixing that you should come up with to complement this action of course these two functions are the two functions that you would immediately write down and You have to see where this thing comes from. So if we compute the second variation of the sky up here Here is the list in terms of component fields So then you see and you notice that not all of the fields acquire a typical dispersion relation, which is energy squared plus spatial momentum squared In fact, this will fail for most of the component fields. So if you look at the VV It doesn't work if you look at the EE it doesn't work and so on Moreover, you get square roots of the Laplacian, which you also don't like So what's the gauge fixing that I wrote down then actually does is It equips all these components fields by Propagators that are actually well defined of the form energy squared plus momentum squared Moreover, you see all the propagators come with a time direction and spatial derivatives So in this setting you are at a very good position to actually big rotate your theory to real time and then Run your flow equation Actually having a real time direction around So upshot we have a unique gauge fixing which leads to a well-defined Hessian for all the component fields That are sitting up here Okay Based on this you can run the usual procedure of constructing your beta functions and Determining the corresponding flow diagram and then it came to our surprise So we recover the fixed point that we have seen in asymptotic safety based on Covariant variables, but there is a new fixed point appearing up here Saying this fixed point D if you trace where this fixed point actually comes from it comes down here from the singularity So at lower dimension it splits up from the singular point and it moves up there This has the effect That in particular this sector if we start from this ag flow flow down We end up collectively in this endpoint C down here so in this case we have a Completion of the flow at low energies, which is provided by this particular point all the other things stayed as previously so while Adding the causal structure doesn't seem to affect the microscopic structure of our space time It might be of relevance if we'll ask questions about long-distance physics Okay, this is what I wanted to say about Bringing time into the game. So given the discussion that we had in parallel session. I wanted to make Two comments about unitarity So we set up a framework which allows us to precisely test this question So we started from Einstein Hilbert We added a single scalar field to the game and what we allow here is a momentum dependent structure function and in the end The structure of this function in terms of momentum Actually allows you to determine the decrease of freedom that are sitting in your theory So ideally you want to determine the form of this function at the fixed point So currently due to technical limitations. We cannot do the full functional flow of this So currently we are limited to approximating this fk to a polynomial And then you see that if you are at the level of the polynomials you run into a problem So if this Polynomial has more than one real root then you run into the problems that you encounter in Perturbative quantization. So that would link The theory to Otsukowski in stabilities This can be avoided if your function f star actually is the product of a p squared Times an analytic function, which is actually positive. So in this case your decrease of freedom are precisely one massless particle and Having this extra momentum Dependence will not introduce extra decrease of freedom So how far are we in the computation? So we are at the level of having the beta functions for a Piece p to the fourth term we integrated the better function. We saw that our Asymptotically safe fixed point still persists. It still has the same crossover behavior, but there are two additional fixed points Now what is remarkable of this diagram is if you are sitting at these UV fixed points You see that the mass of the additional decrease of freedom is shifted to infinity So despite working with a higher derivative theory, you don't get any extra decrease of freedom sitting up here However, if we trace the flow to the infrared we see that currently we cannot take Approach a region where the mass of this extra states Becomes infinite so we can make it arbitrary high, but not yet infinite. So that's the status here This brings me to the conclusions So in terms of analyzing fixed points, we are in good shape So we have seen fixed points which could provide the UV completion of gravity at transplanting scales They have a transparent connection to classical low-energy physics At the level where we could test it. They can extend to gravity meta systems And by using the ADM decomposition, we are now in the position of making real-time correlation functions and cosmological investigations based on the FHG Okay, this brings me to the outlook and the final announcement So we will have a road map workshop on precisely these questions to be held in the Netherlands in Leiden So this will be February next year and Announcement will follow over the asymptotic safety mailing list. So if you are not on this list in scribe Thank you